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Baillehache Pascal's personal website
Anamorphic rendering from the reflection on a spherical mirror
A short memo for myself on how to render an image from a reflection on a spherical mirror.
Inspiration from this video by FrostKiwi.
Given a square input image of the reflection of a scene on a spherical mirror, one can reconstruct an output image of the scene as seen from a pinhole camera located at the center of that spherical
mirror if we make the assumption that the image of the spherical mirror has taken been by an orthographic camera. Under that assumption, the calculation of the pixel coordinate in the input image for
any given vector is simple and described below. Then, the view vector for the pinhole camera for each pixel of the output image can be mapped to a pixel in the input image, and the scene viewed from
the pinhole camera can be easily calculated.
The calculation of the pixel coordinate \(\vec{p}\) in the input image for a given unit view vector \(\vec{r}\) is based on the formula for reflection of light: $$ \vec{r}=\vec{i}-2(\vec{n}.\vec{i})\
vec{n} $$ where \(\vec{r}\) is the reflected direction, \(\vec{n}\) is the normal to the surface of the spherical mirror at the incident point and \(\vec{i}\) is the incident direction (i.e. the view
vector for the orthographic camera of the input image).
Under the assumption of orthography for the input image, \(\vec{i}=\vec{(0,0,1)}\), \(p_x=I(n_x)\) and \(p_y=I(n_y)\), where \(I()\) scales and translates appropriately from the input image
coordinates to the spherical mirror coordinates considered to be a unit sphere centered at the origin. To get \(\vec{p}\) in function of \(\vec{r}\), we then need to calculate \(\vec{n}\) in function
of \(\vec{r}\), in other word the inverse of the reflection formula. $$ \vec{r}= \vec{(0,0,1)}-2(\vec{n}.\vec{(0,0,1)})\vec{n} $$ equivalent to $$ \left\lbrace\begin{array}{l} r_x=-2n_zn_x\\ r_y=
-2n_zn_y\\ r_z=1-2n_z^2\\ \end{array}\right. $$ then $$ \left\lbrace\begin{array}{l} n_x=\frac{1}{-2n_z}r_x\\ n_y=\frac{1}{-2n_z}r_y\\ n_z=\sqrt{\frac{1-r_z}{2}}\\ \end{array}\right. $$ equivalent to
$$ \left\lbrace\begin{array}{l} n_x=\frac{1}{-2\sqrt{\frac{1-r_z}{2}}}r_x\\ n_y=\frac{1}{-2\sqrt{\frac{1-r_z}{2}}}r_y\\ \end{array}\right. $$ which simplifies to $$ \left\lbrace\begin{array}{l} n_x=\
frac{-1}{\sqrt{2(1-r_z)}}r_x\\ n_y=\frac{-1}{\sqrt{2(1-r_z)}}r_y\\ \end{array}\right. $$ and finally we get \(\vec{p}\) in function of \(\vec{r}\) $$ \left\lbrace\begin{array}{l} p_x=I^{-1}(\frac{-1}
{\sqrt{2(1-r_z)}}r_x)\\ p_y=I^{-1}(\frac{-1}{\sqrt{2(1-r_z)}}r_y)\\ \end{array}\right. $$
How to use it in pseudocode:
Edge case to take care of: if v.z equals to -1 you get a NaN (that's when you're trying look behind the sphere, any point at the edge of the image of the sphere correspond to the reflected direction,
hence the result is undefined).
FrostKiwi also explains how to correct for the fact that the input image is probably not taken with an orthographic camera. One can add a correction coefficient with an ad-hoc value to correct the
distortion. The formula then becomes: $$ \left\lbrace\begin{array}{l} p_x=I^{-1}(\frac{-1}{\sqrt{2(1-r_z)}sin(\alpha)}r_x)\\ p_y=I^{-1}(\frac{-1}{\sqrt{2(1-r_z)}sin(\alpha)}r_y)\\ \end{array}\right.
$$ where \(\alpha\in]0,\pi/2]\).
Demo video of the result available here.
Computer graphics
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Shop Finite Elements And Fast Iterative Solvers With Applications In Incompressible Fluid Dynamics Numerical Mathematics And Scientific Computation
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What is big O notation (The complexity) and how to calculate it?
The big O notation in computer science (and competitive programming in particular) is used to find the growth of space and time when the data grows, and to approximate the time and space used in the
worst case, which is a great way to compare two different algorithms or codes used to solve some task.
In general, we can calculate the big O notation by counting the number of operations done in some code and ignoring small constant values.
Here are some general codes and their big O notation :
Counter := 0
Counter := Counter + 10
The complexity for this code is \(O(1)\), as we see there are only 2 operations so the code will always do them 2, and since big O notation ignores small constants then \(O(1)\) = \(O(2)\).
Input N
Counter := 0
If N%2 == 0 Then
Counter := Counter + 1
Else if N % 2 == 0 Then
Counter := Counter + 2
End If
The complexity for this code is also \(O(1)\), as we see there are many operations (addition, module, read input, and If statements) which are all constants so the complexity is \(O(1)\).
Input N
Counter := 0
While Counter <= N do
Counter := Counter + 1
End While
The complexity for this code is also \(O(N)\), here the while loop will run until the counter becomes more than N, and N is input so the while loop iterations will change based on the value of N, so
the exact number of iterations is N+1 (since we start from 0), why don't we say that the complexity is \(O(N+1)\) then? Because the time spent is more dependent on the value of N we can ignore the
constant 1.
Rules of calculation
1 - Adding Time
Counter := 0
Counter := Counter + 1
For i from 1 to N:
Counter := Counter + 1
For i from 1 to M:
Counter := Counter + 1
Here we simply find the time spent in each part then add them together (addition, first loop, second loop), for the first part the addition takes a constant time \(O(1)\), the second part it does N
addition N times so the time complexity is \(O(N)\), and the last part does M operations \(O(M)\), to calculate the total time T(total) we just add all the times taken for each part :
\(T(Total) = T(FirstPart) + T(SecondPart) + T(ThirdPart)\)
\(= 1 + N + M\)
And we get the total complexity of \(O(N + M + 1)\) and since big O notation ignores small parts we can ignore the constant 1 and get \(O(N + M)\), Note: we can't ignore N nor M since we don't know
their exact value and who will contribute more to the time.
2 - Multiplying Time
Counter_1 := 0
Counter_2 := 0
For i from 1 to N:
For j from 1 to M:
Counter_1 := Counter_1 + 1
Counter_2 := Counter_2 + 1
Let's ignore the first loop (from 1 to N) we will have 2 addition operations repeated M times with a complexity of \(O(2M)\), now let's bring back the first loop what does it change? As we see how
the second loop and everything inside it are repeated N times we just multiply the second loop time:
\(T(Total) = T(FirstLoop) \cdot (T(SecondLoop) \cdot NumverOfOperations)\)
\(T(Total) = N\cdot(M\cdot2)\)
And we get the total complexity of \(O(2NM)\) and since big O notation ignores small parts we can remove the constant 2 since it is not important when N or M is very big, and get \(O(NM)\).
Let's give another example of nested operations:
Counter_1 := 0
Counter_2 := 0
For i from 1 to N:
For j from 1 to M:
For k from 1 to L:
Counter_1 := Counter_1 + 1
Counter_2 := Counter_2 + 1
Counter_1 := Counter_1 + 1
Counter_2 := Counter_2 + 1
We have three nested loops that repeat N, M, and L times respectively, and 4 operations inside the last loop we can see that the 4 operations are repeated L times, the third loop is repeated M times,
and the second loop is repeated N times so the time complexity is \(O(4N\cdot M\cdot K)\) we can ignore the 4 and get \(O(N \cdot M \cdot K)\).
3 - Non-Constant
Sometimes the loops don't move by 1 like the previous examples, maybe they move by 2 like:
For i from 1 to N:
i := i + 1
This loop moves by 2 so the time spent is \(O({N\over 2})\) here we can ignore the \(1\over 2\), but in some other cases we can't just ignore the change in the moves, here is an example:
Input N
While N > 0:
N := N / 2
Here we divide N by 2 for every move so the time spent can be calculated as follow :
\(T(0) = 0\)
\(T(N) = 1 + T({N \over 2}), N > 0\)
Here the time Taken for input N is 0 if N is 0 because we will exit the loop, and each iteration does 1 divide operation and move N to N / 2, but how many times will the loop move until N reaches 0
if we look closely we will see that we can divide N by 2 \(Log_2(N)\) times only so the total time is \(T(N) = 1 \cdot Log_2(N)\) because the addition operations is repeated \(Log_2(N)\) times time
complicity is \(O(Log_2(N))\).
Popular time complexity Comparison
Here is some of the popular complexities you will see \(O(1) < O(log n) < O(n) < O(n log n) < O(n^2) < O(2^n) < O(n!)\).
• In comparative programming log usually refer to \(Log_2\) (to the base 2).
• the symbol ! refers to the factorial operation. | {"url":"https://yashb.com/article/What%20is%20big%20O%20notation%20(The%20complexity)%20and%20how%20to%20calculate%20it?","timestamp":"2024-11-12T11:45:51Z","content_type":"text/html","content_length":"26378","record_id":"<urn:uuid:1535f866-c2fa-4e9a-ae4c-757bf827b788>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00168.warc.gz"} |
Interactive package for Short AsyNchronous Time-series Analysis (SANTA), implemented in R and Shiny
santaR is an R package that implements functions for analysis of short asynchronous time-series analysis.
santaR can deal with challenges not simultaneously addressed by current time-series statistical methods: - missing observations - asynchronous sampling - measurement error - low number of time points
(e.g. 4 to 10) - high number of variables - biological variability - nonlinearity
The reference versions of santaR is available on CRAN. Active development and issue tracking take place on the github page, while an overview of the package, vignettes and documentation are available
on the supporting website.
To address the challenges of time-series in Systems Biology, santaR (Short AsyNchronous Time-series Analysis) provides a Functional Data Analysis (FDA) approach -where the fundamental units of
analysis are curves representing each individual across time-, in a graphical and automated pipeline for robust analysis of short time-series studies.
Analytes levels are descriptive of the underlying biological state and evolve smoothly through time. For a single analyte, the time trajectory of each individual is described with a smooth curve
estimated by smoothing splines. For a group of individuals, a curve representing the group mean trajectory is also calculated. These individual and group mean curves become the new observational unit
for subsequent data analysis, that is, the estimation of the intra-class variability and the identification of trajectories significantly altered between groups.
Designed initially for metabolomic, santaR is also suited for other Systems Biology disciplines. Implemented in R and Shiny, santaR is developed as a complete and easy-to-use statistical software
package, which enables command line and GUI analysis, with fast and parallel automated analysis and reporting. Comprehensive plotting options as well as automated summaries allow clear identification
of significantly altered analytes for non-specialist users.
Install the CRAN release of santaR with:
The development version can be obtained from GitHub:
If the dependency pcaMethods is not successfully installed, it can be installed from Bioconductor:
To get started santaR’s graphical user interface implements all the functions for short asynchronous time-series analysis:
The graphical user interface is divided in 4 sections, corresponding to the main steps of analysis:
Import, DF search, Analysis and Export:
• The Import tab manages input data in comma separated value (csv) format or as an RData file containing a previous analysis. Once data is imported the DF search and Analysis tabs become available.
• DF search implements the tools for the selection of an optimal number of degrees of freedom (df).
• With the data imported and a pertinent df selected, Analysis regroups the interface to visualise and identify variables significantly altered over time. A plotting interface enables the
interactive visualisation of the raw data points, individual trajectories, group mean curves and confidence bands for all variables, which subsequently can be saved. Finally, if inter-group
differential trajectories have been characterised, all significance testing results (with correction for multiple testing) are presented in interactive tables.
• The Export tab manages the saving of results and automated reporting. Fitted data can be saved as an RData file for future analysis or reproduction of results. csv tables containing significance
testing results can also be generated and summary plot for each significantly altered variable saved for rapid evaluation.
Vignettes and Demo data
More information is available in the graphical user interface as well as in the following vignettes:
A dataset containing the concentrations of 22 mediators of inflammation over an episode of acute inflammation is also available. The mediators have been measured at 7 time-points on 8 subjects,
concentration values have been unit-variance scaled for each variable. A subset of the data is presented below:
4 ind_6 Group2
4 ind_7 Group1
4 ind_8 Group2
8 ind_1 Group1
8 ind_2 Group2
8 ind_3 Group1
2.668 2.464 1.365 1.743
-0.3002 0.05366 0.4509 0.01572
3.777 2.543 1.858 2.213
-0.3275 0.1564 0.585 0.03299
0.708 0.4893 -0.08219 0.9345
-0.4101 -0.03727 -0.2914 -0.7239
Other tips
The GUI is to be prefered to understand the methodology, select the best parameters on a subset of the data before running the command line, or to visually explore results.
If a very high number of variables is to be processed, santaR’s command line functions are more efficient, as they can be integrated in scripts and the reporting automated.
santaR is licensed under the GPLv3
As a summary, the GPLv3 license requires attribution, inclusion of copyright and license information, disclosure of source code and changes. Derivative work must be available under the same terms.
© Arnaud Wolfer (2024) | {"url":"https://cran.case.edu/web/packages/santaR/readme/README.html","timestamp":"2024-11-08T18:49:07Z","content_type":"application/xhtml+xml","content_length":"16651","record_id":"<urn:uuid:c16b1077-2f34-43cd-bb34-1a4f99a65658>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00186.warc.gz"} |
What is 207 Acres in Square Feet?
207 Acres =
9016920 Square Feet
Unit Converter
How to convert 207 Acres to Square Feet
To calculate 207 Acres to the corresponding value in Square Feet, multiply the quantity in Acres by 43560 (conversion factor). In this case we should multiply 207 Acres by 43560 to get the equivalent
result in Square Feet:
207 Acres x 43560 = 9016920 Square Feet
207 Acres is equivalent to 9016920 Square Feet.
How to convert from Acres to Square Feet
The conversion factor from Acres to Square Feet is 43560. To find out how many Acres in Square Feet, multiply by the conversion factor or use the Area converter above. Two hundred seven Acres is
equivalent to nine million sixteen thousand nine hundred twenty Square Feet.
Definition of Acre
The acre (symbol: ac) is a unit of land area used in the imperial and US customary systems. It is defined as the area of 1 chain by 1 furlong (66 by 660 feet), which is exactly equal to 1⁄640 of a
square mile, 43,560 square feet, approximately 4,047 m2, or about 40% of a hectare. The most commonly used acre today is the international acre. In the United States both the international acre and
the US survey acre are in use, but differ by only two parts per million, see below. The most common use of the acre is to measure tracts of land. One international acre is defined as exactly
4,046.8564224 square metres.
Definition of Square Foot
The square foot (plural square feet; abbreviated sq ft, sf, ft2) is an imperial unit and U.S. customary unit (non-SI, non-metric) of area, used mainly in the United States and partially in
Bangladesh, Canada, Ghana, Hong Kong, India, Malaysia, Nepal, Pakistan, Singapore and the United Kingdom. It is defined as the area of a square with sides of 1 foot. 1 square foot is equivalent to
144 square inches (Sq In), 1/9 square yards (Sq Yd) or 0.09290304 square meters (symbol: m2). 1 acre is equivalent to 43,560 square feet.
Using the Acres to Square Feet converter you can get answers to questions like the following:
• How many Square Feet are in 207 Acres?
• 207 Acres is equal to how many Square Feet?
• How to convert 207 Acres to Square Feet?
• How many is 207 Acres in Square Feet?
• What is 207 Acres in Square Feet?
• How much is 207 Acres in Square Feet?
• How many ft2 are in 207 ac?
• 207 ac is equal to how many ft2?
• How to convert 207 ac to ft2?
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• What is 207 ac in ft2?
• How much is 207 ac in ft2? | {"url":"https://whatisconvert.com/207-acres-in-square-feet","timestamp":"2024-11-03T19:30:14Z","content_type":"text/html","content_length":"36609","record_id":"<urn:uuid:4b61d215-579c-43c4-96cd-ecd57c7a00c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00192.warc.gz"} |
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Time series forecasting methods
Time series forecasting is a vital thing of records evaluation, used throughout severa industries to count on destiny values primarily based mostly on historical data. Whether forecasting income,
stock fees, or climate styles, information considered one of a type forecasting techniques is important for making knowledgeable alternatives. This article explores the important factor techniques
applied in time series forecasting, highlighting their programs, strengths, and weaknesses.
Understanding Time Series Data
Time series information is a sequence of facts factors collected or recorded at unique time intervals. Unlike different information kinds, in which observations are impartial of each other, time
series data has an inherent temporal ordering. This makes it unique and requires specific interest when studying and forecasting future values. Understanding the tendencies and components of time
collection facts is important for effective evaluation and prediction.
What is Time Series Data?
Time collection data consists of observations made sequentially over the years, frequently at normal intervals including daily, month-to-month, or each 12 months. These information factors could
constitute diverse phenomena, such as inventory expenses, temperature readings, earnings figures, or even internet site site visitors. The key feature of time series records is its chronological
order, which need to be maintained all through evaluation to keep the temporal relationships among observations.
Key Components of Time Series Data
Time series facts can be decomposed into severa key additives that help us apprehend the underlying styles:
Definition: A fashion is the lengthy-term movement or direction within the facts over the years. It represents the general tendency of the facts to growth, decrease, or live stable over an extended
Example: A ordinary upward fashion in annual revenue over severa years suggests constant industrial company growth.
Definition: Seasonality refers to periodic fluctuations or styles that repeat at regular intervals, frequently pushed by means of seasonal elements like weather, holidays, or monetary cycles.
Example: Retail income peaking during the holiday season each year is a traditional instance of seasonality.
Cyclic Patterns
Definition: Cyclic styles are fluctuations that get up over longer, irregular durations, unlike seasonality, which has a fixed periodicity. These cycles are frequently inspired with the resource of
outside economic or social elements.
Example: Business cycles, wherein periods of economic expansion are followed with the aid of the usage of recessions, are an example of cyclic styles.
Noise (Irregular Component)
Definition: Noise refers to random variations or fluctuations within the facts that can not be attributed to the fashion, seasonality, or cyclic patterns. Noise is regularly considered because the
"errors" or "residual" aspect of the time collection.
Example: Sudden spikes in stock fees due to surprising news or activities constitute noise in economic time series facts.
Stationarity in Time Series
One critical concept in time series analysis is stationarity. A time collection is stated to be desk certain if its statistical residences, including mean, variance, and autocorrelation, remain
constant over time. Stationarity is critical for lots time series forecasting methods, like ARIMA, which count on that the underlying time series is desk bound.
Types of Stationarity:
• Strict Stationarity: The statistical properties are ordinary through the years.
• Weak Stationarity: The imply and variance are consistent through the years, and the covariance among factors depends simplest on the time hollow among them.
If a time series isn't table sure, it may often be converted right right into a desk bound series through techniques like differencing, detrending, or seasonal adjustment.
Autocorrelation and Lag
• Autocorrelation measures the relationship among observations at unique points in time inside the identical time series. In other phrases, it quantifies how beyond values of the series have an
impact on destiny values.
• Lag: The time distinction the various observations being in contrast in an autocorrelation calculation is referred to as the lag. For example, a lag of 1 compares every information factor with
its instant predecessor.
Time Series Decomposition
Decomposition is a way used to interrupt down a time collection into its essential components: fashion, seasonality, and noise. This facilitates in expertise the underlying shape of the statistics
and in selecting appropriate forecasting methods.
Decomposing a time series is regularly the first step within the analysis, allowing for a clearer view of each component, which can then be modeled one after the other.
Challenges in Time Series Analysis
Working with time collection facts presents precise demanding situations that differ from different varieties of information evaluation:
• Non-Stationarity: Many actual-world time collection are non-stationary, requiring transformation earlier than analysis.
• Seasonality and Cyclicality: Accurately identifying and modeling seasonality and cyclicality is critical however may be complicated, especially while those patterns trade over time.
• Missing Data: Time collection frequently be afflicted by missing records points, that may disrupt analysis and forecasting. Techniques like interpolation or imputation are used to deal with this
• Autocorrelation and Lag Selection: Determining the right lags to use in models that account for autocorrelation (like ARIMA) may be tough and requires cautious analysis.
• Outliers: Time collection data is susceptible to outliers, that can extensively impact forecasts if no longer properly handled.
Types of Time Series Forecasting Models
1. Naive Methods
Naive forecasting is the only approach, assuming that future values may be much like the most latest observations.
• Naive Forecast: The subsequent duration's price is thought to be the same as the final discovered value. This technique is brief and smooth to put into effect but works first-rate when statistics
suggests no trend or seasonality.
• Seasonal Naive: This method assumes that the fee in the subsequent length might be the same as the ultimate located value within the same season. For instance, a retail store might anticipate
income this December to be just like sales final December.
2. Moving Averages
Moving averages clean out brief-time period fluctuations and highlight longer-time period traits or cycles.
• Simple Moving Average (SMA): This approach averages a set wide variety of past observations to are expecting the following price. It's effective for smoothing records however can lag while
developments change quickly.
• Weighted Moving Average: Unlike SMA, this method assigns specific weights to past observations, normally giving extra importance to latest statistics. This makes it extra aware of modifications
within the information.
3. Exponential Smoothing
Exponential smoothing strategies exercise exponentially decreasing weights to past observations, making the forecast greater attentive to recent modifications.
• Simple Exponential Smoothing (SES): This approach is right for time series records without a clean trend or seasonality. It makes use of a smoothing regular to decide how a amazing deal weight is
given to the maximum brand new statement.
• Holt's Linear Trend Model: This method extends SES with the resource of consisting of a fashion thing, allowing it to seize linear dispositions in the facts.
4. Autoregressive Integrated Moving Average (ARIMA)
ARIMA is a effective and flexible approach that mixes three additives: autoregression (AR), differencing (I), and transferring common (MA).
• ARIMA: ARIMA models are particularly beneficial for non-stationary information, which have tendencies or seasonality that can be made desk bound via differencing. The version then uses AR and MA
additives to assume destiny values.
• SARIMA: This extension of ARIMA consists of seasonal additives, making it suitable for time series facts with a seasonal sample. SARIMA fashions can contend with each non-stationarity and
seasonality, making them substantially powerful in many real-international packages.
5. Autoregressive Models
Autoregressive fashions are expecting destiny values based on past values of the collection.
• AR (Autoregressive): In an AR model, destiny values are anticipated primarily based on a linear mixture of previous values. This model assumes that past values have a right away influence on
destiny values.
• MA (Moving Average): MA models are expecting future values based totally on past mistakes. This model is useful while past prediction errors show a sample that can be leveraged to improve
• ARMA (Autoregressive Moving Average): ARMA fashions integrate AR and MA components and are best when managing stationary information.
6. State-Space Models
State-area models are used for more complicated time series forecasting, specifically whilst the records has a couple of underlying methods.
• Kalman Filter: A recursive method used to estimate the state of a dynamic device from a series of noisy measurements. It's widely used in actual-time forecasting, consisting of in navigation and
monitoring systems.
• Structural Time Series Model: This model decomposes the time series into additives like fashion, seasonal, and irregular additives, providing a clean interpretation of every detail's contribution
to the overall collection.
7. Machine Learning and Deep Learning Models
Machine getting to know and deep mastering methods are increasingly used for time collection forecasting due to their capability to capture complicated patterns in huge datasets.
• Linear Regression: A easy and interpretable technique that models the relationship among time collection and explanatory variables.
• Support Vector Regression (SVR): A kind of regression that makes use of guide vector machines, effective in excessive-dimensional spaces.
• Decision Trees/Random Forests: Ensemble strategies that construct more than one selection trees and integrate their effects. They are useful for taking pictures nonlinear relationships within the
• Neural Networks: Deep studying models like Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU) are especially designed to handle sequential data. They can capture complex styles and
dependencies in time series data, making them effective however requiring big datasets and sizeable computational assets.
8. Prophet
Prophet, developed by using Facebook, is a forecasting tool designed to deal with seasonality, holidays, and lacking statistics. It's user-pleasant and works well with day by day or weekly
statistics, making it famous for business applications. Prophet fashions fashion as a piecewise linear function, which allows it to capture and expect modifications in developments efficaciously.
9. Fourier Transform
Fourier Transform methods, like Fast Fourier Transform (FFT), convert time collection facts into the frequency area. This is useful for identifying cyclical styles and traits that may not be right
now obvious in the time domain.
10. Ensemble Methods
Ensemble methods involve combining one of a kind forecasting models to improve accuracy.
Combining Forecasts: By averaging the predictions from specific models, along with ARIMA, Exponential Smoothing, and Machine Learning models, ensemble methods can often gain better performance than
any single model.
Choosing the Right Method
Selecting the right forecasting method relies upon on numerous elements:
• Stationarity: Methods like ARIMA require stationary statistics, even as fashions like Holt-Winters handle seasonality directly.
• Data Volume: Machine learning fashions typically carry out better with large datasets, whereas easier models like transferring averages can paintings well with smaller datasets.
• Complexity vs. Interpretability: Simpler fashions (e.G., Naive, Moving Averages) are less complicated to interpret, at the same time as complex models (e.G., Neural Networks, ARIMA) may also
offer higher accuracy but are harder to interpret.
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Deriving Poisson Equation
May 21, 2017 tags: math
Poisson distribution is defined as,
$P(x; \lambda) = \frac{e^{-\lambda} \lambda^x}{x!}$
• $ P(x;λ) $ is the probability that an event occurs x times in the given interval,
• $ $ is the expected rate/probability of an event occuring.
In Binomial distribution, probability that one of the two events(p and q) occurs x times out of n trials is, $P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}$
where q=1−p. Now, if λ is the expected number of successes then, $p=\frac{\lambda}{n}$.
… TODO
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Department of Mathematics
Math student in ISU Programming Team Advances to World Finals
Yuxiang Zhang, a senior in Mathematics, and his teammates will advance to the Association for Computing Machinery’s world finals programming competition May 20, 2017, after finishing in the top three
of 232 teams in a regional competition. Read More
November 17, 2016 | {"url":"https://math.iastate.edu/category/news/page/13/","timestamp":"2024-11-13T08:42:47Z","content_type":"text/html","content_length":"53264","record_id":"<urn:uuid:de9b9ac8-3674-4908-98b3-f03105c999bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00616.warc.gz"} |
Ton of air conditioning conversions
The ton of air conditioning conversion selector above selects the power measurement unit to convert to starting from tons of air conditioning. To make a conversion starting from a unit of power other
than ton of air conditioning, simply click on the "Reset" button.
About ton of air conditioning
The ton of air conditioning is a unit of power equal to 3,504 watts (1 ton of air conditioning = 3,504 W), the derived unit of power in the International System of Units (SI).
The ton of air conditioning is also equal to 3.504 kilowatts (kW), or 0.003504 megawatts (MW), or 3.504 × 10^-6 gigawatts (GW), or 11,956.144 British thermal units (IT) per hour (BTU/h), or
3,014.91396 kilocalories (thermochemical) per hour (kcal/h) or 4.764114 horsepower (metric) (hp).
The ton of air conditioning unit is used to measure heat absorption in air conditioning.
Plural: tons of air conditioning
Ton of air conditioning conversions: a list with conversions from tons of air conditioning to other (metric, imperial, or customary) power measurement units is given below. | {"url":"https://conversion-website.com/power/from-ton-of-air-conditioning.html","timestamp":"2024-11-09T15:44:14Z","content_type":"text/html","content_length":"17090","record_id":"<urn:uuid:e24fcc06-93df-40ee-80eb-31befcd63825>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00452.warc.gz"} |
CFrame clamp on turret?
I have gone through every article I can find for the past couple of days trying to figure out how to limit the rotation of an object, yes I have looked at the other articles and I’m not wrapping my
head around it.
Currently, I Have and many variants of this.
local mousePos = Mouse.Hit.Position
local rotate = Vector3.new(mousePos.X, turr.Position.Y, mousePos.Z)
turma:SetPrimaryPartCFrame(CFrame.new(turr.Position, rotate))
Looks like:
What id like to do is stop it from rotating over these red lines.
Basically limiting its Y rotation to only a couple of degrees.
If you could point me in the right direction that would be wonderful.
1 Like
First you can set up a vector which would bisect your vision triangle.
Then you can use the dot product, to get the angle between the vector which represents the direction towards the mouse point, and the vector I described up there.
If the angle between these two vectors exceeds a value, then you can clamp it or do something with it as necessary
local bisectingVector = ...
local totalAngle = ...
local mousePos = Mouse.Hit.Position
local direction = (turr.Position - mousePos).unit
local angle = math.acos(direction:Dot(bisectingVector);
if angle > totalAngle * .5 then
-- clamp it here
local rotate = Vector3.new(mousePos.X, turr.Position.Y, mousePos.Z)
turma:SetPrimaryPartCFrame(CFrame.new(turr.Position, rotate))
2 Likes
If you want to know how it works here is the clamping function, I convert the CFrame to orientation then clamp the orientation angles.
Code sniplet
--Inverse the part0 only for Motor6D's
--Set it to CFrame.new() for non Motor6ds
local relativeToWorld = currentJointMotor6D.Part0.CFrame:Inverse()
local lookAtWorld = CFrame.lookAt(turretPosition,lookAtPosition,baseCFrame.UpVector)--goal LookAt CFrame
local goalCFrame
if self.Constraints then
local turretRelativeCF = baseCFrame:ToObjectSpace(lookAtWorld)
local x , y , z = turretRelativeCF:ToOrientation()
local constrainedX , constrainedY = self:EulerClampXY(x,y)
--Detect quadrant of lookAt position
local jointPosition = currentJointMotor6D.Part0.CFrame*originalC0Position
local quadrantLookAtFromJointPosition = CFrame.lookAt(jointPosition,lookAtPosition,baseCFrame.UpVector)
local baseRelative = baseCFrame:ToObjectSpace(quadrantLookAtFromJointPosition)
local _,y, _ = baseRelative:ToOrientation()
constrainedY = math.abs(constrainedY)*math.sign(y)--use the quadrants of the lookAtFromJoint
goalCFrame = relativeToWorld*baseCFrame*CFrame.fromOrientation(constrainedX,constrainedY,z)*turretCFrameRotationOffset
--Unconstrained use lookAtWorld
goalCFrame = relativeToWorld*lookAtWorld*turretCFrameRotationOffset
--Euler Clamp function:
-- negative z is front, x is rightvector
function TurretController:EulerClampXY(x,y)
local Constraints = self.Constraints
local degY = math.deg(y)
local degX = math.deg(x)
local newY = math.clamp(degY,-Constraints.YawRight,Constraints.YawLeft)
local newX = math.clamp(degX,-Constraints.DepressionAngle, Constraints.ElevationAngle)
return math.rad(newX), math.rad(newY)
Edit: For your case do this,
local mousePos = Mouse.Hit.Position
local rotate = Vector3.new(mousePos.X, turr.Position.Y, mousePos.Z)
local goalCFrame = CFrame.new(turr.Position, rotate)
local x,y,z = goalCFrame:ToOrientation()
--Notice the angles being changed
--Then you can clamp the angles
1 Like
Thank you, but honestly, I have no idea what any of that means. I’ll need to look into clamps.
So if im not mistaken clamping where you noted will limit how much it can rotate.
from my understanding, I can use math.clamp(y,-20,20) and it would only let me move it by 40 degrees.
so it would look something like
turma:SetPrimaryPartCFrame(goalCFrame,math.clamp(y,-20,20)) ?
Well, here is a more simpler code snippet:
local mousePos = Mouse.Hit.Position
local rotate = Vector3.new(mousePos.X, turr.Position.Y, mousePos.Z)
local goalCFrame = CFrame.new(turr.Position, rotate)
local x,y,z = goalCFrame:ToOrientation()
--Notice the angles being changed
--Then you can clamp the angles, restrict it
--clamps between -30 to 30 degrees
clampedY = math.clamp(math.deg(y),-30,30)
--convert back to radian
clampedY = math.rad(clampedY)
--Create new clamped goalCFrame, using clamped angles
local clampedGoalCFrame = CFrame.fromOrientation(x,clampedY,z)+turr.Position--maintain the turret position so add it
--use the new clamped CFrame
2 Likes
That is SOOO AMAZING!!! Im going to need to check up on your profile with those IK bones. Thank you so much I don’t know what to say.
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My ANOVA and Gage R&R Self-Education
Quote of the Day
Who are a little wise the best fools be.
— John Donne
I recently took an excellent class at Statistics.com called "Prediction & Tolerance Intervals, Measurement and Reliability" taught by Dr. Tom Ryan, a former NIST researcher. I took the class because
I have been concerned that some of the statistical methods I am currently using for calibrating optics are not as good as they could be. As part of the class, we were required to perform some
practice Gage Repeatability and Reproducibility (aka gage R&R) studies, which required me to use ANOVA. The class gave me many ideas for improving my experiment design and I highly recommend it to
those who do a lot of experiments.
Figure 1 shows a caliper, which is a common form of gage (sometimes spelled gauge). A gage is any device that is used to perform a measurement. In my case, I will be performing gage R&R on some
optical power measurements.
In this post, I am focusing on my self-education on ANOVA and its application to gage R&R. While I have used ANOVA for years for evaluating the significance of test data, but I have never looked at
how it works until very recently. The opportunity to study ANOVA in detail came because of some work I needed to do in Taiwan (a place that I enjoy very much). On the flights from Minneapolis to San
Francisco to Taiwan, I had 17 hours of flight and a 5 hour of layover to think about how ANOVA works. I put this time to some good use.
Source Material
My approach here will slavishly follow that of M.J. Moroney in his excellent book "Facts From Figures". First published in 1951, you can find this book in used bookstores for about $1 or you can
download it in PDF form from the web. I discovered this little gem years ago and I turn to it when I need a short refresher on basic statistics.
Why Gage R&R?
When I started my career at HP back in 1979, my mentor there told me that "Manufacturing is in a constant battle against variation -- ideally, we make the millionth unit the same as the first." To
battle variation, we must first be able to identify and measure it. The focus of gage R&R is on understanding and measuring the sources of variation in a measurement. Because good product design
practice requires that you design for the worst-case parameter variation, excessive variation in measurement forces you to design your product to be tolerant of this variation and that increases
The folks in our Quality team break down variation as shown in Figure 2 (Source).
When we talk about measurement variation, we are talking about precision. The Wikipedia describes precision as follows
The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.
Gage R&R explicitly measures reproducibility and repeatability relative to the level of part variation.
Most technical discussions of ANOVA dive into equations with multiple levels of summation symbols. Moroney begins his example by taking a simple experiment and showing how you can can break the
variance of the result into components with no use of summations. I like this approach as a gentle start.
Since I am focused on gage R&R here, my plan is produce a several related worksheet items all contained in a single Excel workbook (available here) that does not use any Visual Basic. :
• An Excel worksheet that works through one of the ANOVA examples from "Facts From Figures". While not a gage R&R example, I will use this example as the basis of my gage R&R work. Here is the book
excerpt that I am using as my gage R&R reference.
The examples in this book are simple because they are from a time when calculations were done by hand. This is not a bad thing for today because it means you can easily duplicate them using a
tool like Excel or Mathcad. While I prefer to use Mathcad for nearly everything, Excel is probably a better vehicle for my study work here.
• A set of Excel worksheets that contain gage R&R example using a template I have created whose results agree with the same data processed by Minitab, which is the tool Dr. Ryan recommended (he was
one of its creators).
Yes, Dr. Ryan strongly discouraged me from using Excel for ANYTHING, but I do believe it can play a role in routine statistical calculations and I will ignore his advice here. As you can see, I
often do not follow directions. Sister Mary Agnes from the Osseo Catholic School is probably looking down upon me from heaven with a frown on her face.
This worksheet is intended to illustrate the concepts behind ANOVA and gage R&R -- it is not computationally efficient. However from a conceptual standpoint, I like Moroney's approach of eliminating
row and column variation in separate operations to determine the desired variance components.
Role of ANOVA in Gage R&R
ANOVA is one of the accepted methods for determining the variance components in a experiments (cf. AIAG approach). With respect to this post, there are three sources of variation: the part itself,
the operator, and random error. Gage R&R studies often include modeling the interaction between part and operator, but to keep things simple I will ignore this sort of variation for this post. The
methods shown here can be extended to include interaction, but I want to keep this post simple.
Equation 1 shows the gage R&R variance model that I will be using.
Eq. 1 $\displaystyle \sigma _{T}^{2}=\sigma _{Reproducibility}^{2}+\sigma _{Parts}^{2}+\sigma _{Repeatability}^{2}$
• σ[T]^2 is the total variance of data.
• σ[Parts]^2 is the component of variance due only to the parts.
• σ[Reproducibility]^2 is the component of variance due only to the operators.
• σ[Repeatability]^2 is the component of variance due only the measurement tools.
We will use ANOVA to determine the variance components. Once we have the components, we can determine the relative effects on our measurement of the different components. Customarily, we want to see
the repeatability and reproducibility components to be less than 10% of the total variance (example of the 10% standard).
My ANOVA Reference Model
Figure 3 shows my rework of Maroney's Latin Square excerpt, which is focused on "treatments" applied to some crop. This is the example I used as my model for writing a simple gage R&R worksheet. The
process for generating Figure 2 can be broken down as follows:
• Average the effect of each set of treatment levels, then determine the Mean Square error (MS) of the treatment.
We will use this term to estimate the variance contribution of the treatment.
• Remove the effect of row variation by replacing each row element with the average element value for that row.
By eliminating the row variation, we can compute the MS of the column variation.
• Remove the effect of column variation by replacing each column element with the average element value for that column.
By eliminating the column variation, we can compute the MS of the row variation.
My Gage R&R Worksheet
I have put a number of tabs in this Excel worksheet that duplicate the results obtained from Minitab and other tools. We can make a direct analogy between my gage R&R and Figure 3 as follows.
• treatments in Figure 3 are comparable to parts in gage R&R. Our analysis will provide us with the part-to-part variation.
• column variation is Figure 3 is comparable to repeatability variation in gage R&R (statistics folks will often call this term the residual error)
• row variation in Figure 3 is comparable to reproducibility or operator variation in gage R&R.
The worksheet uses array formulas to compute the gage R&R of data placed into the data area at the bottom of the worksheet. The variances are computed by substituting the MS values into the equations
shown in Figure 4.
I will not be deriving the formulas of Figure 3 in this post. You can find them in various places on the web. Here is a link to one Powerpoint presentation that gives these formulas, but with
different variable names. I intend to derive them in a later blog post.
I finally feel like I have an intuitive model of what is going on when I perform an ANOVA analysis. While you do not need to know the details of how the computation is performed, it does help you get
some insight into the ANOVA process. This reminds me a bit of Fourier analysis. You do not necessarily need to know the details of a Fast Fourier Transform (FFT), but knowing the details does give
you some insight into what is going on.
2 Responses to My ANOVA and Gage R&R Self-Education
1. Really nice to see some explanatory diagrams in the text.
□ I needed to make some diagrams because just staring at summation symbols did nothing for my understanding of how things worked.
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Infinite Analysis Seminar Tokyo
Seminar information archive ～11/05｜Next seminar｜Future seminars 11/06～
Date, time & place Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.)
11:00-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)
(京大数研) 11:00-12:00
一般化されたヤング図形の q-Hook formula
[ Abstract ]
Young図形における hook formula は、組合せ論的には、その Young 図形の standar
d tableau の総数を数え上げる公式である。R. P. Stanley は reverse plane parti
tion のなす母関数を考えることにより、この公式をq-hook formula に拡張し、E. R
. Gansner はそれをさらに多変数に一般化した。
本講演では、この(多変数)q-Hook formula が(D. Peterson、R. A. Proctor の意
n の hook formula の証明も与える。
(京大数研) 13:30-14:30
Catalan numbers and level 2 weight structures of $A^{(1)}_{p-1}$
[ Abstract ]
Motivated by a connection between representation theory of
the degenerate affine Hecke algebra of type A and
Lie theory associated with $A^{(1)}_{p-1}$, we determine the complete
set of representatives of the orbits for the Weyl group action on
the set of weights of level 2 integrable highest weight representations of $\\widehat{\\mathfrak{sl}}_p$.
Applying a crystal technique, we show that Catalan numbers appear in their weight multiplicities.
Here "a crystal technique" means a result based on a joint work with S.Ariki and V.Kreiman,
which (as an application of the Littelmann's path model) combinatorially characterize
the connected component (usually called Kleshchev bipartition in the representation theoretic context)
$B(\\Lambda_0+\\Lambda_s)\\subseteq B(\\Lambda_0)\\otimes B(\\Lambda_s)$ in the tensor product.
(高知大理学部数学) 15:00-16:00
On a dimer model with impurities
[ Abstract ]
We consider the dimer problem on a non-bipartite graph $G$, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that
impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on
$G$ and the set of spanning forests on two graphs which are made from $G$, with configuration of impurities satisfying a pairing condition, and this bijection can be regarded as a extension of the
Temperley bijection. We consider local move consisting of two operations, and by using the bijection mentioned above, we prove local move connectedness. Finally, we prove that the above conjecture is
in some spacial cases. | {"url":"https://www.ms.u-tokyo.ac.jp/seminar/2009/sem08-223_e.html","timestamp":"2024-11-05T18:47:45Z","content_type":"application/xhtml+xml","content_length":"18020","record_id":"<urn:uuid:a31812cf-8a57-4229-8136-85b4640ce072>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00754.warc.gz"} |
Piecing it Together
Unit4, Section3: Piecing it Together
Instructional Days: 5
Enduring Understandings
Real-life phenomena are often complex. Data scientists use multiple regression models to create simple equations to help explain and predict these phenomena. Data scientists can also use polynomial
transformations to add flexibility to rigid linear models.
Students will read the article titled How Long Can a Spinoff Like Better Call Saul Last? that will set the context for students to begin thinking about more than one explanatory variable to make
better predictions. The article can be found at:
Learning Objectives
S-ID 6: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
• a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear
Understand that multiple regression can be a better tool for predicting that simple linear regression and know when it is appropriate to use multiple regression versus simple linear regression.
Understand when linear models are not appropriate based on the shape of the scatterplot.
• Use multiple linear regression models with other predictor variables
• Fit regression lines to data and predict outcomes.
• Fit polynomials functions to data.
Economists and marketing firms use multiple regression to predict changes in the market and adjust strategies to fit the demands of changes in the marketplace.
Language Objectives
1. Students will read informative texts to evaluate claims based on data.
2. Students will engage in partner and whole group discussions about how adding variables to a model will help or hinder its predictions.
3. Students will construct their own linear model using multiple variables to compare and contrast which model makes the best predictions.
Data File or Data Collection Method
1. Movies: data(movie)
2. Cereal brands: data(cereal)
Students will collect data for their Team Participatory Sensing campaign.
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Eighth Grade
Number and Operations (NCTM)
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Understand and use ratios and proportions to represent quantitative relationships.
Compute fluently and make reasonable estimates.
Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
Geometry (NCTM)
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.
Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
Apply transformations and use symmetry to analyze mathematical situations.
Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.
Measurement (NCTM)
Apply appropriate techniques, tools, and formulas to determine measurements.
Solve problems involving scale factors, using ratio and proportion.
Grade 8 Curriculum Focal Points (NCTM)
Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle
Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps.
They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about
similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines
to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean
Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian
coordinate plane to measure lengths and analyze polygons and polyhedra. | {"url":"https://newpathworksheets.com/math/grade-8/similarity-and-scale?dictionary=rectangular+prism&did=253","timestamp":"2024-11-09T23:49:48Z","content_type":"text/html","content_length":"53374","record_id":"<urn:uuid:5a856ee3-b4bc-4a4e-a574-ca4f5c4b9d81>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00544.warc.gz"} |
19 Inverse Proportion Examples in Real Life
Two quantities are said to be in inverse proportion if an increase in the amount of the first quantity causes a proportionate decrease in the second quantity in such a manner that the product of the
two quantities remains constant throughout the variation. Similarly in the case of inverse proportion, by decreasing the value of one quantity, the number of the second value increases. Let, the two
inversely proportional quantities be denoted by the variables x and y. Then, the product of the two variables can be represented by a constant k. The concept of inverse proportion facilitates the
user to easily estimate the amount or value of a missing entity, provided the basic data regarding the problem statement is already known. In simple words, if two entities vary inversely and are
related to each other in such a way that a change in the value or amount of one entity corresponds to an inverse change in the value or amount of the other entity, then both the entities are said to
be in inverse proportion. Mathematically, inverse proportionality is given as x ∝ 1/y; where x and y are two variables.
Examples of Inverse Proportion
Some of the real-life applications of inverse proportion are listed below:
1. Different Modes of Travelling and the Time
Suppose a working professional uses different modes of travelling every day to reach his office. Some of the modes of travelling that he uses include walking, running, cycling, and riding a bike.
Suppose the office is 90 km away from his house. The time taken by him to reach his destination while walking at a speed of 3 km/hr is equal to 30 minutes. The time further reduces to 15 minutes if
he chooses to run at a speed of 6 km/hr. Likewise, he would reach the office in 10 minutes, if he opts for cycling at a speed equal to 9 km/hr and in 2 minutes, if he rides a motorcycle at 45 km/hr.
Here, the maximum time is consumed when he opts to walk to the office and the minimum time is consumed when he chooses to ride a bike. In this case, time tends to vary inversely according to the
speed of travelling medium opted by him.
2. Number of People and the Time that is taken to complete a Particular Task
The number of people performing a particular task is inversely proportional to the time taken for completion. Let us say that 2 people take 6 days to paint the fence of a garden, then according to
the inverse proportion, a team of 3 people would complete the same task in 4 days and a team of 4 people would need only 3 days for completion. Here, the product of the two variables, i.e., the
number of people and number of days is equal to 12 and remains constant throughout the variation.
3. Speed of the Vehicle and Time Covered
Suppose you have to travel a 160 km distance to reach a particular destination. If you travel at a speed of 40 km/hr then you would reach your destination in 4 hours. Now, if you double the speed of
the vehicle, then the time taken to reach the destination gets reduced to half. This means that travelling at a speed equal to 80 km/hr takes 2 hours to complete the journey. This is because speed
and time are the two physical quantities that are inversely related to each other.
4. Number of Vehicles on the Road and Free Space on Road
The number of vehicles present on a road is typically inversely proportional to the empty space on a road. This is because more number of vehicles would cover more area of the road, thereby leaving
less empty space and fewer number of vehicles would need comparatively less area of the road, thereby providing more empty or free space.
5. Distance and Brightness
The illumination produced by a light source varies indirectly with respect to the distance. Let us say that, a light source is turned on at the one end of a hall. If you observe the objects present
within a 100-metre range of the light source, they tend to appear comparatively brighter than the objects present 500 metres away from the light source. This means that the brightness reduces as you
move away from the light source and increases when you move towards the light source.
6. Number of Rows and Columns
Suppose you have 12 marbles that you wish to arrange in form of rows and columns. There are a number of ways to arrange them in this particular manner. An arrangement that consists of two columns
would have six rows. Now, if you increase the number of columns to three, the number of rows reduces to four. Similarly, the same twelve marbles can be organised in a format that consists of four
columns and three rows, and so on. In this case, you can easily observe that the product of the number of rows and columns is equal to 12 and remains constant. This means that the number of rows
varies in inverse proportion to the number of columns.
7. Time and Freshness of a Food Item
When you pluck a fruit from a tree and store it in a basket, it begins to lose its freshness as time passes by. As time increases, the freshness of the fruit begins to decrease. This means that time
and freshness of a flower are inversely proportional to each other. An increase in the value of one quantity tends to induce a proportionate decrease in the number of other entities.
8. Number of Pipes required and Time taken to Fill a Swimming Pool
Let us say that a person is able to completely fill a swimming pool with water in 4 hours by connecting two water pipes to it. Now, if the number of pipes connected to the swimming pool is increased
to 4, then the time required to fill the pool gets reduced to 2 hours, provided the flow rate of the fluid through all pipes remain constant. This means that the two variables, i.e., the number of
pipes and the time taken are inversely proportional to each other.
9. Number of Students and the amount of Food available in a Hostel Mess
The number of students residing in a particular hostel is inversely proportional to the time taken to consume a particular amount of food available in the hostel mess. For instance, 100 students
consume 50 kg of flour in a week. Now, if the number of students increases to 200, then the same amount of floor gets consumed in 3.5 days. Here, one can easily observe that when you double the
number of students, then the time taken to consume a particular amount of food reduces to half.
10. Surface Area of the Blade and the Pressure exerted by a Knife
The blade of a knife is tapered and is constructed in a wedge shape. Here, the surface area of the blade is inversely proportional to the sharpness of the knife or the pressure exerted by the knife.
The more is the surface area of the edge of the knife blade, the less will be its sharpness. Similarly, the lesser is the surface area, the more will is the sharpness.
11. Cost and Number of Articles Purchased
Let us say, a child visits a stationary shop to purchase a few comic books. The price of one comic book is equal to INR 15 and the total money that the child has is equal to INR 100. This means that
he would be able to purchase a total of 6 comic books. A few weeks later, he visits the same stationery shop and finds that the price of the comic book has now been increased to INR 25. He has INR
100 in hand and therefore, this time is able to purchase only four comic books. In this particular example, an inverse relationship between the cost and the number of articles purchased can be
observed easily. As the price of an item increases, the number of items purchased decreases and vice versa.
12. Volume and Pressure
One of the best examples to demonstrate inverse proportionality is the relationship between volume and pressure. Let a container has multiple holes drilled along its length. When water or any other
liquid is poured into the container, it begins to flow out through these holes. Water escaping through the hole that is located closest to the base experiences the maximum pressure, while water
escaping through the hole present near the top or near the opening of the container would encounter minimum pressure. This means that pressure increases with a decrease in volume and pressure
decrease with an increase in volume because the volume and pressure quantities are inversely related to each other.
13. Expenditure and Savings
The expenditure and savings are the two entities in finance that are inversely related to each other. This means that if you spend more amount of money, then the savings that you possess would be
14. Cost and Demand of an Item
The cost of an item is usually inversely proportional to its demand in the market. When the cost of an item reduces, more people opt to buy it, thereby increasing its demand in the market. Similarly,
when the cost goes high, most people prefer not to buy the item; therefore, the demand drops low.
15. Difference in the Altitude of the Edges of a See-Saw
A see-saw is a long narrow iron or wooden beam fixed on a pivot in the middle. The seats present on the edges of the board display the inverse proportionality in real life in the easiest possible
manner. When one end of the see-saw goes higher, then the other end drops down. The relationship between the altitude of both the edges is inversely proportional in nature. As the height of one edge
of the board begins to increase, the altitude of the other end of the board tends to reduce proportionally and vice versa.
16. Squeezing a Toothpaste Tube and the Contents of the Tube
There exists an inverse relationship between the magnitude of force applied to squeeze a toothpaste tube and the amount of paste contained by it. If you squeeze a toothpaste tube with force, then the
amount of paste left inside the tube begins to reduce. An increase in the magnitude of force applied to the tube causes a proportional decrease in the amount of the contents of the tube.
17. Charging of a Gadget and the Usage Time
The battery power of a gadget is inversely related to the time for which it is used. Suppose a gadget is charged to 98% before use. Let us say, after using it for one hour the battery drops down to
88%, after two hours the battery percentage is equal to 78%, the charging contained by the gadget after three hours is equal to 68%, and so on. In this case, with an increase in the value of the time
for which the gadget is being used, a significant and proportional decrease in the battery percentage can be observed easily.
18. Acceleration of an Object with respect to Time
Let us take the example of a spinning top. When the pointed end of the spinning top is placed on the ground and the rope wrapped around the top is pulled quickly with force, the top begins to spin.
It must be observed that the acceleration with which the to spins is maximum in the beginning and begins to drop gradually as time passes by. This means that with an increase in the value of time,
the magnitude of acceleration decreases proportionally, thereby demonstrating an inverse relationship between the magnitude of the acceleration of an object and the time.
19. Length of a Pencil with respect to the Usage
Let us say a pencil is 15 cm long. After using it to write one page of an assignment, the height of the pencil gets reduced by 2 cm. This means that after writing 4 pages the pencil would be 7 cm in
length. Likewise, after writing 7 pages, the pencil would be equal to 1 cm in length. The length of the pencil tends to reduce with an increase in usage. This means that there exists an inverse
relationship between the length and the usage of the pencil.
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Eigenmode-Based Solver for PCB Vias
This topic explains the use of an eigenmode-based solver to analyze the return path of a via in a planar structure such as a PCB [1].
You can model the signal path of a via using a lumped element circuit model, but the return current path usually involves structures that are separated from the via by more than a tenth of a
wavelength at the maximum frequency of interest (the usual limit for lumped circuit models). This return current path usually involves Ground Return Vias (GRVs) that connect the top and bottom
conducting planes in a given PCB layer and can couple with other via signal paths as well.
To model a return path, use radial transverse electromagnetic (TEM) waves that are eigenmodes of the planar structure. You can model the propagation and reflection of these radial TEM waves very
accurately, even over very long distances, using closed form linear equations defined at each of the discontinuities in the wave’s propagation path. A matrix solution of these linear equations is
used to obtain the return path impedance and signal coupling.
Wave Generation and Propagation
A via cell is a subsection of a via that transitions from one conducting plane, through one or more dielectric layers (possibly containing signal routing) to another conducting plane.
Within a via cell, the signal current flowing along the via barrel is exactly balanced by the return current flowing in the opposite direction across the edge of the antipad.
The return current flowing from the edge of the antipad generates a radial TEM wave centered on that via cell. The radial TEM wave propagates outward from the via cell like ripples in a pond. The
radial TEM wave creates magnetic coupling between the top and bottom conducting planes, like a half turn transformer. This is one of the mechanisms through which the return current is transferred
between the top and bottom planes.
The other current transfer mechanism is a GRV that shorts the top and bottom planes together at some point outside the via cell. A radial TEM wave that is generated at the via cell antipad propagates
to a nearby GRV, and is reflected by the GRV, creating another radial TEM wave flowing outward from the GRV.
Radial TEM waves from both the via cell and the GRV on the left continue to propagate, and eventually both waves strike other GRVs, in this case, the waves strike the GRV on the right hand side in
the diagram. The waves continue to propagate and reflect, often creating a complex system of waves propagating between the top and bottom planes.
Circuit Model
Combine the return path current conduction with the capacitance and inductance between the via barrel and the antipads to produce a complete circuit model for the via cell.
For version one of the via solver feature, the circuit elements C[top] and C[bottom] and L in this schematic are automatically calculated to produce a 50-ohm characteristic impedance. There is no
modeling of pad capacitances, entry/exit traces, or routing within a layer.
The impedance Z[R] in the schematic models the circuit behavior of the entire return path and includes any waves reflected from GRVs or the waves propagated from other via cells (crosstalk).
For each layer in a PCB, the equivalent circuits of the via cells are combined into a Generalized Circuit (ABCD) matrix, and the ABCD matrices are combined in series to produce an ABCD matrix for the
complete PCB.
Equations and Solutions
This section presents the derivation of the linear matrix equation that the voltages and currents in a given layer must satisfy. This equation is solved for each frequency of interest to obtain the
elements of the layer’s ABCD matrix as a function of frequency.
Variables and Functions
Variable Name Descritpion
n Via Cells Number of via cells in the layer
m Radial TEM Waves Number of radial TEM waves in the layer, m≥n
Voltages and
J Return currents n✕1 vector of via cell return currents
V[out] Outgoing voltage m✕1 vector of outgoing wave voltages, one for each wave, as measured at the antipads of the via cells and the via barrels of the GRVs
V[in] Incoming voltage m✕1 vector of incoming wave voltages, one for each wave, as measured at the antipads of the via cells and the via barrels of the GRVs
V[return] Total return voltage m✕1 vector containing the total return path voltages, as measured at the antipads of the via cells and the via barrels of the GRVs
B Generic amplitude Amplitude coefficient for a zero order outgoing radial TEM wave
P Propagation constants m✕m matrix of propagation constants between the different via cells and GRVs in the layer
Z Wave output impedances m✕n matrix of output source impedances for via cell outgoing waves
$\Gamma$ Reflection coefficients m✕m matrix of reflection coefficients. The reflection coefficients for via cells is zero and the reflection coefficients for GRVs is -1.
Z[return] Return path impedances m✕n matrix of impedances for the complete return path response, including reflections and crosstalk. The first rows of this matrix are used to populate the
ABCD matrix for the layer.
I Identity matrix The m✕m identity matrix
Speeds and
k Propagation constant $k=\omega \sqrt{\mu \epsilon }=\frac{2\pi }{\lambda }$ is the propagation constant at the frequency of interest.
$\eta$ Dielectric impedance $\eta =\sqrt{\frac{\mu }{\epsilon }}$ is the impedance of a dielectric
r[k] Source radius Radius at which the outgoing wave voltage is defined – antipad radius for via cells and barrel radius for GRVs
R[jk] Propagation distance Center to center distance between the via cells or GRVs
d Dielectric thickness The total distance between the top and bottom return planes
H[0]^(2) Zero order Hankel function The Hankel function is a combination of Bessel functions for describing the cylindrical geometry of radial TEM waves. A Hankel function of the second kind
of the second kind describes an outgoing wave. A zero order Hankel function describes voltage of a wave that does not vary with azimuth.
H[1]^(2) First order Hankel function The first order Hankel function happens to be the derivative of the zero order Hankel function, and is therefore useful for describing the radial current of
of the second kind a wave that does not vary with azimuth.
The voltage and current for a zero order radial TEM wave as a function of radius are [2]
$I\left(r\right)=2\pi r{H}_{\varphi }=2\pi r\frac{j}{\eta }B{H}_{1}^{\left(2\right)}\left(k{r}_{j}\right)$
The waves from each via cell and GRV propagate to produce incoming waves at the other via cells and GRVs.
In this equation, the propagation constants are approximately
$P\left(j,k\right)=\frac{{H}_{0}^{\left(2\right)}\left(k{R}_{j}k\right)}{{H}_{0}^{\left(2\right)}\left(k{r}_{k}\right)},je k,=0\text{\hspace{0.17em}}otherwise$
The via solver feature improves the accuracy of this equation by integrating the average around the destination antipad or GRV barrel.
Define the entries in V[out] either as being driven by via cell return current or as being the reflection of incoming waves. The general equation is a combination of the two sources.
${V}_{out}=ZJ+\Gamma {V}_{in}=ZJ+\Gamma P{V}_{out}$
The mode output impedance Z is the ratio of the TEM wave voltage to current at the edge of the antipad. Each GRV is a nearly perfect short circuit reflector at the edge of its via barrel. If the via
cell waves are in rows 1 through n of V[out], then the entries of Z and $\Gamma$ are
$Z\left(j,j\right)=\frac{jd\eta }{2\pi {r}_{j}}\frac{{H}_{0}^{\left(2\right)}\left(k{r}_{j}\right)}{{H}_{1}^{\left(2\right)}\left(k{r}_{j}\right)}\text{\hspace{0.17em}}\text{\hspace{0.17em}}j\le n,\
$\Gamma \left(j,k\right)=-1\text{\hspace{0.17em}}\text{\hspace{0.17em}}j>n,\text{\hspace{0.17em}}\text{\hspace{0.17em}}=0\text{\hspace{0.17em}}otherwise$
The solution for V[out] is
${V}_{out}={\left(I-\Gamma P\right)}^{\left(-1\right)}ZJ$
The solution for the complete return path response is
${V}_{return}={V}_{out}+{V}_{in}=\left(I+P\right){\left(I-\Gamma P\right)}^{\left(-1\right)}ZJ\equiv {Z}_{return}J$
[1] Steinberger, Telian, Tsuk, Iyer and Yanamadala, “Proper Ground Via Placement for 40+ Gbps Signaling”, DesignCon 2022, April 2022.
[2] Ramo, Whinnery and Van Duzer, Fields and Waves in Communications Electronics, third edition, section 9.3, John Wiley and Sons Inc., copyright 1994 | {"url":"https://de.mathworks.com/help/rfpcb/ug/eigenmode-based-solver-for-pcb-vias.html","timestamp":"2024-11-10T19:32:01Z","content_type":"text/html","content_length":"88271","record_id":"<urn:uuid:d8a51c6b-8978-4916-a55e-aac6d325fae4>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00354.warc.gz"} |
Quantum Field Theory
the path of q(t); S is called the "functional" of q(t). In other words, a single value of S is evaluated from a path of q(t). It is sometimes written as S[q(t)] to emphasize its functional
dependence of q(t). Figure 01a1 shows three different paths of q(t) (blue, green, red), and the corresponding Lagrangian L, which is also determined by the path of q(t). The action S[q(t)] would have
three different numbers equal to the areas under L indicated by the blue, green or the red curve.
Figure 01a1 [] Functional of Paths
Figure 01a2 S[q(t)] Functional of Paths []
Figure 01a2 shows a set of more specific paths represented by the formula q = bt - (a/2)t^2 with corresponding value of S[q(t)] (a ~ acceleration, b ~ velocity). | {"url":"https://universe-review.ca/R15-12-QFT.htm","timestamp":"2024-11-10T12:23:57Z","content_type":"text/html","content_length":"12432","record_id":"<urn:uuid:afcac488-4507-47e7-b3ba-a45bc7026ad2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00876.warc.gz"} |
Kadane’s Algorithm—(Dynamic Programming)—How and Why does it Work?Data Science News
Kadane’s Algorithm—(Dynamic Programming)—How and Why does it Work?
So the next time the same sub-problem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time.Those who cannot remember the
past are condemned to repeat it.—Dynamic ProgrammingHere’s a brilliant explanation on the concept of Dynamic Programming on Quora—Jonathan Paulson’s answer to How should I explain dynamic
programming to a 4-year-old?Though there’s more to dynamic programming, we would move forward to understand the Maximum Subarray Problem.Maximum Subarray ProblemThe maximum subarray problem is the
task of finding the largest possible sum of a contiguous subarray, within a given one-dimensional array A[1…n] of numbers.Maximum Sum Subarray (In Yellow)For example, for the array given above, the
contiguous subarray with the largest sum is [4, -1, 2, 1], with sum 6..We would use this array as our example for the rest of this article..Also, we would assume this array to be zero-indexed,
i.e..-2 would be called as the ‘0th’ element of the array and so on..Also, A[i] would represent the value at index i.Now, we would have a look at a very obvious solution to the given problem.Brute
Force ApproachOne very obvious but not so good solution is to calculate the sum of every possible subarray and the maximum of those would be the solution..We can start from index 0 and calculate the
sum of every possible subarray starting with the element A[0], as shown in the figure below..Then, we would calculate the sum of every possible subarray starting with A[1], A[2] and so on up to A
[n-1], where n denotes the size of the array (n = 9 in our case)..Note that every single element is a subarray itself.Brute Force Approach: Iteration 0 (left) and Iteration 1 (right)We will call the
maximum sum of subarrays starting with element A[i] the local_maximum at index i..Thus after going through all the indices, we would be left with local_maximum for all the indices..Finally, we can
find the maximum of these local_maximums and we would get the final solution, i.e..the maximum sum possible..We would call this the global_maximum.But you might notice that this is not a very good
method because as the size of array increases, the number of possible subarrays increases rapidly, thus increasing computational complexity.. More details
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3 Digit Times 3 Digit Multiplication Worksheets
Math, particularly multiplication, develops the keystone of countless scholastic disciplines and real-world applications. Yet, for many learners, understanding multiplication can pose an obstacle. To
resolve this difficulty, teachers and moms and dads have welcomed an effective tool: 3 Digit Times 3 Digit Multiplication Worksheets.
Intro to 3 Digit Times 3 Digit Multiplication Worksheets
3 Digit Times 3 Digit Multiplication Worksheets
3 Digit Times 3 Digit Multiplication Worksheets -
Use this variety worksheet to help students practice multiplying 3 digit numbers by 2 digit numbers using various styles of questions 4th through 6th Grades View PDF Math Riddle 12 Inch Nose Why can
t a nose be twelve inches long To find the answer to the riddle solve the three digit by two digit multiplication problems 4th through 6th Grades
What is K5 K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Multiplication
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20 2 Digit By 3 Digit Multiplication Worksheets Coo Worksheets
20 2 Digit By 3 Digit Multiplication Worksheets Coo Worksheets
Vertical Format This Multiplication worksheet may be configured for 2 3 or 4 digit multiplicands being multiplied by 1 2 or 3 digit multipliers You may vary the numbers of problems on each worksheet
from 12 to 25 This multiplication worksheet is appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade
These math worksheets should be practiced regularly and are free to download in PDF formats 3 Digit by 3 Digit Multiplication Worksheet 1 Download PDF 3 Digit by 3 Digit Multiplication Worksheet 2
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1 Digit Multiplication Worksheets
1 Digit Multiplication Worksheets
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your answers and use the answer key below to check if they are right
There are several variants of each class of worksheet to allow for plenty of practice These two digit and three digit multiplication worksheets gradually introduce long multiplication problems to
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11 Best Images Of Three Digit Multiplication Worksheets 2 Digit By 1 Digit Multiplication
Check more of 3 Digit Times 3 Digit Multiplication Worksheets below
Multiplying 3 Digit By 2 Digit Numbers X
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Three Digit Multiplication Worksheet Have Fun Teaching
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Multiply 3 x 3 digits worksheets K5 Learning
What is K5 K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Multiplication
practice with all factors under 1 000 column form Free Worksheets Math Drills Multiplication Printable
Worksheets Multiplication by 3 Digit Numbers Super Teacher Worksheets
With these multiplication worksheets student can practice multiplying by 3 digit numbers example 491 x 612 Multiplying by 3 Digit Numbers Multiplication 3 digit by 3 digit FREE Graph Paper Math
Drills 3 digits times 3 digits example 667 x 129 4th through 6th Grades View PDF Horizontal 3 Digit x 3 Digit
What is K5 K5 Learning offers free worksheets flashcards and inexpensive workbooks for kids in kindergarten to grade 5 Become a member to access additional content and skip ads Multiplication
practice with all factors under 1 000 column form Free Worksheets Math Drills Multiplication Printable
With these multiplication worksheets student can practice multiplying by 3 digit numbers example 491 x 612 Multiplying by 3 Digit Numbers Multiplication 3 digit by 3 digit FREE Graph Paper Math
Drills 3 digits times 3 digits example 667 x 129 4th through 6th Grades View PDF Horizontal 3 Digit x 3 Digit
Three Digit Multiplication Worksheet Have Fun Teaching
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FAQs (Frequently Asked Questions).
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What Percent Is 35 Of 50? Here's The Perfect Answer! - Yearly Magazine
What Percent Is 35 Of 50: Math can be a tricky subject, and when it comes to percentages, it’s no different. Whether you’re trying to figure out what percent of a number is another number or what
percent one number is of another, the answers don’t always come easily.
So if you’re looking for an explanation on how to figure out the answer to ‘What percent is 35 of 50?’ then you’ve come to the right place!
In this blog post, we’ll look at the equation involved in finding this answer and provide some tips on how you can work out other percentage questions.
What Percent Is 35 Of 50?
What Percent Is 35 Of 50: 35% of 50 is 17.5.
To calculate 35% of 50, simply multiply 35% by 50. The answer, 17.5, is 35% of 50.
What Percent Is 35 Of 50: How to calculate percentages
To calculate a percent of a quantity, you need to know two things: the whole quantity and the portion that is being referred to as the percent.
For example, let’s say you have 50 candy bars and you want to figure out what percentage 25 candy bars is of the total. In other words, you want to know what percent 25 is of 50.
You could set up a proportion and cross multiply to solve this problem, but an easier method is to use a short cut formula for calculating percentages. To use the shortcut method, simply take the
quantity that is being referred to as the percent (25) and divide it by the whole (50). Then, multiply that answer by 100%. Doing so gives you the following:
25 ÷ 50 = 0.50
0.50 × 100% = 50%
This means that 25 candy bars constitutes 50% of the total number of candy bars.
Other examples of percentages
There are plenty of other examples of percentages in real life. Here are a few more common ones:
-A tip at a restaurant is usually around 15% of the bill
-In many countries, the value-added tax (VAT) is around 20%
-The average interest rate for a credit card is about 18%
-The average interest rate for a mortgage is around 4%
-If you’re caught speeding, you may have to pay a fine that’s a certain percentage of your annual income
As we can see, 35% of 50 is 17.5. Knowing this percentage calculation was incredibly useful for figuring out our answer! We hope you found the explanation helpful and now understand exactly how to
calculate a percent from a given number. With these new skills, you’re one step closer to becoming an expert mathematician! | {"url":"https://yearlymagazine.com/what-percent-is-35-of-50/","timestamp":"2024-11-10T02:14:56Z","content_type":"text/html","content_length":"183791","record_id":"<urn:uuid:ca8cd0dc-a8ce-45ef-beae-e7b47f373e10>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00705.warc.gz"} |
Top 55+ Data Science Interview Questions and Answers in 2023 | Data Science Interview Questions and Answers for Beginners- Devduniya - Dev Duniya
Top 55+ Data Science Interview Questions and Answers in 2023 | Data Science Interview Questions and Answers for Beginners- Devduniya
Sure, I’d be happy to help you understand some advanced data science interview questions and provide some example answers. Here are a few questions that you might encounter in a data science
Q1. How do you handle missing or corrupted data in your analyses?
Answer: One approach to handling missing data is to use imputation techniques to estimate the missing values. Another approach is to exclude samples with missing values from the analysis. Handling
corrupted data can involve identifying the source of the corruption and fixing it, or excluding the corrupted data from the analysis.
Q2. What is overfitting and how can it be avoided?
Answer: Overfitting occurs when a model is too complex and captures the noise in the training data, leading to a poor generalization of new data. One way to avoid overfitting is to use simpler
models, such as linear models or decision trees with limited depth. Another way is to use regularization techniques, such as L1 or L2 regularization, which add constraints to the model to prevent
overfitting. Cross-validation can also be used to evaluate the model’s performance on unseen data and ensure that it generalizes well.
Q3. How do you evaluate the performance of a classification model?
Answer: There are many metrics that can be used to evaluate the performance of a classification model, including accuracy, precision, recall, and F1 score. It’s important to consider the context of
the problem and select the appropriate metric. For example, in a medical diagnosis task, it may be more important to prioritize recall (the ability to identify all positive cases) over precision (the
proportion of predicted positive cases that are actually positive).
Q4. What is a confusion matrix and how is it used?
Answer: A confusion matrix is a table that is used to evaluate the performance of a classification model. It displays the number of correct and incorrect predictions for each class. For example, in a
binary classification problem, the confusion matrix will have four cells: true negatives, false negatives, false positives, and true positives. The diagonal cells represent the number of correct
predictions, while the off-diagonal cells represent the number of incorrect predictions. The confusion matrix can be used to compute various evaluation metrics, such as precision, recall, and F1
Q5. What is the bias-variance tradeoff?
Answer: The bias-variance tradeoff is the balance between the bias and variance of a model. A model with high bias will make consistent but inaccurate predictions, while a model with high variance
will make accurate but inconsistent predictions. In general, increasing the complexity of a model will reduce the bias but increase the variance. Finding the right balance between bias and variance
is important for producing good models.
Q6. What is regularization and how does it work?
Answer: Regularization is a technique used to prevent overfitting in complex models, such as neural networks and polynomial regression. It works by adding a penalty term to the objective function
that the model is trying to optimize. The penalty term discourages the model from fitting the noise in the training data and encourages it to find a more generalizable solution. There are two main
types of regularization: L1 regularization, which adds a penalty proportional to the absolute value of the model weights, and L2 regularization, which adds a penalty proportional to the square of the
model weights.
Q7. What is cross-validation and how is it used?
Answer: Cross-validation is a technique used to evaluate the performance of a machine-learning model. It works by dividing the training set into a number of folds, training the model on some of the
folds, and evaluating it on the remaining folds. The performance measure is then averaged across all the folds. Cross-validation is useful for selecting hyperparameters, comparing different models,
and assessing the generalization performance of the model.
Q8. What is the difference between a generative and discriminative model?
A generative model is a model that learns the joint distribution of the input data and the target variables. Given a set of input data, the model can generate samples of the target variables.
Examples of generative models include hidden Markov models and mixture models.
A discriminative model, on the other hand, is a model that learns the conditional distribution of the target variables given the input data. Given a set of input data, the model directly predicts the
corresponding target variables. Examples of discriminative models include logistic regression and support vector machines.
Q9. What is feature selection and why is it important?
Answer: Feature selection is the process of selecting a subset of the most relevant features for building a machine learning model. It is important because it can reduce the complexity of the model,
improve the interpretability of the model, and improve the model’s performance by reducing overfitting and the curse of dimensionality. There are various techniques for feature selection, such as
backward elimination, forward selection, and Lasso regression.
Q10. What is a decision tree and how does it work?
Answer: A decision tree is a type of supervised machine-learning model that can be used for classification or regression tasks. It works by dividing the feature space into regions, called nodes,
using decision rules based on the features. The model makes a prediction by traversing the tree from the root node to a leaf node, where the predicted class or value is stored. Decision trees are
easy to interpret and can handle categorical and numerical data.
Q11. What is a random forest and how does it work?
Answer: A random forest is an ensemble machine-learning model that consists of a collection of decision trees trained on different subsets of the training data. The prediction of the random forest is
the average or majority vote of the individual decision trees. Random forests are used for classification and regression tasks and are known for their good performance and ability to handle
high-dimensional data.
Q12. What is a support vector machine and how does it work?
Answer: A support vector machine (SVM) is a type of supervised machine learning model used for classification tasks. It works by finding the hyperplane in the feature space that maximally separates
the classes. The data points closest to the hyperplane, called support vectors, have the greatest influence on the position of the hyperplane. SVMs are effective for high-dimensional data and can be
used with kernels to handle nonlinear relationships.
Q13. What is K-means clustering and how does it work?
Answer: K-means clustering is an unsupervised machine learning algorithm used for partitioning a dataset into K clusters. It works by randomly initializing K centroids, then iteratively assigning
each data point to the nearest centroid and updating the centroids to the mean of the assigned points. The algorithm converges when the centroids no longer change. K-means clustering is sensitive to
the initial centroid assignments and can be computationally expensive for large datasets.
Q14. What is a Gaussian mixture model and how does it work?
Answer: A Gaussian mixture model (GMM) is a probabilistic model that assumes that the underlying data is generated from a mixture of K Gaussian distributions. It is used for clustering tasks and
works by estimating the parameters of the Gaussian distributions and the mixing weights that indicate the probability of a data point belonging to each cluster. GMMs are more flexible than K-means
clustering because they can handle non-spherical clusters and allow for overlapping clusters.
Q15. What is a neural network and how does it work?
Answer: A neural network is a machine learning model inspired by the structure and function of the brain. It consists of layers of interconnected nodes, called neurons, that process and transmits
information. Neural networks are used for a variety of tasks, such as classification, regression, and generation. They work by adjusting the weights of the connections between the neurons based on
the input data and the desired output, using an optimization algorithm such as stochastic gradient descent.
Q16. What is deep learning and how does it differ from traditional machine learning?
Answer: Deep learning is a subfield of machine learning that uses neural networks with many layers (deep networks) to learn features and make decisions. It differs from traditional machine learning
in that it can learn features automatically from raw data, rather than requiring manual feature engineering. Deep learning has been successful in a wide range of applications, such as image and
speech recognition, natural language processing, and machine translation.
Q17. What is a convolutional neural network and how does it work?
Answer: A convolutional neural network (CNN) is a type of neural network used for processing data that has a grid-like topology, such as an image. It works by applying a series of filters to the
input data to extract features, which are then passed through a series of fully connected layers for classification or regression. CNN’s are particularly effective for image recognition tasks because
they can learn translation-invariant features and handle variations in the appearance of the input data.
Q18. What is a recurrent neural network and how does it work?
Answer: A recurrent neural network (RNN) is a type of neural network used for processing sequential data, such as time series or natural language. It works by using hidden states that are passed
through a series of time steps and are updated based on the current input and the previous hidden state. This allows the RNN to capture dependencies between the elements in the sequence and make
predictions based on the entire sequence.
Q19. What is a recommendation system and how does it work?
Answer: A recommendation system is a system that suggests items to users based on their past interactions or preferences.
There are two main types of recommendation systems:
content-based recommendation systems and collaborative filtering recommendation systems.
Content-based recommendation systems recommend items based on the characteristics of the items and the user’s past preferences. For example, if a user has previously rated action movies highly, a
content-based recommendation system might recommend other action movies with similar characteristics.
Collaborative filtering recommendation systems recommend items based on the past preferences of users with similar tastes. For example, if user A and user B have rated similar movies highly, a
collaborative filtering recommendation system might recommend movies that user B has rated highly to user A.
Q20. What is an autoencoder and how does it work?
Answer: An autoencoder is a type of neural network used for unsupervised learning. It consists of two parts: an encoder that maps the input data to a latent space and a decoder that maps the latent
representation back to the original space. The goal of the autoencoder is to learn a compact and informative representation of the input data. Autoencoders can be used for dimensionality reduction,
feature learning, and anomaly detection.
Q21. What is a gradient descent algorithm and how does it work?
Answer: A gradient descent algorithm is an optimization algorithm used to minimize a loss function. It works by iteratively taking steps in the opposite direction of the gradient of the loss function
with respect to the model parameters. The size of the steps is determined by the learning rate. The algorithm converges when the loss function reaches a minimum.
Q22. What is an ensemble method and how does it work?
Answer: An ensemble method is a machine learning technique that combines the predictions of multiple models to improve the performance of the final model. Ensemble methods can be used for both
classification and regression tasks. There are two main types of ensemble methods: boosting and bagging.
Boosting algorithms, such as AdaBoost, work by training a series of weak models and weighting them based on their performance. The final prediction is made by summing the weighted predictions of the
individual models.
Bagging algorithms, such as random forests, work by training a number of models independently on different subsets of the training data and averaging or voting their predictions. The final prediction
is made by aggregating the predictions of the individual models.
Q23. Can you explain how a Random Forest model works?
Answer: A Random Forest is an ensemble learning method for classification and regression that uses multiple decision trees and combines their predictions to make a final decision. Each tree in the
forest is trained on a different random sample of the data, and the final prediction is made by averaging (for regression) or majority voting (for classification) the predictions of the individual
trees. This approach helps to reduce overfitting and improve the generalization of the model.
Q24. How do you handle missing values in a dataset?
Answer: There are several strategies for handling missing values in a dataset, including:
• Removing rows or columns with missing values
• Imputing missing values using statistical measures such as the mean, median, or mode
• Using algorithms that are capable of handling missing values, such as decision trees or k-nearest neighbors
• The appropriate approach depends on the specific context and the amount of missing data.
Q25. Can you explain how a neural network works?
Answer: A neural network is a type of machine-learning model inspired by the structure and function of the human brain. It consists of layers of interconnected “neurons,” which process and transmit
information. Each neuron receives input from other neurons, combines these inputs using weights that represent the strength of the connections between neurons, and then applies an activation function
to produce an output. The output of one layer serves as the input to the next layer, and the process continues until the final output is produced. Neural networks can be trained to perform a variety
of tasks by adjusting the weights of the connections between neurons based on the input data and the corresponding desired output.
Q26. How do you choose the appropriate evaluation metric for a machine-learning model?
Answer: The appropriate evaluation metric depends on the specific goals of the project and the characteristics of the data. Some common evaluation metrics for classification tasks include accuracy,
precision, recall, and F1 score. For regression tasks, common evaluation metrics include mean absolute error, mean squared error, and root means squared error. It is important to consider the
trade-offs between different metrics and choose the one that is most relevant to the problem at hand.
Q27. Can you explain how gradient descent works?
Answer: Gradient descent is an optimization algorithm used to find the values of parameters (coefficients) of a function (model) that minimizes a cost function. The algorithm starts with initial
estimates of the parameters and iteratively improves them by computing the gradient of the cost function with respect to the parameters and moving in the direction that reduces the cost. The size of
the step taken in each iteration is determined by the learning rate, which controls the speed at which the algorithm converges to the optimal solution.
Q28. How would you handle a large dataset that doesn’t fit in memory?
Answer: One possible solution to this problem would be to use a database or data storage solution that is designed to handle large datasets, such as a distributed database or a data lake.
Alternatively, you could try using a tool like Apache Spark, which is designed to process large datasets in a distributed manner. Another option might be to sample the data and work with a smaller
subset of the data, or to use techniques like feature selection to reduce the amount of data you need to work with.
Q29. How would you approach building a recommendation system?
Answer: To build a recommendation system, you would first need to determine the goal of the system and the type of recommendations you want to provide. For example, you might want to recommend
products to customers based on their purchase history or recommend content to users based on their interests. You would then need to collect data about the items you want to recommend and the users
you want to recommend them to and use this data to train a machine learning model that can predict which items a user is likely to be interested in. You could then use this model to generate
recommendations for each user.
Q30. How would you handle missing or corrupted data in a dataset?
Answer: There are several strategies you could use to handle missing or corrupted data in a dataset. One approach might be to simply remove any rows or columns that contain missing or corrupted data.
Another option might be to impute the missing values using techniques like mean imputation or linear interpolation. You could also try using machine learning models that are robust to missing data,
such as decision trees or random forests. Finally, you could try to identify the root cause of the missing or corrupted data and take steps to fix the problem at the source.
Q31. How do you handle missing values in a dataset?
Answer: One way to handle missing values is to simply remove any rows or columns that contain missing values. This is not always possible or desirable, however. Another option is to impute the
missing values, either using a simple approach like replacing the missing value with the mean or median of the other values in that column or using more advanced techniques such as linear regression
or matrix completion.
Q32. How do you handle categorical variables in a dataset?
Answer: There are several ways to encode categorical variables for use in a machine learning model. One common approach is to use one-hot encoding, where a new column is created for each category and
a binary value (0 or 1) is entered into the column to indicate the presence or absence of that category. Another option is to use integer encoding, where each category is assigned a unique integer
Q33. What is overfitting in the context of machine learning?
Answer: Overfitting occurs when a machine learning model is trained too well on the training data, and as a result, it does not generalize well to new, unseen data. This can happen if the model is
too complex or if there is too little training data. Overfitting can be mitigated by using techniques such as regularization or by increasing the amount of training data.
Q34. How do you evaluate the performance of a machine-learning model?
Answer: There are several ways to evaluate the performance of a machine learning model. One common approach is to split the available data into a training set and a test set, and use the training set
to train the model and the test set to evaluate its performance. Other evaluation metrics include accuracy, precision, recall, and F1 score. It is important to use the appropriate metric for the
specific task and to also consider the business objectives of the model.
Q35. What is regularization and why is it important?
Answer: Regularization is a technique used to prevent overfitting in machine learning models. It does this by adding a penalty term to the objective function that the model is trying to optimize.
This penalty term increases as the model become more complex, which encourages the model to find simpler solutions that generalize better to new data.
Q36. What is the bias-variance tradeoff in the context of machine learning?
Answer: The bias-variance tradeoff is the balance between underfitting (high bias) and overfitting (high variance) in a machine learning model. A model with high bias will make consistent but
potentially incorrect predictions, while a model with high variance will make widely varying predictions, but potentially capture more of the underlying pattern in the data. Finding the right balance
between bias and variance is important for building a good model.
Q37. What is the difference between supervised and unsupervised learning?
Answer: In supervised learning, the model is trained on labeled data, where the correct output is provided for each example in the training set. Common applications of supervised learning include
regression and classification tasks. In unsupervised learning, the model is not provided with labeled training examples and must discover the structure of the data through techniques such as
Q38. What is a neural network and how does it work?
Answer: A neural network is a type of machine-learning model inspired by the structure and function of the human brain. It consists of layers of interconnected “neurons,” which process and transmit
information. Each neuron receives input from other neurons, processes it using an activation function, and passes the output on to other neurons in the next layer. Neural networks are particularly
good at learning complex, nonlinear relationships in data.
Q39. What is cross-validation and why is it important?
Answer: Cross-validation is a resampling procedure used to evaluate the performance of machine learning models. It works by dividing a dataset into a number of “folds,” and then training the model on
a different subset of the data each time while evaluating the model on the remaining folds. This allows you to use the entire dataset for training and testing, which can be especially useful when you
have a limited amount of data.
Cross-validation is important because it helps you to get an estimate of the performance of your model that is more reliable than using a single train/test split. This is because it gives you a
better idea of how well your model will generalize to new data. When you train and test your model on the same data, it can give you an overly optimistic estimate of its performance, because the
model has “seen” the data it is being tested on. This can lead to overfitting, where the model performs well on the training data but poorly on new, unseen data.
By using cross-validation, you can get a more accurate estimate of the performance of your model and avoid overfitting. It is a key part of the model selection process and is widely used in machine
Q40. What is the difference between a supervised and unsupervised learning algorithm?
Answer: Supervised learning algorithms require labeled training data. The algorithm learns from this data to make predictions about unseen data. Examples include linear regression and k-nearest
Unsupervised learning algorithms do not require labeled training data. The algorithm learns by discovering patterns in the data. Examples include k-means clustering and principal component analysis.
Q41. What is a confusion matrix?
Answer: A confusion matrix is a table that is used to evaluate the performance of a classification algorithm. It compares the predicted class labels with the true class labels and summarizes the
results in a table. The rows represent the predicted class labels and the columns represent the true class labels. The diagonal elements represent the number of correct predictions, while the
off-diagonal elements represent the number of incorrect predictions.
Q42. What is cross-validation?
Answer: Cross-validation is a technique used to evaluate the performance of a machine learning algorithm. It involves dividing the data into a training set and a testing set, training the model on
the training set, and evaluating the model on the testing set. This process is repeated a number of times with different splits of the data to get an estimate of the model’s performance.
Cross-validation is useful because it helps to ensure that the model generalizes well to unseen data.
Q43. What are a false positive and a false negative?
Answer: A false positive is a prediction made by a classification algorithm that an instance belongs to a certain class, when in fact it does not. For example, a false positive in a medical test
might be a test result that indicates a person has a disease when they are actually healthy.
A false negative is a prediction made by a classification algorithm that an instance does not belong to a certain class, when in fact it does. For example, a false negative in a medical test might be
a test result that indicates a person is healthy when they actually have the disease.
Q44. Can you explain the bias-variance tradeoff?
Answer: The bias-variance tradeoff is a fundamental concept in machine learning. It refers to the balance between two sources of error in a model: bias and variance.
Bias is the error that is introduced by approximating a real-life problem with a simplified model. A model with high bias tends to be oversimplified and may not capture the complexity of the data.
This leads to underfitting, where the model is not able to accurately capture the trends in the data.
Variance is the error that is introduced by sensitivity to small fluctuations in the training data. A model with high variance tends to be very complex and may be sensitive to small changes in the
training data. This leads to overfitting, where the model performs well on the training data but poorly on unseen data.
The bias-variance tradeoff refers to the fact that it is often not possible to simultaneously minimize both bias and variance. In practice, this means that it is important to find a balance between
the two sources of error in order to build a model that generalizes well to unseen data.
Q45. How have you improved the accuracy of a model in your previous work?
Answer: In my previous role, I worked on a classification model that was predicting whether or not a customer would churn. Initially, the model had an accuracy of around 75%. I tried a few different
approaches to improve the accuracy, including:
Tuning the hyperparameters of the model using cross-validation
Adding additional features to the training data, such as customer demographics and account history
Ensemble learning, where I trained several models and combined their predictions using techniques like boosting or voting
Through these efforts, I was able to improve the accuracy of the model to around 85%.
Q46. Can you describe a time when you had to deal with an imbalanced dataset, and how you addressed it?
Answer: I recently worked on a project where we were trying to predict whether or not a patient had a certain disease, but the dataset was highly imbalanced – there were far more patients without the
disease than with it. This can cause problems with model performance, because the classifier may become biased toward predicting the majority class.
To address this issue, I tried a few different techniques:
• Undersampling the majority class to match the size of the minority class
• Oversampling the minority class to match the size of the majority class
• Using class weights to penalize the model for misclassifying the minority class more heavily
• Using a different evaluation metric, such as precision or AUC, which are less sensitive to imbalanced class distributions
• Ultimately, I found that using class weights in combination with undersampling gave the best performance.
Q47. Can you discuss a recent project you worked on that required feature engineering?
Answer: In my previous role, I worked on a project to predict housing prices in a particular region. One of the challenges we faced was that the raw data contained a lot of missing values and
categorical variables that needed to be encoded.
To address these issues, I did the following:
• For missing values, I used techniques like imputation to fill in the missing values with reasonable estimates
• For categorical variables, I used one-hot encoding to convert them into numerical form
• I also created additional features by combining or transforming existing features, such as taking the log of continuous variables or multiplying two categorical variables together
• Through these efforts, I was able to improve the performance of the model significantly.
Q48. What is the difference between supervised and unsupervised learning?
Answer: Supervised learning involves training a model on a labeled dataset, where the correct output is provided for each example in the training set. The model makes predictions based on this
input-output mapping. Unsupervised learning involves training a model on an unlabeled dataset, allowing the model to discover patterns or relationships on its own.
Q49. What is a decision tree?
Answer: A decision tree is a flowchart-like tree structure used for predicting the outcome of a decision based on certain conditions. It breaks down a dataset into smaller and smaller subsets while
at the same time, an associated decision tree is incrementally developed. The final result is a tree with decision nodes and leaf nodes.
Q50. What is linear regression?
Answer: Linear regression is a statistical method used to model the linear relationship between a dependent variable and one or more independent variables. It estimates the mean value of the
dependent variable for the given values of the independent variables.
Q51. What is regularization?
Answer: Regularization is a technique used to prevent overfitting in models. Overfitting occurs when a model is excessively complex, such as having too many parameters relative to the number of
training examples. This can result in poor generalization of new data. Regularization introduces a penalty term in the optimization objective that encourages the model to be simpler and reduces the
risk of overfitting.
Q52. What is cross-validation?
Answer: Cross-validation is a resampling procedure used to evaluate the performance of machine learning models. It involves dividing the dataset into equal portions, using one portion as the test set
and the other portions as the training set, and evaluating the model on the test set. This process is repeated multiple times, with a different portion of the dataset used as the test set each time.
The average performance across all iterations is used as an estimate of the model’s generalization performance.
Q53. What is overfitting?
Answer: Overfitting occurs when a model is overly complex and is able to fit the training data very well, but generalizes poorly to new data. This can happen when there are a large number of
parameters relative to the number of training examples, or when the model is too flexible. Overfitting is a problem because it means the model is not generalizing well to new examples and is unlikely
to perform well on unseen data.
Q54. What is deep learning?
Answer: Deep learning is a subfield of machine learning that is inspired by the structure and function of the brain, specifically the neural networks that make up the brain. It involves training
artificial neural networks on a large dataset, allowing the network to learn and extract features from the data automatically. Deep learning has been successful in a number of areas, including image
and speech recognition, natural language processing, and playing games.
Q55. What is the difference between a generative and discriminative model?
A generative model learns to model the joint distribution of input and output variables, while a discriminative model learns to model the conditional distribution of the output given the input. In
other words, a generative model learns to generate new examples that are similar to the training data, while a discriminative model makes predictions about the label of a given input example.
Q56. What is a support vector machine?
A support vector machine (SVM) is a type of supervised learning algorithm that can be used for classification or regression tasks. The idea behind SVMs is to find the hyperplane in a high-dimensional
space that maximally separates the different classes.
In the case of a linear SVM, the hyperplane is a linear decision boundary that separates the classes. Nonlinear SVMs can also be used by using the so-called kernel trick, which maps the input data
into a higher-dimensional space in which a linear decision boundary can be found.
SVMs are known for their good generalization performance, meaning that they can often achieve good accuracy on unseen data. They are also relatively robust to overfitting, especially when using the
kernel trick.
I hope these examples give you a better idea of the types of questions you might encounter in a data science interview, and how you might approach answering them.
If you have any queries related to this article, then you can ask in the comment section, we will contact you soon, and Thank you for reading this article.
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Expanded Method Multiplication Worksheet
Math, particularly multiplication, creates the cornerstone of various scholastic disciplines and real-world applications. Yet, for many students, understanding multiplication can pose an obstacle. To
address this obstacle, educators and parents have actually welcomed an effective tool: Expanded Method Multiplication Worksheet.
Introduction to Expanded Method Multiplication Worksheet
Expanded Method Multiplication Worksheet
Expanded Method Multiplication Worksheet -
These comprehensive differentiated activity sheets allow children to practise multiplying 4 digit numbers by 2 digit numbers using the expanded method for long multiplication You can find a teacher
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Motivating consistent practice, supplying support, and creating a positive knowing atmosphere are beneficial actions. | {"url":"https://crown-darts.com/en/expanded-method-multiplication-worksheet.html","timestamp":"2024-11-12T05:16:36Z","content_type":"text/html","content_length":"28611","record_id":"<urn:uuid:9f9faddf-d715-45d9-905f-3ec3537b0f16>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00710.warc.gz"} |
Bra-ket notation
Bra-ket or bracket (or even bra-c-ket) notation was formulated by Dirac^[1] to provide a concise method for performing and describing the linear algebra used throughout the matrix mechanics
formulation of quantum mechanics. The notation is in wide use in the field today, and although developed with quantum mechanics in mind it can be employed more generally when working with any vector
space. In this notation vectors are represented by kets, such as ${\displaystyle |\psi \rangle }$, while their corresponding dual vectors are given by bras, ${\displaystyle \langle \psi |}$. In the
context of quantum mechanics the state of a system corresponds to a vector in a Hilbert space, so the state ${\displaystyle |\psi \rangle }$ is analogous to the wave function ${\displaystyle \psi
Mathematical description
Let ${\displaystyle {\mathcal {H}}}$ be a Hilbert space and ${\displaystyle {\mathcal {H}}^{*}}$ its dual space (which is isomorphic to ${\displaystyle {\mathcal {H}}}$ if the space is
finite-dimensional). Elements of ${\displaystyle {\mathcal {H}}}$ are then labelled by kets and elements of ${\displaystyle {\mathcal {H}}^{*}}$ are labelled by bras. Together a bra and a ket can
form a Dirac bracket, ${\displaystyle \langle \cdot |\cdot \rangle }$, which is equal to the inner product between them. The bracket then is a map from ${\displaystyle {\mathcal {H}}^{*}\times {\
mathcal {H}}}$ to a field ${\displaystyle F}$ (in quantum mechanics the field is the complex numbers, ${\displaystyle \mathbb {C} }$).
When the order of the bra and ket is reversed the resulting object is an operator, sometimes called a ket-bra, ${\displaystyle |\cdot \rangle \langle \cdot |}$. This operator is given by the outer
product of the ket with the bra, and is a map from ${\displaystyle {\mathcal {H}}}$ onto itself since ${\displaystyle \left(|\cdot \rangle \langle \cdot |\right)|\cdot \rangle =|\cdot \rangle \langle
\cdot |\cdot \rangle =\alpha |\cdot \rangle \in {\mathcal {H}}}$, where ${\displaystyle \alpha =\langle \cdot |\cdot \rangle \in F}$ is a scalar. By convention, duplicated vertical bars in an
expression are dropped as we have done here (i.e. writing ${\displaystyle \langle \cdot |\cdot \rangle }$ instead of ${\displaystyle \langle \cdot ||\cdot \rangle }$).
Uses in quantum mechanics
Suppose that ${\displaystyle {\mathcal {H}}}$ corresponds to the state space for a quantum system. For example, if the system was a particle in a box then ${\displaystyle {\mathcal {H}}}$ would
contain every possible state that the particle could occupy. Now let the state of the system be ${\displaystyle |\psi \rangle \in {\mathcal {H}}}$, with ${\displaystyle |\psi \rangle }$ normalized
(that is, ${\displaystyle \langle \psi |\psi \rangle =1}$) and let ${\displaystyle {\hat {A}}}$ be an operator corresponding to the observable ${\displaystyle A}$.
Expectation value
The expected result of a measurement of ${\displaystyle A}$ is given by ${\displaystyle \langle {\hat {A}}\rangle =\langle \psi |{\hat {A}}|\psi \rangle }$.
Overlap and probability
The overlap between the state of the system and another state ${\displaystyle |\phi \rangle \in {\mathcal {H}}}$ is ${\displaystyle \langle \phi |\psi \rangle }$, which means that the probability of
finding the system in state ${\displaystyle |\phi \rangle }$ is given by ${\displaystyle |\langle \phi |\psi \rangle |^{2}}$. This can also be seen as the expectation value of the projection operator
${\displaystyle {\hat {P}}_{\phi }=|\phi \rangle \langle \phi |}$, since this yields ${\displaystyle \langle {\hat {P}}_{\phi }\rangle =\langle \psi |\phi \rangle \langle \phi |\psi \rangle =\langle
\psi |\phi \rangle \langle \psi |\phi \rangle ^{*}=|\langle \phi |\psi \rangle |^{2}}$
Resolution of the identity
If the states ${\displaystyle \{|\varphi _{i}\rangle \}}$ are the (normalized) eigenstates of ${\displaystyle {\hat {A}}}$ then the identity operator can be expressed as
${\displaystyle I=\sum _{i}|\varphi _{i}\rangle \!\langle \varphi _{i}|}$.
This result holds if the ${\displaystyle |\varphi _{i}\rangle }$ are any complete set of orthonormal vectors, which is guaranteed to be the case for the eigenvectors of a Hermitean matrix.
1. ↑ P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press (1930). Fourth edition 1958. Paperback 1981. | {"url":"https://en.citizendium.org/wiki/Bra-ket_notation","timestamp":"2024-11-04T02:55:46Z","content_type":"text/html","content_length":"78823","record_id":"<urn:uuid:a56f5c54-66da-41e1-8c1a-2ef05edb5890>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00809.warc.gz"} |
Quilter’s Match- Part 3A Charting
Quilter’s Match- Part 3A Charting
Now that you understand the concept of breaking down blocks to their grid, you can more easily understand what it would be to break the block down to the patches of fabric that make it up.
We will break down a traditional block to the patches. As you see the list of patches there are large and small of the same patch. That is because there are different sizes of the same shape. These
all must be accounted for. Luckily, the large is twice the size of the small, so it will be pretty straight forward when we chart it all out.
Now that we have the parts that make up the whole, we can start to fill in the chart.
As we move into the Charting section, please do not be scared off by the numbers.
It is all very simple arithmetic, but a calculator will take away any math issues.
If you follow the instructions, step by step, you’ll be able to do this with no problems.
Please, give it a shot and have faith. It is SOOOO much easier than it looks.
As you break down a block, you will add information to a worksheet that I have created so everything you need is right there in front of you.
The work sheet you are looking at right now, has already been completely filled out with the information for this particular block. As we move through each diagram the highlighted area is what I
would like you to focus on. the shaded area will not be relevant at that time. For the purpose of this exercise, I will concentrate on the yellow fabric, and the yellow patches in this block. The
chart would be completed and used for all the fabrics and patches in the same way I am about to demonstrate.
In the first column, you see I have taken note of each patch of fabric in the block and noted its shape. I have used the abbreviation of HST for half square triangle and QST for quarter square
triangle. Note that I have not given this column an identifying letter at the top.
Working one patch at a time:
In the second column, Column A, I have noted how many of each patch shape is used in 1 block. In the third column, Column B, I have noted how many blocks I plan to make.
In the fourth column, Column C, I have multiplied Column A and Column B to determine the total number of patches of that particular shape that will be necessary for all the blocks in the quilt top.
We will write in the desired FINISHED size of the patch in Column D.
In Column E we will add the necessary seam allowances to the finished measurements in column D. ½ for squares or rectangles, 7/8” for half square triangles, and 1 ¼” for quarter square triangles.
In Column F is where we will note how many of the desired shape we will get from the cut square. A square would be 1: for half square triangles, you will get 2 from a single square: and you will get
4 quarter square triangles from a single square.
Column G will tell us the number of squares needed to be cut, in order to yield the number patches of a specific shape for all the blocks in the quilt top. We get this number by dividing the number
of patches needed, by the number of patches we would get from a single square. Column C divided by Column F.
So the first patch is a square, we would get only 1 from each square cut, we will need a total of 36 squares, so we will need a total of 36 cut squares.
The second patch is a half square triangle. We would get 2 from each square cut, and we need a total of 72 triangles. Column C : 72 divided by column F: 2 = 36 cut squares.
The third patch is a quarter square triangle. We would get 4 from each square cut, and we need a total of 36 triangles. Column C : 36 divided by Column F: 4 = 9 cut squares.
Now we know what we need to cut, but we haven’t figured out the fabric amounts yet or how we will cut all these squares.
We’ll continue with the Chart in the next segment of Quilter’s Math Part 3B. | {"url":"https://smartplatequilting.com/quilters-match-part-3a-charting/","timestamp":"2024-11-14T13:15:53Z","content_type":"text/html","content_length":"42359","record_id":"<urn:uuid:a0165dea-4740-4eec-900e-8a6b931ab8f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00328.warc.gz"} |
Multicore Sorter
Continuing on from my last post, I'll start with a review of the simpler of the three multicore abstractions: the Sorter. Sorting is a great candidate for parallel processing as many sorting
algorithms already divide and conquer their work. My approach was to borrow from two of these algorithms, QuickSort and MergeSort, to do just enough sorting just in time. Here's how:
1. partition the data to be sorted into evenly sized shards, one for each cpu core
2. launch a worker thread to quicksort each shard
3. when all the workers are complete, merge the sorted shards back together
This still leaves a large merge in a single thread as the last step. Luckily, the WideFinder 2 Benchmark doesn't require us to completely sort all the data. We just need the top ten items in each
category. So given a set of sorted shards (say that fast a few times!), it's easy to iterate over them to produce a list of the overall top ten items. The implementation below does the parallel
quicksort, and then, acting as an Iterator over the entire sorted collection, amortizes the work of merging the shards back together over the sequential access of the entire collection. Given that we
only need 10 items, most of this work goes undone in our use of Sorter. Here's the code:
package net.waldin.mc
import scala.actors.Actor._
import Config.numSorters
class Sorter[T](iter: Iterator[T], size: Int, comp: (T,T)=>Int) extends Iterator[T] {
val shardSize = Math.ceil(size.toDouble / numSorters).toInt
val shardIdx = List.range(0, size, shardSize).toArray
val array = new Array[T](size)
var sorted = false
final def shardOffset(shard:Int) = shard * shardSize
object Comparator extends java.util.Comparator[T] {
override def compare(a: T, b: T) = comp(a,b)
def sort {
//extract array
// create actors with array shards for quicksort
val sorters = for(shard <- (0 until numSorters).toList) yield {
scala.actors.Futures.future {
val start = shardOffset(shard)
val end = size min (start + shardSize)
java.util.Arrays.sort(array, start, end, Comparator)
// await completion of futures
for(s <- sorters) s()
sorted = true
override def hasNext: Boolean = {
if(!sorted) sort
private def firstShardWithData = {
var shard = 0
while(shard < numSorters &&
shardIdx(shard) == (size min shardOffset(shard + 1))) {
shard += 1
if(shard == numSorters) None else Some(shard)
override def next(): T = {
if(!sorted) sort
firstShardWithData match {
case None => throw new NoSuchElementException
case Some(n) =>
var shard = n
var tryShard = shard + 1
while(tryShard < numSorters) {
if(shardIdx(tryShard) < (size min shardOffset(tryShard + 1)) &&
comp(array(shardIdx(tryShard)), array(shardIdx(shard))) < 0){
shard = tryShard
tryShard += 1
shardIdx(shard) += 1
array(shardIdx(shard) - 1)
No comments: | {"url":"http://blog.waldin.net/2008/06/multicore-sorter.html","timestamp":"2024-11-14T07:37:24Z","content_type":"application/xhtml+xml","content_length":"34840","record_id":"<urn:uuid:e94f59be-151c-4ab1-a4b9-147094caf40d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00379.warc.gz"} |
Johannes Marti
On modal fixpoint logics:
On coalgebraic modal logic, partially based on my MSc thesis:
• Uniform Interpolation for Coalgebraic Fixpoint Logic (preprint, doi),
with F. Seifan and Y. Venema, at CALCO 2015.
• Lax Extensions of Coalgebra Functors and Their Logic (preprint, doi),
with Y. Venema, in the Journal of Computer and System Sciences, 2015.
This paper is an extended version of the earlier:
• Lax Extensions of Coalgebra Functors (preprint, doi),
with Y. Venema, at CMCS 2012.
On unification in modal logic:
On description logic:
On nonmonotonic and conditional logic:
On epistemic game theory:
• Choice Structures in Games (arXiv, doi),
with P. Galeazzi, in Games and Economic Behavior, 2023.
On representation theorems for belief and meaning:
• Interpreting Linguistic Behavior with Possible World Models (UvA),
PhD Dissertation at the University of Amsterdam, 2016.
02 Jul Modal unification step by step (slides), UNIF 2023 (International Workshop on Unification), Rome, with Sam van Gool.
04 Aug Size measures and alphabetic equivalence in the mu-calculus, LICS 2022, Haifa, Israel.
13 Apr A focus-style proof systems for the alternation-free mu-calculus, LLAMA seminar, ILLC, University of Amsterdam.
21 Dec Comparative similarity and natural properties, Research Forum, Department of Philosophy, University of Bayreuth, Germany.
07 Sep A focus system for the alternation-free mu-calculus, TABLEAUX 2021, online.
E-mail: johannes.marti@gmail.com | {"url":"http://johannesmarti.com/","timestamp":"2024-11-07T05:44:39Z","content_type":"application/xhtml+xml","content_length":"12846","record_id":"<urn:uuid:f4eb5109-3f49-41cc-8b16-56bf1ed73114>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00678.warc.gz"} |
Head Tracking with Kinect v2
This is yet another post in my series about the new Kinect using the November 2013 developer preview SDK. Today we’re going to have some fun by combining the color, depth and body data streams
(mentioned in my last few posts, here, here and here) and some interesting math to create an image that magically tracks the user’s head.
This is an early preview of the new Kinect for Windows, so the device, software and documentation are all preliminary and subject to change.
If you recall from my last post, I used the CoordinateMapper to translate the coordinates of the user’s joint information on top of the HD color image. The magic ingredient converts the Joint’s
Position to a ColorSpacePoint.
Joint headJoint = body.Joints[JointType.Head];
ColorSpacePoint colorSpacePoint =
If we take the X & Y coordinates from this
and the wonderful
extension methods of the WriteableBitmapEx project
, we can quickly create a cropped image of that joint.
int x = (int)Math.Floor(colorSpacePoint.X + 0.5);
int y = (int)Math.Floor(colorSpacePoint.Y + 0.5);
int size = 200;
WriteableBitmap faceImage = _bmp.Crop(new Rect(x,y,size,size));
Wow, that was easy! Although this produces an image that accurately tracks my head, the approach is somewhat flawed as it doesn’t scale based on the user’s position from the camera: if you stand too
close to the camera you’ll only see a portion of your face; Stand too far and you’ll see my face and torso. We can fix this by calculating the desired size of the image based on the depth of the
joint. To do this, we’ll need to obtain a
for the Joint and a simple trigonometric formula…
by itself doesn’t contain the depth data. Instead, it contains the X & Y coordinates from the depth image which we can use to calculate the index in the array of depth data. I’ve
outlined this in a previous post
, but for convenience sake here’s that formula again:
// get the depth image coordinates for the head
DepthSpacePoint depthPoint =
// use the x & y coordinates to locate the depth data
FrameDescription depthDesc = _sensor.DepthFrameSource.FrameDescription;
int depthX = (int)Math.Floor(depthPoint.X + 0.5);
int depthY = (int)Math.Floor(depthPoint.Y + 0.5);
int depthIndex = (depthY * depthDesc.Width) + depthX;
ushort depth = _depthData[depthIndex];
To calculate the desired size of the image, we need to determine the width of the joint's pixel in millimeters. We do this using a blast from the past, our best friend from high-school trigonometry,
Given that the Kinect’s Horizontal Field of View is 70.6°, we bisect this in half to form a right-angle triangle. We then take the depth value as the length of the adjacent side in millimeters. Our
goal is to calculate the
side in millimeters, which we can accomplish using the TOA portion of the mnemonic:
tan(0) = opposite / adjacent
opposite = tan(0) * adjacent
Once we have the length of the opposite, we divide it by the number of pixels in the frame which gives us the length in millimeters for each pixel. The algorithm for calculating pixel width is shown
private double CalculatePixelWidth(FrameDescription description, ushort depth)
// measure the size of the pixel
float hFov = description.HorizontalFieldOfView / 2;
float numPixels = description.Width / 2;
/* soh-cah-TOA
* TOA = tan(0) = O / A
* T = tan( (horizontal FOV / 2) in radians )
* O = (frame width / 2) in mm
* A = depth in mm
* O = A * T
double T = Math.Tan((Math.PI * 180) / hFov);
double pixelWidth = T * depth;
return pixelWidth / numPixels;
Now that we know the length of each pixel, we can adjust the size of our head-tracking image to be a consistent “length”. The dimensions of the image will change as I move but the amount of space
around my head remains consistent. The following calculates a 50 cm (~19”) image around the tracked position of my head:
double imageSize = 500 / CalculatePixelWidth(depthDesc, depth);
int x = (int)(Math.Floor(colorPoint.X + 0.5) - (imageSize / 2));
int y = (int)(Math.Floor(colorPoint.Y + 0.5) - (imageSize / 2));
WriteableBitmap faceImage = _bmp.Crop(new Rect(x,y, imageSize, imageSize));
Happy Coding.
9 comments:
Good ol' Soh-Cah-Toa
Interesting post! The youtube video really helped me understand the problem at hand. Your solution seems to work really well.
Looks like the 2.0 SDK has better/quicker skeleton tracking compared to 1.x. Does it track skeletons when a user walks by (facing perpendicularly to the sensor)?
Good ol' Soh-Cah-Toa
Interesting post! The youtube video helped me understand the problem at hand. The solution seems to work really well.
Looks like the 2.0 SDK has much better/quicker skeleton tracking compared to 1.x. Does it track skeletons when the user walks by (perpendicular to the sensor)?
Great work! Exactly what I was looking for. Do you think you can release the code for this? I'm working on a school project and this would really help me speed up my development. Cheers!
Hi bryan, Its a very good and simple tutorial to understand ,Kudos to that.
My query is.Can the colorstream be centered according to the user , like the head image which align always at the center. how can the whole body be aligned at the center.in the colorstream,.
Thanks in advance
Hi bryan, Its a very good and simple tutorial to understand ,Kudos to that.
My query is.Can the colorstream be centered according to the user , like the head image which align always at the center. how can the whole body be aligned at the center.in the colorstream,.
Thanks in advance
@kinectJockey I think you could easily adjust this sample to center on the user by changing the joint to MidSpine instead of the Head joint. The joints are listed here: https://msdn.microsoft.com
You would likely want to capture about 100cm on either side of the joint.
Hi bryan , Thanks for the solution it worked great. can you tell me how to place a image between the joints,(not on the joint) also how to rotate it according to joins movement
@kinectJockey -- you'd want to take a look at my post where I render the joints: http://www.bryancook.net/2014/03/drawing-kinect-v2-body-joints.html
To render images between joints, you need to know the relationship between the joints and traverse the skeleton to determine the distance and center point between them. To rotate the image, you'd
have to consider that the outside joint rotates around the inside joint -- eg hand rotates around the elbow -- you can then calculate a right angle triangle between the joints to determine the
angle of rotation.
Joints also provide an orientation https://pterneas.com/2017/05/28/kinect-joint-rotation/
If you're looking for source code, most of this is adapted from the SDK Samples | {"url":"https://www.bryancook.net/2014/03/head-tracking-with-kinect-v2.html?showComment=1423203550622","timestamp":"2024-11-13T01:38:22Z","content_type":"application/xhtml+xml","content_length":"89819","record_id":"<urn:uuid:99b0de08-0f90-4b40-9813-defa268193a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00230.warc.gz"} |
Entanglement by Identity, or Interaction Without Ever Touching
Identity of particles entails their entanglement, which can also be observed in pure form without interaction. (Source: Shutter2U/Vecteezy)
What is interaction and when does it occur? Intuition suggests that the necessary condition for the interaction of independently created particles is their direct touch or contact through physical
force carriers. In quantum mechanics, the result of the interaction is entanglement–the appearance of non-classical correlations in the system. It seems that quantum theory allows entanglement of
independent particles without any contact. The fundamental identity of particles of the same kind is responsible for this phenomenon.
“The whole is other than the sum of its parts.” – Aristotle (Metaphysics, Book VIII)
Quantum mechanics is currently the best and the most accurate and sophisticated theory used by physicists to describe the world around us. Its characteristic feature, however, is the abstract
mathematical language notoriously leading to serious interpretational problems. The view of reality proposed by this theory is still a subject of scientific dispute, which, instead of expiring over
time, is becoming hotter and more interesting. The new motivation and intriguing questions are brought forth by a fresh perspective resulting from the standpoint of quantum information and the
enormous progress of experimental techniques. This allows verification of the conclusions drawn from subtle thought experiments directly related to the problem of interpretation. Moreover, we are now
witnessing enormous progress in the field of quantum communication and quantum computer technology, which significantly draws on non-classical resources offered by quantum mechanics.
The work by Pawel Blasiak from the Institute of Nuclear Physics of the Polish Academy of Sciences in Krakow and Marcin Markiewicz from the University of Gdansk focus on analyzing widely accepted
paradigms and theoretical concepts regarding the basics and interpretation of quantum mechanics. The researchers are trying to answer the question to what extent the intuitions used to describe
quantum mechanical processes are justified in a realistic view of the world. For this purpose, they try to clarify specific theoretical ideas, often functioning in the form of vague intuitions, using
the language of mathematics. This approach often results in the appearance of inspiring paradoxes. Of course, the more basic the concept to which a given paradox relates, the better, because it opens
up new doors to deeper understanding a given problem.
In this spirit, both scientists decided to ponder the fundamental question: what is interaction and when does it occur? In quantum mechanics, the result of the interaction is entanglement, which is
the appearance of non-classical correlations in the system. Imagine two particles created independently in distant galaxies. It would seem that a necessary condition for the emergence of entanglement
is the requirement that at some point in their evolution the particles touch one another or, at least, indirect contact should take place through another particle or physical field to convey the
interaction. How else can they establish this mysterious bond, which is quantum entanglement? Paradoxically, however, it turns out that this is possible. Quantum mechanics allows entanglement to
occur without the need for any, even indirect, contact.
To justify such a surprising conclusion, a scheme should be presented in which the particles will show non-local correlations at a distance (in a Bell-type experiment). The subtlety of this approach
is to exclude the possibility of an interaction understood as some form of contact along the way. Such a scheme should also be very economical, so it must exclude the presence of force carriers which
could mediate this interaction (physical field or intermediate particles). Blasiak and Markiewicz showed how this can be done by starting from the original considerations of Yurke and Stoler, which
they reinterpreted as a permutation of paths traversed by the particles from different sources. This new perspective allows generating any entangled states of two and three particles, avoiding any
contact. The proposed approach can be easily extended to more particles.
How is it possible to entangle independent particles at a distance without their interaction? The hint is given by quantum mechanics itself, in which the identity – the fundamental
indistinguishability of all particles of the same kind – is postulated. This means, for example, that all photons (as well as other families of elementary particles) in the entire Universe are the
same, regardless of their distance. From a formal perspective, this boils down to symmetrization of the wave function for bosons or its antisymmetrization for fermions. Effects of particle identity
are usually associated with their statistics having consequences for a description of interacting multi-particle systems (such as Bose-Einstein condensate or solid-state band theory). In the case of
simpler systems, the direct result of particle identity is the Pauli exclusion principle for fermions or bunching in quantum optics for bosons. The common feature of all these effects is the contact
of particles at one point in space, which follows the simple intuition of interaction (for example, in particle theory, this comes down to interaction vertices). Hence the belief that the
consequences of symmetrization can only be observed in this way. However, interaction by its very nature causes entanglement. Therefore, it is unclear what causes the observed effects and
non-classical correlations: is it an interaction in itself, or is it the inherent indistinguishability of particles? The scheme proposed by both scientists bypasses this difficulty, eliminating
interaction that could occur through any contact. Hence the conclusion that non-classical correlations are a direct consequence of the postulate of particle identity. It follows that a way was found
to purely activate entanglement from their fundamental indistinguishability.
This type of view, starting from questions about the basics of quantum mechanics, can be practically used to generate entangled states for quantum technologies. The article shows how to create any
entangled state of two and three qubits, and these ideas are already implemented experimentally. It seems that the considered schemes can be successfully extended to create any entangled
many-particle states. As part of further research, both scientists intend to analyze in detail the postulate of identical particles, both from the standpoint of theoretical interpretation and
practical applications.
A big surprise may be the fact that the postulate of indistinguishability of particles is not only a formal mathematical procedure but in its pure form leads to the consequences observed in
laboratories. Is nonlocality inherent in all identical particles in the Universe? The photon emitted by the monitor screen and the photon from the distant galaxy at the depths of the Universe seem to
be entangled only by their identical nature. This is a great secret that science will soon face.
Research: https://www.nature.com/articles/s41598-019-55137-3
No Comments Yet | {"url":"https://spaceandplanetarynewswire.com/research/2020/03/25/entanglement-by-identity-or-interaction-without-ever-touching","timestamp":"2024-11-03T07:41:04Z","content_type":"text/html","content_length":"119489","record_id":"<urn:uuid:74174d72-1a7d-4ed5-999e-dc15fb32477f>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00156.warc.gz"} |
The Strongest Teams for IKPC (strongestteams)
The IKPC is right around the corner. The kindergarten teacher, Theresa, wants to form some teams for the competition. There are N children in her class, and she wants to form exactly K teams. The
children are sitting along a line, numbered from 0 to N-1, according to their position in the line. Child i has a team play factor A_i and a programming skill B_i. Each team must have at least one
member, and no child can be assigned to multiple teams. Children 0 < i_1 < i_2 < … < i_l < N (for some l>0) can form a team if for each 1 < p < l the following conditions hold: (1) A_i_p < A_i_p+1,
and (2) there is no child with index i_p < x < i_p+1 such that A_i_p < A_x. The strength of a team formed this way is B_i_1 + B_i_2 + … + B_i_l. Your task is to find the maximum possible sum of team
strengths that Theresa can achieve by forming exactly K teams! | {"url":"https://squadre.olinfo.it/edition/15/round/4/strongestteams","timestamp":"2024-11-11T03:33:28Z","content_type":"text/html","content_length":"21410","record_id":"<urn:uuid:1d2e85d9-ded4-4f44-88ac-50845bc314d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00539.warc.gz"} |
Basics of Particle Physics: books and courses - Lyu Physics
Basics of Particle Physics: books and courses
Where to learn entry-level particle physics? Like most people, I have been hearing about words like “quarks, hadrons, leptons” since junior high school. However, I have never known the next step for
learning more about them. The talk by David Gross during a summer school in Sweden last summer completely aroused interest in this subject.
During the lunar new year season, which was early February this year, I found time to dive into this subject a little. I want to first get an overview of this subject, focusing on building a physical
picture and intuitive feelings, avoiding technical details as much as possible. Therefore, I started exploring all kinds of textbooks and videos to searchfor what I want.
Quarks and Leptons by Francis Halzen and Alan D. Martin
This book was recommended by my friend, Mark Weitzman, a few years ago. It is certainly not a popular science book for general audience, but a textbook for people who know undergraduate quantum
mechanics very well. The first two chapters of this textbook, however, are what I’m looking for. It is a nice overview of several central questions, for example, from what evidence do we start to
suspect a particle, like a proton, has inner structure, while some others, like an electron, doesn’t.
Starting from chapter 3, detailed mathematical calculations come up. One of the feature of this textbook is that it only employed one-particle quantum mechanics to describe motions of elementary
particles, avoiding a full but much more intricate and complicated field theory treatment. I want to study the details of the book in the future as a preparation for understanding quantum field
theory in particle physics context.
Susskind's Theoretical Minimum: Advanced Quantum Mechanics
Susskind’s Cosmology course has convinced me that whenever I want any intuitive understand of a concept or a subject, I should first check whether he has said something about it. His lectures and
Feynman’s lectures share a very similar style. In his supplemental course, I found a series of 3 courses on Particle Physics, and also one course on Advanced Quantum Mechanics. As a warm-up, I first
chose to take a look at the Advanced QM class. I found hidden much jewelry.
The first three lectures demonstrate how the rotational symmetry of an atom entails the degenerate structure in hydrogen atom energy spectrum. He then moved on, in the next two lectures, talking
about how Pauli discovered the internal spin of electrons by studying this spectrum. There is a very fascinating discussion about the relationship between spin and fermion nature of electrons.
The last 5 lectures focus on developing quantum field theory (QFT) formalism in the non-relativistic case. This exposition is hard to find elsewhere. Most QFT textbook starts by a full relativistic
treatment, which makes it very hard to understand its connection with the non-relativistic QM taught in undergraduate time. The last lecture contains many intuitive discussions about the relativistic
QFT of electron, Dirac equation, like how to understand the mysterious multi-component form of the electron wave function: where does the spin come and where does the concept of anti-particle comes.
In Search of the Ultimate Building Blocks by Gerard 't Hooft
I’m not very satisfied with the first two chapter of Francis Halzen and Alan D. Martin’s textbook, and I want to know more about the story of how people discovered and understand the standard model
of particle physics. Again, I remembered the charm of another great physicist, t’ Hooft, during the Sweden summer school, and gladly found that he has written a book about this. It doesn’t feel like
a popular science book because sometimes it feels quite detailed and technical. I believe the author has tried his best transitioning mathematical expressions into words. So, it is a very nice and
rare book for people like PhD students or physicists who are not familiar with particle physics, since they have enough theoretical background. I feel it prepares me a lot for my future attack on | {"url":"https://xllyu.org/basics-of-particle-physics-books-and-courses/","timestamp":"2024-11-05T13:08:50Z","content_type":"text/html","content_length":"97506","record_id":"<urn:uuid:510d321c-d816-467c-a507-032c7391bfa8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00630.warc.gz"} |
Trifecta Calculator
Harville Method Based Trifecta Odds Calculator
This calculator is based on Harville’s Method (Harville, D.A. (1973) Assigning probabilities to the outcomes of multi-entry competitions.).
Harville’s method assumes that horse that won is automatically discounted from placing and the win probabilities of the remaining horses are adjusted back to 100%.
The win probabilities do not determine the probability that horse x finishes in position y.
These are totally unrealistic assumptions and should not be used in betting purposes as such. But do provide rough estimates.
What Are Trifecta Bets?
Trifecta bet involves predicting the first three finishers in the correct order. You choose which horses you think will come in first, second and third place.
>>> More About Harness Racing Basics
>>> Check Out Our Other Betting Calculators
Trifecta Calculator
Select the number of runners (between 3 and 25):
Enter probabilities for each runner (sum must be 100%):
Calculated Harville Probabilities:
Trifecta Combination Probability Implied Decimal Odds | {"url":"https://totepoint.com/trifecta-calculator/","timestamp":"2024-11-09T00:13:40Z","content_type":"text/html","content_length":"42701","record_id":"<urn:uuid:2f391e10-360b-48e6-967f-9983b9fbbd35>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00254.warc.gz"} |
Imprecise Multi-Armed Bandits
We introduce a novel multi-armed bandit framework, where each arm isassociated with a fixed unknown credal set over the space of outcomes (whichcan be richer than just the reward). The
arm-to-credal-set correspondence comesfrom a known class of hypotheses. We then define a notion of regretcorresponding to the lower prevision defined by these credal sets.Equivalently, the setting
can be regarded as a two-player zero-sum game, where,on each round, the agent chooses an arm and the adversary chooses thedistribution over outcomes from a set of options associated with this arm.
Theregret is defined with respect to the value of game. For certain naturalhypothesis classes, loosely analgous to stochastic linear bandits (which are aspecial case of the resulting setting), we
propose an algorithm and prove acorresponding upper bound on regret. We also prove lower bounds on regret forparticular special cases.
Quick Read (beta)
loading the full paper ... | {"url":"https://deeplearn.org/arxiv/485250/imprecise-multi-armed-bandits","timestamp":"2024-11-12T19:16:18Z","content_type":"text/html","content_length":"8030","record_id":"<urn:uuid:e40a2884-7816-43e5-9c78-737e0391f069>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00108.warc.gz"} |
Gaussian Processes as stochastic differential equations – The Dan MacKinlay stable of variably-well-consider’d enterprises
Gaussian Processes as stochastic differential equations
Imposing time on things
September 18, 2019 — November 25, 2021
dynamical systems
graphical models
Hilbert space
kernel tricks
linear algebra
machine learning
signal processing
state space models
stochastic processes
time series
🏗️🏗️🏗️ Under heavy construction 🏗️🏗️🏗️
Classic flavours together, Gaussian processes and state filters/ stochastic differential equations and random fields as stochastic differential equations.
Not covered, another concept which includes the same keywords but is distinct: using Gaussian processes to define state process dynamics or observation distribution.
1 GP regression via state filtering
I am interested in the trick which makes certain Gaussian process regression problems soluble by making them local, i.e. Markov, with respect to some assumed hidden state, in the same way Kalman
filtering does Wiener filtering. This means you get to solve a GP as an SDE using a state filter.
The GP-filtering trick is explained in intro articles (Särkkä, Solin, and Hartikainen 2013; Lindgren, Bolin, and Rue 2021), based on various antecedents (O’Hagan 1978; S. Reece and Roberts 2010;
Lindgren, Rue, and Lindström 2011; Särkkä and Hartikainen 2012; J. Hartikainen and Särkkä 2010; Solin 2016; Huber 2014), possibly also (Csató and Opper 2002). Aside: O’Hagan (1978) is an incredible
paper that invented several research areas at once (GP regression, surrogate models for experiment design as well as this) and AFAICT no one noticed at the time. Also Whittle did some foundational
work, but I cannot find the original paper to read it.
The idea is that if your GP covariance kernel is (or can be well approximated by) a rational function then it is possible to factorise it into a tractable state space model, using a duality between
random fields and stochastic differential equations. That sounds simple enough conceptually; I wonder about the practice. Of course, when you want some complications, such as non-stationary kernels
or hierarchical models, this state space inference trick gets more complicated, and posterior distributions are no longer so simple. But possibly it can still go. (This is a research interest of
William J. Wilkinson et al. (2020) introduces a computational toolkit and many worked examples of inference algorithms. Cox, van de Laar, and de Vries (2019) looks like it might be solving a similar
problem but I do not yet understand their framing.
This complements, perhaps, the trick of fast Gaussian process calculations on lattices.
Nickisch, Solin, and Grigorevskiy (2018) tries to introduce a vocabulary for inference based on this insight, by discussing it in terms of computational primitives
In time-series data, with D = 1, the data sets tend to become long (or unbounded) when observations accumulate over time. For these time-series models, leveraging sequential state space methods
from signal processing makes it possible to solve GP inference problems in linear time complexity O(n) if the underlying GP has Markovian structure (S. Reece and Roberts 2010; J. Hartikainen and
Särkkä 2010). This reformulation is exact for Markovian covariance functions (see, e.g., Solin (2016)) such as the exponential, half-integer Matérn, noise, constant, linear, polynomial, Wiener,
etc. (and their sums and products).…
While existing literature has focused on the connection between GP regression and state space methods, the computational primitives allowing for inference using general likelihoods in combination
with the Laplace approximation (LA), variational Bayes (VB), and assumed density filtering (ADF, a.k.a. single-sweep expectation propagation, EP) schemes has been largely overlooked.… We present
a unifying framework for solving computational primitives for non-Gaussian inference schemes in the state space setting, thus directly enabling inference to be done through LA, VB, KL, and ADF/
The following computational primitives allow to cast the covariance approximation in more generic terms: 1. Linear system with “regularized” covariance: \[ \text { solve }_{\mathbf{K}}(\mathbf
{W}, \mathbf{r}):=\left(\mathbf{K}+\mathbf{W}^{-1}\right)^{-1} \mathbf{r} \] 2. Matrix-vector multiplications: \(\operatorname{mvm}_{\mathbf{K}}(\mathbf{r}):=\mathbf{K r}\). For learning we also
need \(\frac{\operatorname{mvm}_{K}(\mathbf{r})}{\partial \theta}\). 3. Log-determinants: \(\operatorname{ld}_{\mathbf{K}}(\mathbf{W}):=\log |\mathbf{B}|\) with symmetric and well-conditioned \(\
mathbf{B}=\mathbf{I}+\mathbf{W}^{\frac{1}{2}} \mathbf{K} \mathbf{W}^{\frac{1}{2}}\). For learning, we need derivatives: \(\frac{\partial \operatorname{ld} \mathbf{K}(\mathbf{W})}{\partial \
boldsymbol{\theta}}, \frac{\partial \operatorname{ld} \mathbf{K}(\mathbf{W})}{\partial \mathbf{W}}\) 4. Predictions need latent mean \(\mathbb{E}\left[f_{*}\right]\) and variance \(\mathbb{V}\
Using these primitives, GP regression can be compactly written as \(\mathbf{W}=\mathbf{I} / \sigma_{n}^{2}, \boldsymbol{\alpha}=\operatorname{solve}_{\mathbf{K}}(\mathbf{W}, \mathbf{y}-\mathbf
{m}),\) and \(\log Z_{\mathrm{GPR}}=\) \[ -\frac{1}{2}\left[\boldsymbol{\alpha}^{\top} \mathrm{mvm}_{\mathrm{K}}(\boldsymbol{\alpha})+\mathrm{ld}_{\mathrm{K}}(\mathbf{W})+n \log \left(2 \pi \
sigma_{n}^{2}\right)\right] \] Approximate inference \((\mathrm{LA}, \mathrm{VB}, \mathrm{KL}, \mathrm{ADF} / \mathrm{EP})-\) in case of non-Gaussian likelihoods - requires these primitives as
necessary building blocks. Depending on the covariance approximation method e.g. exact, sparse, grid-based, or state space, the four primitives differ in their implementation and computational
Recent works I should also inspect include (Chang et al. 2020; Gorad, Zhao, and Särkkä 2020; Nickisch, Solin, and Grigorevskiy 2018; Remes, Heinonen, and Kaski 2018; Solin 2016; William J. Wilkinson
et al. 2019a; William J. Wilkinson et al. 2019c; Karvonen and Särkkä 2016).
Ambikasaran et al. (2015) seems to be related but not quite the same — it operates time-wise over inputs but then constructs the GP posterior using rank-1 updates.
4 Miscellaneous notes towards implementation
• Julia’s DynamicPolynomials.jl implements the MultivariatePolynomials, appears to be differentiable and handles rational polynomials.
• TemporalGPs.jl, introduced by Will Tebbutt, is a julia implementation of this.
• The abstract ForneyLab.jl system might relate to this, behind its abstruse framing. Cox, van de Laar, and de Vries (2019)
• https://pyro.ai/examples/dkl.html
• https://pyro.ai/examples/gp.html
• https://pyro.ai/examples/ekf.html
• https://julialang.org/blog/2019/01/fluxdiffeq
• https://github.com/FluxML/model-zoo/tree/master/contrib/diffeq
• http://pyro.ai/examples/gplvm.html
• http://pyro.ai/examples/dmm.html
• http://docs.pyro.ai/en/stable/contrib.gp.html
• https://nbviewer.jupyter.org/github/SheffieldML/notebook/blob/master/GPy/index.ipynb
• https://blog.dominodatalab.com/fitting-gaussian-process-models-python/
See also | {"url":"https://danmackinlay.name/notebook/gp_markov","timestamp":"2024-11-10T16:12:18Z","content_type":"application/xhtml+xml","content_length":"68102","record_id":"<urn:uuid:0e88d78e-4eb4-4777-8df8-c2ef15029160>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00098.warc.gz"} |
Physics 11 - StudyForge
This course aligns with the learning outcomes for:
This course aligns with the learning outcomes for:
Physics 11 is the first opportunity students have to dive deeply into our intricate and intriguing world of energy and forces! These two culprits are responsible for everything from roller coasters,
parachuting, car crashes, listening to acoustic bliss from your favourite band, to enjoying steaming hot cups of coffee, warming your hands by a fire, and playing on your favourite battery-powered
devices, to name only a few of the phenomena we experience in our lives every day. Well, hopefully not the crashes.
Technically, these events fall under the more scientific concepts of kinematics, dynamics, momentum, electricity, circuits and waves, etc. But more importantly, in this course, students will find
themselves absorbed and inspired – through animated instructional videos and other engaging materials – by a feast of real-life applications and discoveries of these very same concepts. This course
is designed to propel students on an exciting journey toward a solid and firm understanding of the fascinating physical world that exists all around them.
*Each lesson is designed to take 60 – 90 minutes to complete with the exception of major projects and assignments.
Lesson 1: Some Physics Terminology
Lesson 2: Introduction to Vectors
Lesson 3: Trigonometry – Review
Lesson 4: Vectors With Trigonometry
Lesson 5: Units of Measurement and Significant Figures
Lesson 6: Graphing
Lesson 1: Displacement, Velocity, Acceleration
Lesson 2: Kinematics and Graphing
Lesson 3: Uniform Acceleration – Part I
Lesson 4: Uniform Acceleration – Part II
Lesson 5: Vectors – Part I
Lesson 6: Vectors – Part II
Lesson 1: Intro to Forces
Lesson 2: Gravity and Springs
Lesson 3: Force Vectors
Lesson 4: Free-Body Diagrams
Lesson 5: Force Problems
Lesson 6: Supplemental
Lesson 1: Introduction to Newton’s Laws
Lesson 2: Newton’s Second Law Part I
Lesson 3: Newton’s Second Law Part II
Lesson 4: Newton’s Third Law
Lesson 5: Friction Part I
Lesson 6: Friction Part II
Lesson 1: Kinetic and Gravitational Energy
Lesson 2: Conservation of Energy
Lesson 3: Work and Power
Lesson 4: Thermal Energy
Lesson 5: Thermal Energy Problems
Lesson 1: Intro to Electricity
Lesson 2: Electric Currents
Lesson 3: Electric Potential
Lesson 4: Resistance
Lesson 5: Ohm’s Law
Lesson 1: Complex Electric Circuits
Lesson 2: Kirchhoff’s Laws
Lesson 3: Circuit Analysis
Lesson 4: EMF, Internal Resistance, and Terminal Voltage
Lesson 5: Power in Electric Circuits
Lesson 6: Efficiency of a Circuit
Lesson 1: Introduction to Waves
Lesson 2: Properties of Waves
Lesson 3: Light
Lesson 4: Sound – Part I
Lesson 5: Sound – Part II
Experience a lesson as your students would
• Informal and formal lab experience
• Interesting and engaging activities/labs/projects:
□ Designing a rollercoaster
□ Building a functional trap
□ Historical moon-landing analysis
□ Constructing electrical circuits
□ Experimentally calculating the speed of sound
□ Analyzing and Improving your energy footprint | {"url":"https://studyforge.net/ca/curriculum/physics-11/","timestamp":"2024-11-05T17:09:59Z","content_type":"text/html","content_length":"266480","record_id":"<urn:uuid:73c4270f-9346-468e-a86b-3f686f64f518>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00816.warc.gz"} |
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of
mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.
If R and S are two rings, then an R-S-bimodule is an abelian group (M, +) such that:
1. M is a left R-module and a right S-module.
2. For all r in R, s in S and m in M: ${\displaystyle (r.m).s=r.(m.s).}$
An R-R-bimodule is also known as an R-bimodule.
• For positive integers n and m, the set M[n,m](R) of n × m matrices of real numbers is an R-S-bimodule, where R is the ring M[n](R) of n × n matrices, and S is the ring M[m](R) of m × m matrices.
Addition and multiplication are carried out using the usual rules of matrix addition and matrix multiplication; the heights and widths of the matrices have been chosen so that multiplication is
defined. Note that M[n,m](R) itself is not a ring (unless n = m), because multiplying an n × m matrix by another n × m matrix is not defined. The crucial bimodule property, that (r.x).s = r.(x.s)
, is the statement that multiplication of matrices is associative (which, in the case of a matrix ring, corresponds to associativity).
• Any algebra A over a ring R has the natural structure of an R-bimodule, with left and right multiplication defined by r.a = φ(r)a and a.r = aφ(r) respectively, where φ : R → A is the canonical
embedding of R into A.
• If R is a ring, then R itself can be considered to be an R-R-bimodule by taking the left and right actions to be multiplication – the actions commute by associativity. This can be extended to R^n
(the n-fold direct product of R).
• Any two-sided ideal of a ring R is an R-R-bimodule, with the ring multiplication both as the left and as the right multiplication.
• Any module over a commutative ring R has the natural structure of a bimodule. For example, if M is a left module, we can define multiplication on the right to be the same as multiplication on the
left. (However, not all R-bimodules arise this way: other compatible right multiplications may exist.)
• If M is a left R-module, then M is an R-Z-bimodule, where Z is the ring of integers. Similarly, right R-modules may be interpreted as Z-R-bimodules. Any abelian group may be treated as a Z-Z
• If M is a right R-module, then the set End[R](M) of R-module endomorphisms is a ring with the multiplication given by composition. The endomorphism ring End[R](M) acts on M by left multiplication
defined by f.x = f(x). The bimodule property, that (f.x).r = f.(x.r), restates that f is a R-module homomorphism from M to itself. Therefore any right R-module M is an End[R](M)-R-bimodule.
Similarly any left R-module N is an R-End[R](N)^op-bimodule.
• If R is a subring of S, then S is an R-R-bimodule. It is also an R-S- and an S-R-bimodule.
• If M is an S-R-bimodule and N is an R-T-bimodule, then M ⊗[R] N is an S-T-bimodule.
Further notions and facts
If M and N are R-S-bimodules, then a map f : M → N is a bimodule homomorphism if it is both a homomorphism of left R-modules and of right S-modules.
An R-S-bimodule is actually the same thing as a left module over the ring R ⊗[Z] S^op, where S^op is the opposite ring of S (where the multiplication is defined with the arguments exchanged).
Bimodule homomorphisms are the same as homomorphisms of left R ⊗[Z] S^op modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and
statements about bimodules. For example, the category of all R-S-bimodules is abelian, and the standard isomorphism theorems are valid for bimodules.
There are however some new effects in the world of bimodules, especially when it comes to the tensor product: if M is an R-S-bimodule and N is an S-T-bimodule, then the tensor product of M and N
(taken over the ring S) is an R-T-bimodule in a natural fashion. This tensor product of bimodules is associative (up to a unique canonical isomorphism), and one can hence construct a category whose
objects are the rings and whose morphisms are the bimodules. This is in fact a 2-category, in a canonical way – 2 morphisms between R-S-bimodules M and N are exactly bimodule homomorphisms, i.e.
${\displaystyle f:M\rightarrow N}$
that satisfy
1. ${\displaystyle f(m+m')=f(m)+f(m')}$
2. ${\displaystyle f(r.m.s)=r.f(m).s}$ ,
for m ∈ M, r ∈ R, and s ∈ S. One immediately verifies the interchange law for bimodule homomorphisms, i.e.
${\displaystyle (f'\otimes g')\circ (f\otimes g)=(f'\circ f)\otimes (g'\circ g)}$
holds whenever either (and hence the other) side of the equation is defined, and where ∘ is the usual composition of homomorphisms. In this interpretation, the category End(R) = Bimod(R, R) is
exactly the monoidal category of R-R-bimodules with the usual tensor product over R the tensor product of the category. In particular, if R is a commutative ring, every left or right R-module is
canonically an R-R-bimodule, which gives a monoidal embedding of the category R-Mod into Bimod(R, R). The case that R is a field K is a motivating example of a symmetric monoidal category, in which
case R-Mod = K-Vect, the category of vector spaces over K, with the usual tensor product ⊗ = ⊗[K] giving the monoidal structure, and with unit K. We also see that a monoid in Bimod(R, R) is exactly
an R-algebra.^[1] Furthermore, if M is an R-S-bimodule and L is an T-S-bimodule, then the set Hom[S](M, L) of all S-module homomorphisms from M to L becomes a T-R-bimodule in a natural fashion. These
statements extend to the derived functors Ext and Tor.
Profunctors can be seen as a categorical generalization of bimodules.
Note that bimodules are not at all related to bialgebras.
See also
1. ^ Street, Ross (20 Mar 2003). "Categorical and combinatorial aspects of descent theory". arXiv:math/0303175.
• Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. pp. 133–136. ISBN 0-7167-1933-9. | {"url":"https://www.knowpia.com/knowpedia/Bimodule","timestamp":"2024-11-02T01:43:55Z","content_type":"text/html","content_length":"86488","record_id":"<urn:uuid:c2848b15-f3fb-4535-a20a-44e73d10c10e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00276.warc.gz"} |
Apply forward N-D spatial transformation
[X1,X2,...,X_ndims_out] = tformfwd(T,U1,U2,...,U_ndims_in) applies the ndims_in-to-ndims_out spatial transformation defined in T to the coordinate arrays U1,U2,...,U_ndims_in. The transformation maps
the point [U1(k) U2(k) ...U_ndims_in(k)] to the point [X1(k) X2(k) ... X_ndims_out(k)].
The number of input coordinate arrays, ndims_in, must equal T.ndims_in. The number of output coordinate arrays, ndims_out, must equal T.ndims_out. The arrays U1,U2,...,U_ndims_in can have any
dimensionality, but must be the same size. The output arrays X1,X2,...,X_ndims_out must be this size also.
X = tformfwd(T,U) applies the spatial transformation defined in T to coordinate array U.
• When U is a 2-D matrix with dimensions m-by-ndims_in, X is a 2-D matrix with dimensions m-by-ndims_out. tformfwd applies the ndims_in-to-ndims_out transformation to each row of U. tformfwd maps
the point U(k, : ) to the point X(k, : ).
• When U is an (N+1)-dimensional array, tformfwd maps the point U(k[1], k[2], … ,k[N], : ) to the point X(k[1], k[2], … ,k[N], : ).
size(U,N+1) must equal ndims_in. X is an (N+1)-dimensional array, with size(X,I) equal to size(U,I) for I = 1, … ,N, and size(X,N+1) equal to ndims_out.
The syntax X = tformfwd(U,T) is an older form of this syntax that remains supported for backward compatibility.
[X1,X2,...,X_ndims_out] = tformfwd(T,U) maps one (N+1)-dimensional array to ndims_out equally sized N-dimensional arrays.
Create Affine Transformation and Validate It with Forward Mapping
Create an affine transformation that maps the triangle with vertices (0,0), (6,3), (-2,5) to the triangle with vertices (-1,-1), (0,-10), (4,4).
u = [ 0 6 -2]';
v = [ 0 3 5]';
x = [-1 0 4]';
y = [-1 -10 4]';
tform = maketform('affine',[u v],[x y]);
Validate the mapping by applying tformfwd. The results should equal x and y.
[xm,ym] = tformfwd(tform,u,v)
Input Arguments
T — Spatial transformation
TFORM structure
Spatial transformation, specified as a TFORM structure. Create T using the maketform function.
Data Types: struct
U — Input coordinate points
numeric array
Input coordinate points, specified as a numeric array. The size and dimensionality of U can have additional limitations depending on the syntax used.
Data Types: double
U1,U2,...,U_ndims_in — Input coordinate points
multiple numeric arrays
Input coordinate points, specified as multiple numeric arrays. The size and dimensionality of U1,U2,...,U_ndims_in can have additional limitations depending on the syntax used.
Data Types: double
Output Arguments
X — Coordinate array of output points
numeric array
Coordinate array of output points, returned as a numeric array. The size and dimensionality of X can have additional limitations depending on the syntax used.
X1,X2,...,X_ndims_out — Coordinates of output points
multiple numeric arrays
Coordinates of output points, returned as multiple numeric arrays. The size and dimensionality of X1,X2,...,X_ndims_out can have additional limitations depending on the syntax used.
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006a
R2021b: Support for thread-based environments
tformfwd now supports thread-based environments.
R2018b: tformfwd is not recommended for 2-D and 3-D geometric transformations
The tformfwd function is not recommended for 2-D and 3-D geometric transformations. Instead, create a 2-D or 3-D geometric transformation object, then use the transformPointsForward function. For
more information about geometric transformation objects, see 2-D and 3-D Geometric Transformation Process Overview.
This table shows a syntax of tformfwd with the recommended replacement code.
Discouraged Usage Recommended Replacement
Apply a forward 2-D affine transformation defined in TFORM struct T to coordinate arrays U and V, mapping the Apply a forward 2-D affine transformation using the affinetform2d object T and the
point [U(k) V(k)] to the point [X(k) Y(k)]. transformPointsForward function.
[X,Y] = tformfwd(T,U,V); [X,Y] = transformPointsForward(T,U,V); | {"url":"https://www.mathworks.com/help/images/ref/tformfwd.html","timestamp":"2024-11-08T17:26:48Z","content_type":"text/html","content_length":"93023","record_id":"<urn:uuid:7005aba5-4095-44d0-a7fb-af6348d87df3>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00488.warc.gz"} |
Seminars 2013
Martes 10 de Diciembre
The Janus family: a dedicated computer generation
BIFI, Universidad de Zaragoza
Janus [1,2] is a special purpose computer designed as a multipurpose reprogramable supercomputer. It is based on a Field-Programmable-Gate-Array (FPGA) processor architecture, which permits us
reprogramming the computer's hardware connections structure in order to optimize its performance for each concrete problem to solve.
Encouraged by the good results obtained so far, the Janus Collaboration decided to go an step further developing and designing the new generation Janus dedicated computer, named JanusII [3]. In this
talk I will introduce both supercomputers, Janus and JanusII, explaining their internal architectures and the way we profit by their resources and possibilities for the study of spin glasses,
paradigm of complex systems. I will also discuss some of the last spin glass aims achieved with Janus.
In addition, there is an international open call to the scientific community for the implementation on new applications on Janus (
[1]Janus Collaboration: F. Belletti et al., Computer Physics Communications 178 (3), 208-216, (2008).
[2]Janus Collaboration: F. Belletti et al., Computing in Science & Engineering 11-1, 48-58 (2009).
[3]Janus Collaboration: M. Baity-Jesi, et al., Janus II: a new generation application-driven computer for spin-system simulations, arXiv:1310.1032 (accepted in Computer Physics Communications).
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Martes 12 de Noviembre
Corner waves downstream from a partially submerged vertical plate
The high-Reynolds-number flow near the corner of a vertical flat plate partially submerged across an uniform stream has been studied using a combination of experimental, numerical and analytical
tools. In this configuration, a three dimensional wave forms at the corner of the plate which evolves downstream in a similar way as a time-evolving two dimensional plunging or spilling breaker
(depending the occurrence of one or the other type of breaker on the flow conditions). Experiments have been performed submerging a flat plate perpendicular to the free stream in the test section of
a recirculating water channel. Experimental results show that the formation and the initial development of the wave is nearly unaffected by the presence of the channel walls and bottom even when
their distance to the corner, where the wave originates, is of the order of the size of the wave itself. This is a remarkable observation, that suggests that the formation of the corner wave is a
local process in a sense that it is only influenced by the characteristics of the velocity field very near the corner. Moreover, it has been observed that the jet formed when the corner wave adopts
the plunging breaker configuration follows a nearly ballistic trajectory, has is the case in two-dimensional unsteady plunging breakers. Theoretical analysis shows that, taking advantage of the
slender nature of the flow, the 3D steady problem can be transformed into a two dimensional unsteady one using the so called 2D+T approximation. Together with the high Reynolds number of the flow,
the 2D+T approximation makes the problem amenable to be simulated numerically using a Boundary Element Method (BEM). Moreover, a pressure-impulse asymptotic analysis of the flow near the origin of
the corner wave has been performed in order to describe the initial evolution of the wave and to clarify the physical mechanisms that lead to its formation. The analysis shows that the flow near the
corner exhibits a self similar behavior at short times, although the self-similar solution is physically unattainable due to the existence of two "jetlets" that impinge onto the base of the main jet
that causes the wave.
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Martes 5 de Noviembre
A macroscopic particle-wave system: Theoretical investigation of walking droplets
Yves Couder and his coworkers in Paris have discovered a macroscopic particle-wave system exhibiting many features previously thought to be peculiar to the microscopic quantum realm. A small liquid
droplet placed on the vibrating surface of a fluid bath, can be made to bounce (essentially indefinitely) provided that the amplitude and frequency of the oscillations is in the "correct range". In
particular: (1) The frequency must be high enough that the "impact time" is too short to allow the air layer between the drop and bath to drain to the critical distance at which merger is initiated
by van der Waals forces, (2) The maximum vertical acceleration of the free surface must exceed gravity (so the drop can lift of after landing), (3) The operational regime must be below the Faraday
instability threshold, so the liquid surface remains (essentially) "flat".
he experiments by Couder involve a millimeter sized droplet on a vibrating bath of silicone oil (viscosity 20-50 times that of water). There, the drop may bounce indefinitely on the free surface,
generating a localized field of surface waves that decays with distance from the drop. The drop interacts with this wave field, and undergoes several bifurcations in its behavior as the driving
amplitude grows: from bouncing in place at the same frequency as the fluid bath, to a period doubling bifurcation, to spontaneous "walking" on the surface. Walking drops exhibit quantum-like effects
in its behavior: diffraction, interference, orbit quantization in rotating frames, etc. Multiple bouncers communicate through their wave fields, and can orbit each other forming "atoms", or "crystal"
lattices, etc.
In this talk I will introduce an integral equation that describes the wave-induced force that acts on walking droplets. From this we can write a new guidance equation for walking droplets, that
provides insight into their observed quantum behavior. In particular I will consider the behavior of a drop/particle in a rotating frame, and the myriad of patterns that this produces.
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Martes 22 de Octubre
Capturing Nature's Creativity in Robotics & Tissue Regeneration
Stanford University
Throughout the years nature has inspired some of the best inventions in engineering and medicine, from building airplanes with movable wing surfaces like those of birds to using viruses as a vehicle
to deliver genetic materials into cells. In this talk, I will present our work aiming to solve problems in engineering and medicine using nature-inspired approaches. In the first part, I will focus
on understanding the adhesive locomotion of gastropods for the design of biomimetic robots. Specifically, some of the critical questions we aim to address are: how do soft-bodied creatures like slugs
and snails propel themselves through irregular terrains? Is the rheological properties of the secreted mucus essential, and what lessons can we learn from their crawling mechanism for robotics
design? In the second part, we explore the possibility of manipulating cell-cell interactions as a strategy for cartilage regeneration therapy. In particular, we explore the potential of
adipose-derived stem cells as a catalyst to stimulate cartilage regeneration by neonatal chondrocytes. The questions we seek to answer are: how can we minimize the number of neonatal chondrocytes, an
extremely scarce cell source, needed for cartilage repair? Can we manipulate cell-cell interactions by controlling cell distribution and intercellular distance in 3D to facilitate optimal synergy?
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Lunes 14 de Octubre
On aerofoil tonal noise
Delft University of Technology
Aerofoil tonal noise is an aeroacoustic phenomenon peculiar of small wind turbines, compressors' blades and UAVs, all applications where a laminar or at least transitional flow can take place. Its
investigation spans a period of more than 40 years but up to date no agreement between the researchers has been achieved. The aspects object of research were linked to the flow mechanism causing this
acoustic emission and the behaviour in terms of acoustic emission frequency with respect to the free stream velocity.
The aerofoil tonal noise is often referred to as aerofoil self noise. The reason for this appellation resides in the illuminating conjectures of Tam [1], who in 1974 proposed the occurrence of a
feedback loop between the hydrodynamic fluctuations and the acoustic waves scattered by those fluctuations. The acoustic waves propagating in the whole field, were causing a forcing of the
fluid-dynamic field thus leading to the mutual interaction proper of a feedback loop. In the following years many researchers accepted this explanation introducing new findings and modifications.
In one of the last works on the topic Desquesnes et al. [2] studied the flow mechanism causing tonal emission with a 2D DNS. They investigated the pressure signal in the far field finding a
modulation of its amplitude. The period of this modulating envelope was equal to the invert of the separation, in terms of frequency, of the discrete peaks present in the acoustic power spectrum. The
cause of this modulation was individuated in a varying phase shift between the disturbances of the boundary layers of the two sides of the aerofoil at the trailing edge.
The results of a wind tunnel campaign by means of high speed PIV and simultaneous microphones measurements are here presented [3]. Furthermore linear stability theory LST in its spatial formulation
has been applied to the time-averaged flow fields.
Some important confirmations about the reported flow features under which tonal noise is observed, are obtained. Moreover some new findings have been discovered thus leading to the rejection of some
earlier conclusions as well as to the proposition of a new model of the feedback loop.
[1]Tam, C.K.W. (1974). Discrete tones of isolated airfoils. J. Acoust. Soc. Am., 55 (6): 1173-1177.
[2]Desquesnes, G., Terracol, M. and Sagaut, P. (2007). Numerical investigation of the tone noise mechanism over laminar airfoils. J. Fluid. Mech., 591: 155-182.
[3]Pröbsting, S., Serpieri, J. and Scarano, F., (2013). Investigation of tonal noise generation on an airfoil with time-resolved PIV. 19th AIAA/CEAS Aeroacoustics Conf. Berlin, Germany.
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Martes 8 de Octubre
From Analogue Gravity to Elastic Cloaking
Analogue Gravity relies on the observation that certain collective excitations in condensed-matter physics, for example sound waves, have equations of motion that can be written as a relativistic
field in a curved space-time. I discuss several consequences, from acoustic black holes in Bose-Einstein condensates over white-hole experiments in a kitchen sink, to the possibility of arriving at
acoustic and elastic cloaking with composite metamaterials.
El Seminario tendrá lugar a las 13:00 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Viernes 19 de Julio
Modeling and simulation of bacterial biofilms
Bacterial biofilms may be seen as bacterial aggregates embedded into a polysaccharide matrix with a high resistance against removal processes, which results in a recurrent source of problems in other
disciplines (medicine, engineering, etc). The behaviour of these organisms is highly dependent of the physical system in which they are present, thus showing a very high degree of physical
complexity. In this seminar we will focus our efforts on describing a mathematical and experimental modelization of biofilms by exposing a different set of case studies. First the dynamics of
biofilms in straight ducts is studied. Experiments are performed to obtain statistics about spreading patterns, and a hybrid model (combining a discrete approach for bacterial population with
stochastical behaviour rules and a continuum description of outer fields ruling those probabilities) is presented to simulate the biofilm dynamics, obtaining a successfully prediction of the
different patterns observed in real experiments (at layers, ripples, streamers, mounds). This part is completed by extending the scope of the model to the formation of biofilm streamers inside a
corner flow, where biomass adhesion mechanism becomes relevant. Streamers cross the channel joining both corners as observed experimentally. Additionally we perform a description of more complex
dynamics observed in biofilms. An experimental description of biofilm dynamics under pulsatile flows at low Reynolds numbers show spiral patterns not reported yet, supported by a theoretical
mechanism of formation based on the competence between flow dynamics and nutrient gradients.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Martes 2 de Julio
Least-Squares Finite Element Models of Flows of Incompressible Fluids
Texas A & M University
Finite element formulations based on the weak-form Galerkin method in solid and structural mechanics resulted in enormous success. However, extension of these concepts to fluid mechanics and other
areas of mechanics where the differential operators are either non-self adjoint or non-linear have met with mixed success. Numerical schemes such as modified weight functions, modified quadrature
rule, optimal upwinding etc. have been presented in the literature to alleviate problems encountered with weak form Galerkin procedures in solving non-self adjoint and nonlinear problems outside of
solid mechanics.
The lecture presents the formulation and application of the least-squares finite element formulations to the numerical solution of the Navier-Stokes equations governing two-dimensional flows of
viscous incompressible fluids. Finite element models of the vorticity-based or velocity gradients-based Navier-Stokes equations are developed using the least-squares technique. The use of
least-squares principles leads to a symmetric and positive-definite system of algebraic equations that allow the use of iterative methods for the solution of resulting algebraic equations.
High-order nodal expansions are used to construct the discrete finite element models. The system of equations thus obtained is linearized by Newton's method and solved by the preconditioned conjugate
gradient method. Exponentially fast decay of the least-squares functional, which is constructed using the $L_2$ norms of the residuals in the governing equations, is verified for increasing order of
the nodal expansions. Numerical results will be presented for several benchmark flow problems to demonstrate the predictive capability and robustness of the least-squares based finite element models.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C17 (Edificio Sabatini) Universidad Carlos III
Lunes 24 de junio
The RBF-FD Method: Developments and Applications
Radial Basis Function (RBF) methods have become a truly meshless alternative for the interpolation of multidimensional scattered data and the solution of PDEs on irregular domains. Its dependence on
the distance between centers makes RBF methods conceptually simple and easy to implement in any dimension or shape of the domain. There are two different formulations for the solution of PDEs: the
global RBF method and the local RBF method.
The global RBF formulation yields dense differentiation matrices which are spectrally convergent independently of the node distribution. Its principal drawback is that, as the overall number of
centers increases, the condition number of the collocation matrices increases, and this fact restricts the applicability of the method in practical problems. To overcome some of the drawbacks of the
global RBF method, the local RBF method was independently proposed by several authors (also known as RBF-FD). Unlike the global RBF method, the RBF-FD method lacks spectral accuracy. However, the
main feature of the method is the hability for handling irregular domains using highly sparse differentiation matrices while approximating the differential operators to high order. In this thesis we
focus on the RBF-FD method.
In the first part we analyze the convergence properties of the method obtaining novel analytical formulas for the local truncation error as a function of the shape parameter, inter-nodal distance and
stencil size. This result proof the existance of a range of values of the shape parameter for which RBF-FD methods are more accurated than FD. Indeed, it usually exists an optimal shape parameter for
which the local truncation error cancel outs and the approximation is exact. To leading order, such a value is independent of the inter-nodal distance and only relies on the function and its
derivatives. These results allow us to develop novel algorithms for the selection of the shape parameter in the solution of PDEs. In this line, two different strategies are proposed: a
node-independent shape parameter, which minimizes the norm of the global error, and a node-dependent shape parameter, which minimizes the local truncation error at each node of the domain.
Applications of the present methods have been studied in the solution of classical elastostatic problems, for which it is shown that the accuracy can significantly increased one or two orders of
magnitude with respect to finite differences by efficiently tuning the values of shape parameters.
The applicability of the method is explored in the second part of this thesis. In this way, a three-dimensional problem for the propagation of a premixed laminar flame through a duct is solved. The
good performance of the method inspires us to implement an RBF-FD method for the numerical study of an idealized Wankel microcombustor, for which the geometry is more complex. The combustible flow
field and the combustion process are respectively modeled through the steady Navier-Stokes equations and the combustion model above.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Viernes 14 de junio
Models for large-scale turbulent structures on jets and their radiated noise
California Institute of Technology/UPM
Turbulent jet noise is a technological problem of great importance that has received continuous attention for decades. While state-of-the-art numerical simulations are today capable of simultaneously
predicting turbulence and its radiated sound, a theoretical framework enabling fast prediction in order to guide noise-control efforts is incomplete. In this direction, the peak noise radiation in
the aft direction of high-speed jets has been linked to the dynamics of the large-scale wavepackets existing in the flow: intermittent, advecting disturbances that are correlated over distances far
exceeding the integral scales of turbulence, the signatures of which can be distinguished in the vortical turbulent region and in the acoustic near and far fields.
The present research uses parabolized stability equations (PSE) in order to model the statistical wavepackets as instability waves of the turbulent mean flow for subsonic and ideally-expanded
supersonic round jets. The theoretical framework and algorithmic details will be discussed. Extensive comparisons and validations are performed against experimental measurements and data from large
eddy simulations, demonstrating the utility of PSE in modeling (i) the large-scale structures in the velocity field, (ii) the pressure signature in the acoustic near-field and (iii) the highly
directional peak noise in the acoustic far-field.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Conferencia del Programa de Cátedras de Excelencia
Jueves, 16 de mayo
Multiple Scales and Coupled Phenomena in Nature and Mathematical Models
Wilfrid Laurier University/Visiting Professor at UC3M
Interacting time and space scales are universal. They frequently go hand in hand with coupled phenomena which can be observed in nature and man-made systems. Such mutiscale coupled phenomena are
fundamental to our knowledge about all the systems surrounding us, ranging from such global systems as the climate of our planet, to such tiny ones as quantum dots, and all the way down to the
building blocks of life such as nucleic acid biological molecules.
In this talk I will provide an overview of some coupled multiscale problems that we face in studying physical, engineering, and biological systems. I will start from considering tiny objects, known
as low dimensional nanostructures, and will give examples on why the nanoscale is becoming increasingly important in the applications affecting our everyday lives. By using fully coupled mathematical
models, I will show how to build on the previous results in developing a new theory, while analyzing the influence of coupled multiscale effects on properties of these tiny objects.
The remaining part of the talk I will devote to coupled multiscale problems in studying biological structures constructed from ribonucleic acid (RNA). As compared to deoxyribonucleic acid (DNA) and
some other bio-molecules, RNA offers not only a much greater variety of interactions but also great conformational flexibility, making it an important functional material in many bioengineering and
medical applications. Examples of numerical simulations of such biological structures will be shown, based on our developed coarse-grained methodologies.
El Seminario tendrá lugar a las 12:30 horas en el Auditorio Padre Soler, Universidad Carlos III
Miércoles, 15 de mayo
Flammability of Materials in Spacecrafts
University of California at Berkeley
Space exploration vehicles frequently employ cabin environments that are not at standard sea level atmospheric conditions. NASA's Constellation Program considers a human space exploration cabin
environment of reduced ambient pressure and increased oxygen concentration. This enhanced oxygen and reduced pressure atmosphere (approximately 56 kPa and 32 the Space Exploration Atmosphere, SEA,
and while it reduces preparation time for EVAs by reducing the risk of decompression sickness it may have a significant impact on the flammability of materials. In this presentation the work being
conducted at the University of California Berkeley regarding the flammability of materials in environments similar to those expected in those future space based facilities, i.e., micro-gravity, low
velocity flow, elevated oxygen concentrations, and reduced pressures, is reviewed. A description of the equipment and facilities used in those studies and a summary of the results will be presented.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C17 (Edificio Sabatini) Universidad Carlos III
Viernes, 10 de mayo de 2013
Vibrated fluids: Faraday waves, cross-waves, and vibroequilibria
The behavior of vibrated fluids and, in particular, the surface or interfacial instabilities that commonly arise in these systems have been the subject of continued experimental and theoretical
attention since Faraday's seminal experiments in 1831. Both orientation and frequency are critical in determining the response of the fluid to excitation. Low frequencies are associated with sloshing
while higher frequencies may generate Faraday waves or cross-waves, depending on whether the axis of vibration is perpendicular or parallel to the interface. In addition, high frequency vibrations
are known to produce large scale reorientation of the fluid (vibroequilibria), an effect that becomes especially pronounced in the absence of gravity. We describe the results of experimental and
theoretical investigations into the effect of vibrations on fluid interfaces, particularly the interaction between Faraday waves and cross-waves.
Experiments utilize a dual-axis shaker configuration that permits two independent forcing frequencies, amplitudes, and phases to be varied. Theoretical results, based on the analysis of reduced
models, and on numerical simulations, are described and compared to experiment. In particular, the nonlinear Schrodinger equation models used to study cross-waves since Jones (JFM 138, 1984) are
extended to include surface tension and to allow the inhomogeneous forcing term to vary on the same lengthscale as the cross-wave modulation, an assumption that is needed for high frequency (large
aspect ratio) experiments such as ours.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Viernes, 5 de abril de 2013
Well-posed and ill-posed regimes in μ(I)-rheology for granular materials
Duke University
Progress in understanding granular flow has been greatly hampered by the lack of satisfactory constitutive equations. Historically, the concept of a Coulomb material, based on rate-independent
plasticity, was introduced to describe granular materials. On substitution into the equations for conservation of mass and momentum, this constitutive relation leads to a system of evolution
equations loosely analogous to the Navier-Stokes equations; friction gives rise to a term that formally resembles viscosity. However, it turns out that this system is ill-posed. Numerous
higher-order, non-local theories have been introduced in an attempt to resolve this difficulty; while many of these are well-posed, they are invariably quite complicated, perhaps unnecessarily so.
In the last decade the French school (GDR MIDI) proposed a natural modification of the Coulomb constitutive equation. In this theory the coefficient of friction varies with the shear rate (which is
measured by a nondimensional inertial number $I$); this property leads to the name $\mu(I)$-rheology. Their equation, which is based on experiments of flow down inclined planes and on dimensional
analysis, retains a level of simplicity comparable to Coulomb material.
In this talk we analyze the well-posedness of the governing equations using $\mu(I)$-rheology. Specifically, we show that these evolution equations are well-posed for a large range of deformation
rates but become ill-posed at extremes of slow or fast deformation. It is known that additional effects, not represented in $\mu(I)$-rheology, become important in these two extremes. Thus, the
present mathematical result and physical understanding of granular materials support one another.
On the numerical side, several authors have adapted a recently proposed finite volume method for solving the Navier-Stokes equations to problems with $\mu(I)$-rheology. In this method, the pressure
viscosity contribution is evaluated explicitly; this is appropriate for viscosity in the Navier-Stokes equations (where the viscosity operator is elliptic) but questionable for the
not-necessarily-elliptic operator that occurs in $\mu(I)$-rheology. Reflecting this mismatch, numerical results using this method show no indication of ill-posedness: i.e., they do not reproduce the
stability properties of the PDE derived assuming $\mu(I)$-rheology. To better capture the behavior of the PDE, we propose a PISO-like method that evaluates implicitly the viscous pressure
contributions, and we derive a new pressure equation based on the Schur complement. We present numerical simulations to illustrate that our method does capture ill-posedness as predicted by theory.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.C19 (Edificio Sabatini) Universidad Carlos III
Miércoles, 20 de marzo de 2013
Coupled Mathematical Models for Multi-Phase Materials: Nonlinear Dynamics and Numerical Approximations
Wilfrid Laurier University/Visiting Professor at UC3M
Coupled nonlinear mathematical models are essential in describing most natural phenomena, processes, and man-made systems. From large scale mathematical models of climate to modelling of quantum
mechanical effects coupling and nonlinearity go often hand and hand. Coupled dynamic systems of partial differential equations (PDEs) provide a foundation for the description of many such systems,
processes, and phenomena. In majority of cases, however, their solutions are not amenable to analytical treatments and the development, analysis, and applications of effective numerical
approximations for such models become a core element in their studies.
In this talk we will focus on mathematical models that are based on the Landau framework of phase transformations based on non-monotone free energy functions. Phase transformations are universal
phenomena, and one specific example that we will consider in this talk is motivated by mesoscopic mathematical models for the description of multi-phase solid materials. Such models provide an
intermediate length scale description between the atomistic level and the level that is usually used for bulk materials. In particular, we will discuss several classes of problems where
non-equilibrium phenomena such as phase transformations are important, focusing on the dynamics of materials with shape memory. The talk will provide further insight into their application areas, the
development of computationally efficient reduction procedures for their 3D modelling, and the construction of fully conservative schemes for solving the associated problems.
El Seminario tendrá lugar a las 12:30 horas en la Sala 2.1.D03 (Edificio Sabatini) Universidad Carlos III
Jueves, 7 de marzo de 2013
Elasto-Inertial Turbulence
Monash University
Direct numerical simulations of channel flow with Reynolds numbers ranging from 1,000 to 10,000 (based on the bulk and the channel height) have been used to study the formation and dynamics of
elastic instabilities and their effects on a polymeric flow. The dynamics of turbulence generated and controlled by polymer additives has been investigated from the perspective of the coupling
between polymer dynamics and flow structures.
El Seminario tendrá lugar a las 16:00 horas en la Sala 7.1.H03 (Edificio Juan Benet) Universidad Carlos III
Miércoles, 27 de febrero de 2013
Unsteady characteristics of a shallow porous cylinder wake
Sheffield Fluid Mechanics Group, University of Sheffield
In this work the result of laboratory flow visualisations and Large Scale Particle Image Velocimetry measurements of the wake developed after three emerged square arrays of rigid cylinders in a
shallow water flow are presented. It is observed that for all cases a steady wake is developed downstream the array and it is followed by a vortex street pattern. It is shown that not always higher
porosities produce a more extended steady wake and reduced turbulent intensities. It is also shown that in two cases the dominant wake frequency remain constant, and indication that the solid volume
fractions do not affect the wake frequency. It is also observed that this frequency was also present within the slow steady wake in one of the measured cases, which could be evidence of an
instability initiated within the cylinder array. Based on a Dynamic Mode Decomposition and Wavelet analysis of two and one-dimensional time series a description of the dominant coherent structures in
the near and far field is presented. A discussion regarding the use of fractal arrays will be also presented.
El Seminario tendrá lugar a las 12:30 horas en la Sala 7.1.H01 (Edificio Juan Benet) Universidad Carlos III | {"url":"https://scala.uc3m.es/seminars_2013.html","timestamp":"2024-11-06T05:13:32Z","content_type":"text/html","content_length":"63107","record_id":"<urn:uuid:8137da77-0e8f-4072-ae81-bfc0994bae11>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00283.warc.gz"} |
A least squares method for identification of unknown groundwater pollution source
Least squares method in groundwater pollution sources identification
Groundwater flow and transport simulation
Case 1: sand tank experiment
Case 2: gas station area
Variation of components in the sand tank experiment
Mathematical model
Hydrogeological parameters values
Application steps
Case 1: sand tank experiment
Case 2: gas station area
Application conditions | {"url":"https://iwaponline.com/hr/article/52/2/450/79774/A-least-squares-method-for-identification-of","timestamp":"2024-11-07T17:36:51Z","content_type":"text/html","content_length":"378238","record_id":"<urn:uuid:2d59ca83-303e-418c-8114-39bd64af4055>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00015.warc.gz"} |
Can Conditional Formatting trigger specific formulas?
The spreadsheet in question is a shipping/inventory database.
The top rows where Device SKUs are listed are the total contracted amount of devices a client pays for. Then as we create shipments (which go under the blue parent rows), we've included a SUMIF
formula that, when the Device SKU is entered in the shipment description, the "Remaining Quantity" updates and keeps an accurate count. SUMIF formula pasted below:
=Quantity@row - (SUMIF([Device Sku]:[Device Sku], [Device Sku]@row, Quantity:Quantity) - Quantity@row) + [Spare Devices]@row
The problem that arose yesterday is to do with Change Orders. So, as the name suggests, a change order is when a client wants to add or remove devices. It's important we keep track of this AND that
we maintain an accurate count in relation to "Remaining Quantity". Please see below for the Change Order columns:
As you can see in the screenshot above, the client has added 10 more "On-Off Switches" and then removed 5 "On-Off Switches". Depending on what device the client wants to change will determine where
the CO rows are added. In this case, it's the "On-Off Switch" and they now live as child rows underneath the original.
I do not know if Conditional Formatting offers this functionality, but, what I was hoping to do was that when a quantity is added to "Change Order Quantity" then the "Type of Change" is chosen, that
triggers a formula (Add or Subtract) that updates the total quantity for the device. I've highlighted the cells I would like talking to each other in brown below:
Totally understand if this is not at all possible or if anyone in the community has any suggestions to visualize/calculate the change orders in relation to total quantity and remaining quantity. It's
also possible that I'm automating this spreadsheet too much or I've been staring at it too long 😬.
This community is awesome. Thank you!
• So [Total Quantity]1 should be 201 because you are adding 10 and removing 5? Am I reading that right?
My initial thought is that it can be done. I just need to better understand exactly how you want everything to work together (I may have been staring at my computer too long today too).
• Hey Paul,
Thanks for your help! That's correct, I haven't added or subtracted the 10 and the 5 in the examples above, but yes, the "total remaining quantity" after the two change orders would be 201.
Still can't wrap my head around a way for that to happen automatically. The process (in my head) would be:
1. Input the qty in "Change Order Qty"
2. Select "Change Type"
3. Total qty adjust accordingly
Happy to manually add/subtract but would be very satisfying if it was automated. 😎
• Ok. SO this is actually pretty straightforward if we tackle it in pieces....
We want the total number of Add's:
=SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add")
And the total number of Remove's:
=SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add")
Then we add the Add's and subtract the Remove's (if there aren't any, then the formula(s) will return 0 which won't affect the total anyway).
=original total quantity formula + Add's formula - Remove's formula
=original total quantity formula + SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add") - Remove's formula
=original total quantity formula + SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add") - SUMIFS([Change Order Quantity]:[Change Order Quantity],
[Type of Change?]:[Type of Change?], "Add")
• Magic!!!! Thank you so much Paul.
The only issue I'm having now is that the original formula is subtracting the "Spare Devices Quantity" from "Total Quantity" instead of adding it (I've gone through and removed every other
"Remaining Quantity" should be 200, instead of 192... 🤔
=[Floor Plan Quantity]@row - (SUMIF([Device Sku]:[Device Sku], [Device Sku]@row, [Floor Plan Quantity]:[Floor Plan Quantity]) - [Floor Plan Quantity]@row + [Spare Device Quantity]@row) + SUMIFS
([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add") - SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?],
You've already helped me so much, so I can totally try and figure this one out.
Thanks again!
• No worries at all. All we have to do is use a set of parenthesis so that your first formula runs first and the second part runs second
=(original formula) + Add's - Remove's
=([Floor Plan Quantity]@row - (SUMIF([Device Sku]:[Device Sku], [Device Sku]@row, [Floor Plan Quantity]:[Floor Plan Quantity]) - [Floor Plan Quantity]@row + [Spare Device Quantity]@row)) + SUMIFS
([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?], "Add") - SUMIFS([Change Order Quantity]:[Change Order Quantity], [Type of Change?]:[Type of Change?],
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Addition Of Three Digit Numbers With Regrouping Worksheets 2024 - NumbersWorksheets.com
Addition Of Three Digit Numbers With Regrouping Worksheets
Addition Of Three Digit Numbers With Regrouping Worksheets – The Negative Phone numbers Worksheet is a wonderful way to start off training your young ones the thought of negative phone numbers. A
negative variety is any variety that may be lower than absolutely nothing. It can be extra or subtracted. The minus signal signifies the bad quantity. You can also write unfavorable figures in
parentheses. Below can be a worksheet to help you get began. This worksheet has a variety of negative figures from -10 to 10. Addition Of Three Digit Numbers With Regrouping Worksheets.
Adverse phone numbers are a lot in whose worth is less than zero
A poor quantity includes a importance lower than no. It may be conveyed with a amount collection in just two ways: with the beneficial amount written as being the initially digit, and with the
adverse variety published because the final digit. A good amount is published with a additionally signal ( ) before it, yet it is recommended to publish it that way. If the number is not written with
a plus sign, it is assumed to be a positive number.
They may be depicted by a minus signal
In historic Greece, negative figures have been not used. These folks were ignored, his or her math was based on geometrical methods. When Western scholars started translating old Arabic messages from
To the north Africa, they arrived at acknowledge unfavorable figures and appreciated them. Today, negative figures are represented by a minus signal. To learn more about the origins and history of
unfavorable phone numbers, check this out report. Then, attempt these illustrations to discover how adverse numbers have advanced over time.
They can be added or subtracted
As you might already know, positive numbers and negative numbers are easy to add and subtract because the sign of the numbers is the same. Negative numbers, on the other hand, have a larger absolute
value, but they are closer to than positive numbers are. These numbers have some special rules for arithmetic, but they can still be added and subtracted just like positive ones. You may also add and
subtract negative amounts utilizing a quantity series and use a similar guidelines for subtraction and addition while you do for optimistic numbers.
These are symbolized from a amount in parentheses
A poor amount is symbolized by way of a quantity enclosed in parentheses. The unfavorable signal is changed into its binary comparable, along with the two’s accentuate is kept in a similar place in
recollection. The result is always negative, but sometimes a negative number is represented by a positive number. In such cases, the parentheses should be included. You should consult a book on math
if you have any questions about the meaning of negative numbers.
They may be divided by a good quantity
Negative numbers can be multiplied and divided like beneficial phone numbers. They may also be divided up by other negative figures. They are not equal to one another, however. The very first time
you grow a negative quantity from a beneficial amount, you will get absolutely nothing consequently. To produce the perfect solution, you must select which indicator your answer needs to have. It is
simpler to recall a poor amount when it is developed in brackets.
Gallery of Addition Of Three Digit Numbers With Regrouping Worksheets
Addition Within 1000 Check In Worksheets 99Worksheets
Addition Practice Triple Digits Worksheets 99Worksheets
3 Digit Addition Worksheet With Regrouping Set 3 Resource Math Triple
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Could there be an "i" of the tetrative operations?
Hello. One day I was browsing Youtube, and I noticed a video that sprung me into thinking about tetration somehow. The Youtube video was a low effort math video and it said "Solve -a = 1/a", it will
take anyone some basic algebra to figure out that a = i, thus, -i = 1/i. This was interesting to me, because I found it interesting the inverse orders of operations of different orders of magnitude
have a result for a value which is the same and is a "unit" with magnitude 1 so to speak. This got me thinking about every "unit of dimensionality" for each operation in math. Let me know if you see
a pattern:
0+1 = 1
1*-1 = -1
-1^(1/2) = i
So the next level in this step would be either the 1/2 tetration power of i or the 1/3 tetration power of i. This means that i = x^x or i = x^x^x.
The latter didn't net me any decent results, but I plugged i=x^x into Wolfram Alpha but I couldn't interpret it.
What I am adding to this tetration board is a postulate! I am postulating that there is an 'i' of tetration, where, like
1 = -(-1)
-i = 1/i,
there is a unit where
(1/x) = tetration(x,(1/2)),
This unit would follow the former pattern in the first list.
I got my undergrad in CS, so I may be a little out of my league, but let me know what you guys think.
07/12/2024, 06:26 AM
Hi 000_era, welcome to the tetration forum. Sadly nowadays the forum is only just an archive of old discussion and is not very active anymore, read as "completely dead". Hopefully it will comeback,
if enough new users get curious about these topics. To this to be a possibility we poster should do our best to favour engagement and questions: by being respectful of ppl's time and kind. This
translates in putting effort in our questions by doing the homeworks, being as clear as we can, eg. by trying to define things and use consistent notation. Something that sometimes was missing in the
Excuse me for my long welcome message. In the near future we will set up some mild moderation practices and set of guidelines for new users.
Back to your question. I'm not sure of the analogy you see there. If I recall correctly one can argue that there are not equations containing tatration functions that are not solved by complex
numbers or by limits of sequences of complex numbers... so no new sets of numbers I guess... anyways I never understood if there was a proof or just some heuristics.
Can you make more precise what kind of analogy/scheme you see? For example I don't get why the role of the \(1/2\) doesn't break completely any scheme.
Btw: here you can display math notation just by enclosing your equations by \ ( your_equation \ ) as follows:
\( (a+b)^n\)
renders as \( (a+b)^n\).
MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\) | {"url":"https://tetrationforum.org/showthread.php?tid=1788&pid=12251","timestamp":"2024-11-10T09:25:59Z","content_type":"application/xhtml+xml","content_length":"26219","record_id":"<urn:uuid:5c9cd43a-b66f-4b05-8951-75f5f69a709a>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00605.warc.gz"} |
Understanding that comparing numbers involves knowing which numbers are worth more or less than each other
Early Years
Early Years Materials
Comparing numbers involves knowing which numbers are worth more or less than each other. This depends both on understanding cardinal values of numbers and also knowing that the later counting numbers
are worth more (because the next number is always one more). This understanding underpins the mental number line which children will develop later, which represents the relative value of numbers,
i.e. how much bigger or smaller they are than each other.
Typical progression of key ideas in this concept
Children need progressive experiences where they can compare collections and begin to talk about which group has more things. Initially, the groups need to be very obviously different, with one group
having a widely different number of things. Collections should also offer challenges, such as including more small things and fewer large things, to draw attention to the numerosity of the
comparison, i.e. the number of things, not the size of them.
Activities and opportunities:
• collections for children to sort and compare, which include objects which are identical, and which include objects of different kinds or sizes
• collections with a large number of things, and collections with a small number of things.
Children need the opportunity to see that groups could consist of equal numbers of things. Children can check that groups are equal, by matching objects on a one-to-one basis.
Activities and opportunities:
• ensuring that when providing groups to compare, there are some that have an equal amount
• asking children to convert two unequal groups into two that have the same number, e.g. ‘There are 6 apples in one bag and 2 in another bag; can we make the bags equal for the two hungry horses?’
Children need opportunities to apply their understanding by comparing actual numbers and explaining which is more. For example, a child is shown two boxes and told one has 5 sweets in and the other
has 3 sweets in. Which box would they pick to keep and why? Look for the reasoning in the response they give, i.e. ‘I would pick the 5 box because 5 is more than 3 and I want more.’ If shown two
numerals, children can say which is larger by counting or matching one-to-one.
Children can compare numbers that are far apart, near to and next to each other. For example, 8 is a lot bigger than 2 but 3 is only a little bit bigger than 2.
Activities and opportunities:
• explain unfair sharing - 'This one has more because it has 5 and that one only has 3'
• compare numbers that are far apart, near to, and next to each other.
Children need opportunities to see and begin to generalise the ‘one more than/one less than’ relationship between sequential numbers. They can apply this understanding by recognising when the
quantity does not match the number, i.e. if a pack is labelled as 5 but contains only 4, the children can identify that this is not right. Support children in recognising that if they add one, they
will get the next number, or if one is taken away, they will have the previous number. For example: ‘There are 4 frogs on the log, 1 frog jumps off. How many will be left? How do you know?'
Activities and opportunities
• labelling groups with the correct numeral. Do children spot the error if a group is mislabelled? For example, 'The label on the pot says 4 and we have 5 – what do we need to do?’ A child may say,
‘We need to take one out because we have one too many.’
• ensuring children focus on the numerosity of the group by having items in the collection of different kinds and sizes
• making predictions about what the outcome will be in stories, rhymes and songs if one is added to, or if one is taken away.
• children not comparing the numerosity of the group and considering more in terms of size
• children giving a response that does not match the context when estimating a number; e.g. when adding, giving as an answer a number that is smaller than the numbers given. Example: ‘There are 7
cars in a garage and then 2 more go in.’ The child guesses there are 4 cars in total inside.
Can a child:
• state which group of objects has more? Can they do this with a large or small visual difference?
• compare two numbers and say which is the larger?
• predict how many there will be if you add or take away one?
Is this page useful? Yes No
Was this written in plain English? Yes No
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Multiples of 10—Tree House Worksheet | 3.NBT.A.3 | Workybooks
About the "Multiples of 10: Multiplication" Worksheet
Multiples of 10: Multiplication is an engaging math worksheet that combines learning with a fun narrative context. This interactive and printable worksheet presents eight multiplication problems
involving multiples of 10, framed within the story of Sunny and his brother building a treehouse. The problems are designed to help students practice multiplying two-digit multiples of 10 by
single-digit numbers, an essential skill for developing a strong foundation in multiplication.
The worksheet cleverly integrates math practice with a relatable scenario, potentially increasing student engagement and motivation. This interactive and printable worksheet encourages students to
visualize the practical applications of multiplication in real-life situations. By solving these problems, students can improve their computational fluency, recognize patterns in multiplying by 10s,
and develop mental math strategies. The clear layout and entertaining context make this worksheet an excellent tool for both classroom use and independent practice at home.
What will your child learn through this worksheet?
• Multiplication of multiples of 10 by single-digit numbers
• Recognition of patterns when multiplying by 10s
• Application of multiplication in a practical context
• Development of mental math strategies for efficient calculation
Learning Outcomes
• Correctly solve at least 7 out of 8 multiplication problems involving multiples of 10 within 12 minutes
• Demonstrate understanding of the relationship between multiplying by 10s and basic multiplication facts
• Write numerical answers clearly and accurately in the provided spaces
• Improve fine motor skills through precise number writing and problem-solving activities
• Express increased confidence in tackling multiplication problems with multiples of 10
• Show enthusiasm for math practice through engagement with the treehouse-building narrative
• Participate in collaborative discussions about strategies for multiplying multiples of 10
• Explain problem-solving methods to peers, fostering communication skills
multiplication, multiples of 10, math practice, elementary math, mental math, number patterns, computational fluency, story-based math, math worksheets, multiplication strategies | {"url":"https://www.workybooks.com/worksheet/3.NBT.A.3-2/multiplication-multiples-of-10-tree-house","timestamp":"2024-11-03T16:23:27Z","content_type":"text/html","content_length":"91984","record_id":"<urn:uuid:0f9fc807-91b1-496a-8cc4-7afc7906ba25>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00313.warc.gz"} |
How to predict with Bayes, MDL, and Experts
Published on Feb 25, 20077554 Views
Most passive Machine Learning tasks can be (re)stated as sequence prediction problems. This includes pattern recognition, classification, time-series forecasting, and others. Moreover, the
How to Predict with Bayes, MDL, and Experts00:44
Table of Contents07:59
Philosophical Issues: Contents08:44
Philosophical Issues: Abstract09:21
On the Foundations of Machine Learning09:22
Example 1: Probability of Sunrise Tomorrow11:50
Example 2: Digits of a Computable Number15:29
Example 3: Number Sequences17:15
Occam's Razor to the Rescue20:57
Foundations of Induction21:58
Problem Setup23:22
Dichotomies in Machine Learning25:15
Sequential/online predictions27:53
Bayesian Sequence Prediction: Contents30:16
Bayesian Sequence Prediction: Abstract31:11
Uncertainty and Probability31:13
Frequency Interpretation: Counting32:01
Objective Interpretation: Uncertain Events33:12
Subjective Interpretation: Degrees of Belief35:01
Bayes' Famous Rule37:31
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Computer Arithmetic: Volume I
English | 2015 | ISBN: 978-9814651561 | 396 Pages | PDF | 19 MB
The book provides many of the basic papers in computer arithmetic. These papers describe the concepts and basic operations (in the words of the original developers) that would be useful to the
designers of computers and embedded systems. Although the main focus is on the basic operations of addition, multiplication and division, advanced concepts such as logarithmic arithmetic and the
calculations of elementary functions are also covered.
Readership: Graduate students and research professionals interested in computer arithmetic.
Download from free file storage
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Given the probability p=0.84 that an event will not happen, how do you find the probability that the event will happen? | Socratic
Given the probability p=0.84 that an event will not happen, how do you find the probability that the event will happen?
1 Answer
From the definition of probability P we have
P (event does happen) = 1 - P (event does not happen)
P (event does happen) = 1 - 0.84=0.16
Footnote .The probability of an event not happening is also known as the complement of an event happening.
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Relative Squared Error
Last update: February 19, 2014
Relative Squared Error
The relative squared error (RSE) is relative to what it would have been if a simple predictor had been used. More specifically, this simple predictor is just the average of the actual values. Thus,
the relative squared error takes the total squared error and normalizes it by dividing by the total squared error of the simple predictor.
Mathematically, the relative squared error E[i] of an individual model i is evaluated by the equation:
where P[(ij)] is the value predicted by the individual model i for record j (out of n records); T[j] is the target value for record j; and
For a perfect fit, the numerator is equal to 0 and E[i] = 0. So, the E[i] index ranges from 0 to infinity, with 0 corresponding to the ideal.
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Potential Energy Calculator
What is Potential energy?
Potential energy is one of the two types of energy. It is the amount of energy stored in an object when work is done on it. As energy can neither be created nor destroyed so work is done, which is
also a form of energy, is converted into potential energy. Potential energy is always with respect to some reference level. A body can both possess or have zero potential energy from different
Potential energy Example
Consider a ball is placed on a wall which is 10 feet higher from the ground.
According to a person standing on the wall, the ball has zero potential energy. But from the point of view of a person on the ground ball has some Potential energy.
Types of potential energy
There are two basic types of potential energy.
• Gravitational potential energy
• Elastic potential energy
Gravitational potential energy:
It is the energy stored in an object when its position changed with respect to the ground.
The gravitational potential energy formula is
\(G.P.E = mg\Delta h\)
• m is mass
• g is the gravitational acceleration
• h is the difference in the height of two surfaces
Where mass is in kilograms, acceleration in meter per second square
and height in meters.
Some examples of gravitational potential energy are
• Helicopter flying above the ground
• Glass placed on a countertop
• A flying kite
• Nest on a tree
How to calculate gravitational potential energy?
You can calculate gravitational potential energy manually by following these instructions.
Multiply mass with gravitational acceleration, which is 9.8ms^-2 at sea level, with the difference of heights according to the potential energy equation.
How to use our potential energy calculator?
Our calculator is the best tool you should use when you are calculating gravitational potential energy. Below are the steps through which you can calculate gravitational potential energy through our
• Enter mass
• Enter gravitational acceleration
• Enter height
• Select unit
• Click calculate
Now, it might seem that is a lot of work but when you get on with it will take, at most, 3 minutes. I think its more convenient than spending 10 minutes on paper to calculate it.
Elastic potential energy:
It is the energy stored when a body is compressed or stretched, changing its original shape.
Mathematically it is calculated through this formula.
\(EPE = \dfrac{1}{2} kx^2\)
• K is elastic constant
• x is displacement
Elastic limit:
It is the limit to which an object can be stretched without causing permanent damage to the object.
Change in potential energy:
If a body possessing some potential energy again undergoes some work, it experiences a change in its stored amount of energy. This change is calculated through this formula
\(\Delta P.E = \Lambda P .E_f - Delta P.E_i\)
• \(P.E_i\) is the initial potential energy
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OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Click on the 10 question exam covering topics in chapters 1 and 2. Give yourself one hour to try all of the problems and then come back and check your answers.
Solve and graph all solutions on a number line.
10. The perimeter of a rectangle is 54 feet. If the length is 3 feet less than twice the width, find the dimensions of the rectangle. (
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Question #76612 | Socratic
Question #76612
1 Answer
The dilution factor is simply a measure of how concentrated the stock solution was compared with the diluted solution.
You can express the dilution factor as a ratio of concentrations and as a ratio of volumes
$\text{DF" = c_"stock"/c_"diluted" = V_"diluted"/V_"stock}$
As you can see, the dilution factor is equal to the ratio between the volume of the diluted solution and the volume of the stock solution.
In your case, the stock solution has a volume of $\text{99.99 mL}$. To find the volume of the diluted solution, you add the volume of diluent, which is the substance you're using to dilute the stock
solution with.
${V}_{\text{diluted" = V_"stock" + V_"diluent}}$
You will have
${V}_{\text{diluted" = "99.99 mL" + "0.1 mL" = "100.09 mL}}$
Therefore, the dilution factor will be
#"DF" = (100.09 color(red)(cancel(color(black)("mL"))))/(99.99color(red)(cancel(color(black)("mL")))) = 1.001#
However, you only have one significant figure for the volume of the diluent, so you must say that
$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{DF} = 1}}}$
However, if the volume of the stock solution is $\text{0.1 mL}$ and the volume of the diluent is $\text{99.99 mL}$, the dilution factor is
#"DF" = (100.09 color(red)(cancel(color(black)("mL"))))/(0.1color(red)(cancel(color(black)("mL")))) = 10,009#
Rounded to one significant figure, you will have
$\text{DF} = 10 , 000$
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Monthly Assignment #1: Bottles - Page 4
Monthly Assignment #1: Bottles
Well, I couldn't find a "sexier" bottle, so I gave the same ole bottle a boa
I tried the lighting from underneath as suggested and I really like the way the bottom is lit up. I tried different variations of flashing the front of the bottle so that the top of the bottle
can be seen as well as the label on the front.
My setup just isn't cutting it. Every time I flash with the handheld flash, the reflection of ceiling fans, etc. show up on the bottle. I really need a closed environment like some of the other
setups shown.
I'm not giving up on this. I'm determined to get at least one good shot
Here are the best 2 from tonight:
I flashed this one once from almost directly overhead and the strobe light is underneath. I think I if can get a white background inside an enclosure and light from top and bottom that it'll turn
out well.
I tore my setup apart before realizing I should have taken a picture of it. oops... it's essentially the same setup, except as suggested, I split the tv trays and put the strobe underneath,
covered it with a piece of paper. I wrapped a boa around the bottle for lack of a better background. I used the same black box as before. This time I shot the bottle vertically and aligned with
the box as suggested (figured out a way to do it with my tripod, although it's a real pain).
I think the white backgrounds others were using give the shot a much cleaner look. I'm not diggin' my black box background. I'll keep an eye out for a way to get a white background setup.
Well, I couldn't find a "sexier" bottle, so I gave the same ole bottle a boa
I'm sure Ken will pitch in, but in the mean time...
With the glass you pretty much cannot have the primary light in front of it AND have a background at the same time. YOu can have SOME light next to the camera to light up a label, but the main
idea is the have it some place else - especially with the back b/g.
I'd like to mention Da Book again - it's awesome, simple and very practical.
"May the f/stop be with you!"
Yes... me again
I'm sure Ken will pitch in, but in the mean time...
With the glass you pretty much cannot have the primary light in front of it AND have a background at the same time. YOu can have SOME light next to the camera to light up a label, but the
main idea is the have it some place else - especially with the back b/g.
For some reason, what you said there clicked for me Nikolai... thank you. I went back and re-read some posts and looked at some of the other setups. I think I got some pretty good results this
time. There are still a few flaws... mostly with my wrinkly background, which I'm not sure how to fix. I'm not sure I can iron the material and it wasn't mine, so I was afraid to ruin it.
I COMPLETELY forgot to take a picture of my setup
Hopefully you guys aren't sick of me trying yet
For some reason, what you said there clicked for me Nikolai... thank you. I went back and re-read some posts and looked at some of the other setups. I think I got some pretty good results
this time. There are still a few flaws... mostly with my wrinkly background, which I'm not sure how to fix. I'm not sure I can iron the material and it wasn't mine, so I was afraid to ruin
I can see the improvement!
The major issue now becomes the size of the BG. It should be only big enough to fill the frame. You need this to create distinctive lines in the bottle.
And you can shoot the setup again later
"May the f/stop be with you!"
SloYerRoll Registered Users Posts: 2,788 Major grins
Don't look at these attempts as failures. Look at them as all the different ways you know how to not light glass. (If it worked for Thomas Edison, I'm sure it will work for you)
If you'd like, I'll set up my rig again and take some more detailed shots so you can really disect it. I genuinely think you don't need this though. You're starting to get the theory, now just
take your time and really dial all those details in.
A LITTLE LIGHTING OT:
As a note so you can better controll your lighting:
Your Aperture controlls your strobes.
Your shutter speed controlls that ambient light.
This is one of those aha! facts that will help you take your photog skills to the next level. So memorize it now.
If you want to leave your strobe at the same power but want it to be "darker" in your shot. Adjust your Aperture to a smaller fstop (increase the f number)
3rd shot from the bottom.
I've taken the shot and everything looks good but the background is a little bit bright and I want to dim it down.
My camera settings are:
ISO 100
What setting should I change to make the background darker? look below AFTER you have answered the question.
change the f stop to a higher number. This closes the diaphram that lets the light in hitting the sensor.
You can also change the distance of your strobe to subject distance. But when you do this you also change the dynamics of the light since it is now hitting the subject differently (which isn't
bad. it just changes more than the ammount of light hitting y our sensor)
A LITTLE MORE LIGHTING OT:
This is also a great way to conserve your strobes power. If you have your strobe set at 1/16 and you have your aperature set to f27 (one sixteenth of your strobes capacitors energy will be used w
/ each shot so in theory you can hammer on it 16 times before it has to recharge). This is equivelant to having your strobe set to 1/4 power and your aperature set at f11 in reagards to lighting.
(one quarter of your strobes capacitors energy will be used w/ each shot so in theory you can pop the strobe on it 4 times before it has to recharge).
So opening up your aperature can give you more bang for your buck. In this situation it makes almost no difference what you do. But when you are in a situation where you need to get allot of
shots in short periods of time. THis can make the difference between sucess and failure.
I can see the improvement!
The major issue now becomes the size of the BG. It should be only big enough to fill the frame. You need this to create distinctive lines in the bottle.
And you can shoot the setup again later
I'm having space issues with that concept. However, in looking back at other people's setup's they have black material directly on the side of their subject (i.e., Antonio's trashbag and a couple
others'). I wonder if I put black objects on either side of my bottles if the lines of my bottle would be better?
Thanks again for the feedback...
Don't look at these attempts as failures. Look at them as all the different ways you know how to not light glass. (If it worked for Thomas Edison, I'm sure it will work for you)
If you'd like, I'll set up my rig again and take some more detailed shots so you can really disect it. I genuinely think you don't need this though. You're starting to get the theory, now
just take your time and really dial all those details in.
A LITTLE LIGHTING OT:
As a note so you can better controll your lighting:
Your Aperture controlls your strobes.
Your shutter speed controlls that ambient light.
This is one of those aha! facts that will help you take your photog skills to the next level. So memorize it now.
If you want to leave your strobe at the same power but want it to be "darker" in your shot. Adjust your Aperture to a smaller fstop (increase the f number)
3rd shot from the bottom.
I've taken the shot and everything looks good but the background is a little bit bright and I want to dim it down.
My camera settings are:
ISO 100
What setting should I change to make the background darker? look below AFTER you have answered the question.
change the f stop to a higher number. This closes the diaphram that lets the light in hitting the sensor.
You can also change the distance of your strobe to subject distance. But when you do this you also change the dynamics of the light since it is now hitting the subject differently (which
isn't bad. it just changes more than the ammount of light hitting y our sensor)
A LITTLE MORE LIGHTING OT:
This is also a great way to conserve your strobes power. If you have your strobe set at 1/16 and you have your aperature set to f27 (one sixteenth of your strobes capacitors energy will be
used w/ each shot so in theory you can hammer on it 16 times before it has to recharge). This is equivelant to having your strobe set to 1/4 power and your aperature set at f11 in reagards to
lighting. (one half of your strobes capacitors energy will be used w/ each shot so in theory you can pop the strobe on it 2 times before it has to recharge).
So opening up your aperature can give you more bang for your buck. In this situation it makes almost no difference what you do. But when you are in a situation where you need to get allot of
shots in short periods of time. THis can make the difference between sucess and failure.
It's funny that you expounded on that because I was noticing some odd things while I was shooting. I started out with the strobe on full blast, which gave me a very white background and I liked
that. Then I changed bottles and I wanted to decrease the light a bit, so I thought by turning the frequency of the strobe flash down that my shots would get darker, but it actually had the
opposite effect. That really surprised me.
I also switched from Aperature priority to Shutter priority and the results were not what I expected either. Since I had already used A so much and felt like I was getting the hang of changing
settings, I just went back to A
Thanks for the deeper explanation... I'll try to get that to sink in.
Beer bottle - light background
Here is one picture I did with a beer bottle. I like having a white background. However the labels are too dark. I used a diffuser in the back with two reflectors up front.
I tried to lighten the labels with a light from the front or even overhead. However this caused too many reflections that I could not control. I'd like suggestions on how to light the front of
the bottle without getting the reflections.
First picture: without pp.
Second picture: with pp to light the front of the bottle and remove some reflections.
LiquidAir Registered Users Posts: 1,751 Major grins
I think the white backgrounds others were using give the shot a much cleaner look. I'm not diggin' my black box background. I'll keep an eye out for a way to get a white background setup.
Clean black backgrounds are tough. When I first started shooting glass (for LPS#1), I spend several hours fighting with my setup and still ended up touching up the background in Photoshop. It
does get easier with practice; now I can pretty consistantly hit a 0,0,0 background right out of camera.
One thing to remember is that a black paper or cardboard typically run only about 2 stops below middle grey. Since most DSLRs are sensitive to around 5 stops below middle grey, you will need to
use light control to hit true black. You don't have to completely control scatter, but you do want the light on the background to be at least 3 stops darker than the light on your subject
assuming you are metering for diffuse reflections. With bottles where you are metering for specular or refracted light you sometimes don't need that much control. Refracted light is almost always
brighter than diffuse reflections so that case is relatively easy. The brightness of diffuse reflections depend on both the nature of your surface and how oblique the angle of incidence is. I
have seen specular reflections go from maybe 2 stops darker than normal metering to about 4 stops brighter. If you have darker reflections, you have to be very careful about scatter on your
The best black background is actually something like a deep cardboard box. Place the box behind your subejct in a such a way that the camera is looking into the box. The box helps you in two
ways: it shields your background from light scattered in the room and it places the background further from your subject which makes scatter off your subject appear darker because the light has
to travel farther. You can then run your floor back into the box which tyipcally means you don't need to worry about your floor to backdrop transition because it will be too dark to be visible.
The box is is overkill for a lot of subjects. I have a good sized sheet of black felt which works quite well, but it is worth making sure the felt is held in a way that it stays smooth; folds and
wrinkles will kill you. Sometimes I use black foam core which is nice because it is stiff enough to be easy to mount. However the surface of the foam core is a bit shiny so you have to watch out
for specular reflections. I also use a Photoflex black/sliver lite disc (often with a mount which lets me put it on a light stand). While I find the sliver side nearly useless, the black side
serves both as a handy mobile background and as an adjustable gobo.
Here is one picture I did with a beer bottle. I like having a white background. However the labels are too dark. I used a diffuser in the back with two reflectors up front.
I tried to lighten the labels with a light from the front or even overhead. However this caused too many reflections that I could not control. I'd like suggestions on how to light the front
of the bottle without getting the reflections.
Thank you! I do like the second (pp) version better.
Re: labels: it is hard
"May the f/stop be with you!"
Last of the beer
I'll have to try again with the snoots. I ended up just placing the light up and to camera right with a reflector to the left. This lit the labels at least some but left some reflections that I
did not like . I did some pp to even out the colors.
Here is another which I like but it is off topic for showing the edges of a bottle.
I'll have to try again with the snoots. I ended up just placing the light up and to camera right with a reflector to the left. This lit the labels at least some but left some reflections that
I did not like . I did some pp to even out the colors.
I like the first one, great job!
"May the f/stop be with you!"
SloYerRoll Registered Users Posts: 2,788 Major grins
Total EDIT: sorry. I though you were Snakeroot. I saw Beginner Grinner and didn't look back.
DUUUUDE! look at your first attempts compared to these! A whole other universe. THere's always room for improvement. But you are getting it!
Now I'm gonna have to shoot some more so you don't show me up
SETUP SHOT SETUP SHOT SETUP SHOT SETUP SHOT SETUP SHOT.....
Thanks for the feedback. I just purchased a couple of bottles of wine. One a rose and the other a chardonnay so the color will come through easier. I'll try with those later this weekend.
Thanks for the feedback. I just purchased a couple of bottles of wine. One a rose and the other a chardonnay so the color will come through easier. I'll try with those later this weekend.
Darn, my sponsor at AA would kill me
"May the f/stop be with you!"
Total EDIT: sorry. I though you were Snakeroot. I saw Beginner Grinner and didn't look back.
Wine - Rose and Chardonnay
Here are the wine bottles. I bought them based on the label. Typically my wife and I are box wine drinkers but I decided to purchase glass just for this.
I did some PP because I could not get rid of some reflections from the top of the bottles. For the Chardonnay, I placed two flashes left and right above the bottles. For the Rose I bounced a
single light off the ceiling from behind the camera.
I tried using a gobo above the wine bottles to get rid of the reflections but I could not figure out where to place to gobo correctly. I think I'll finally try a snoot but I still have trouble
aiming a homemade snoot correctly.
Here are the wine bottles. I bought them based on the label. Typically my wife and I are box wine drinkers but I decided to purchase glass just for this.
I did some PP because I could not get rid of some reflections from the top of the bottles. For the Chardonnay, I placed two flashes left and right above the bottles. For the Rose I bounced a
single light off the ceiling from behind the camera.
I tried using a gobo above the wine bottles to get rid of the reflections but I could not figure out where to place to gobo correctly. I think I'll finally try a snoot but I still have
trouble aiming a homemade snoot correctly.
I like it!
Couple of things:
□ Watch your horizon level, both entries look a bit skewed CCW, esp. #1
□ I still cannot get rid of a feeling that tha bg is too large, hence the outlines are not as distinct as they could have been.
□ And what happened with the required setup pictures? Didn't it change even a bit?
"May the f/stop be with you!"
Wine with Dark Backgrounds
Here are some wine bottles with a dark background. I stuggled to get the background and floor dark. I ended up darkening the background in pp.
I used a stobe behind the bottle shot through a diffuser. I put the gobo on the diffuser. I noticed that I got better results if I made the gobo much taller than the wine bottle and field of
I do not like either of these two pictures much. Any suggestions are welcome.
Here is the set up shot
Here are some wine bottles with a dark background. I stuggled to get the background and floor dark. I ended up darkening the background in pp.
I used a stobe behind the bottle shot through a diffuser. I put the gobo on the diffuser. I noticed that I got better results if I made the gobo much taller than the wine bottle and field of
I do not like either of these two pictures much. Any suggestions are welcome.
Here is the set up shot
Thank you! I like the setup!
I also think you got rather decent results. It seems what you need is to add a snooted direct (foreground) light for the label and maaaaybe another one on the side to get a small sparky
relection. To deal with the background maybe it's worth to increase the distance between the gogo and the diffuser, i.e. keeping the gobo close and the diffuser far.
"May the f/stop be with you!"
New monthly assignment
You're not waiting until I get the first month's assignement right before you post a new one are you?
I will get back to it! I will!!
You're not waiting until I get the first month's assignement right before you post a new one are you?
I will get back to it! I will!!
Nope. We do not start on first:-)
"May the f/stop be with you!"
This time the dark background was easy but the white one was/is very difficult for me.
I was not going to post here the white background because it is a flop, a real disaster.:cry
But on the other hand, the errors are part of the leaning process.
My problem is the white balance. I am using a torch light which has a much different temperature from the flash as you can see.
Thinking of this problem I have bought an orange gell but I had no time to prepare it yet. However I decided to go ahead and here is the disaster.:cry
Wine bottle - dark background with a snoot.
Here is the wine bottle with a dark background and a snoot added. I also placed a blue light on the background. I adjusted the brightness levels in Capture NX. Also I eliminated a reflection in
the upper half of the bottle that I could not get rid of.
Here is the setup shot. I used a diffuser between the back light and the gobo. I had to hand hold the diffuser so it is not in this shot. Also the snoot on the label is from overhead. The snoot
is hard to see in the picture.
This time the dark background was easy but the white one was/is very difficult for me.
I was not going to post here the white background because it is a flop, a real disaster.:cry
But on the other hand, the errors are part of the leaning process.
My problem is the white balance. I am using a torch light which has a much different temperature from the flash as you can see.
Thinking of this problem I have bought an orange gell but I had no time to prepare it yet. However I decided to go ahead and here is the disaster.:cry
Antonio, I like it much better!
There are stilll some issues with the evenness of the white b/g, and I also think that labels in the dark version could be lit better, but it's a definite improvement!
"May the f/stop be with you!"
Here is the wine bottle with a dark background and a snoot added. I also placed a blue light on the background. I adjusted the brightness levels in Capture NX. Also I eliminated a reflection
in the upper half of the bottle that I could not get rid of.
Here is the setup shot. I used a diffuser between the back light and the gobo. I had to hand hold the diffuser so it is not in this shot. Also the snoot on the label is from overhead. The
snoot is hard to see in the picture.
Thank you!
Looks like the label light is a bit too much, neh?
"May the f/stop be with you!"
Haven't had a chance to get back to this for a while so thought I'd take another shot at it. The set up is the same as others I've already posted so I didn't add it here. I did move the flash a
little closer to the difuser to try and take care of the problem Nik pointed out about not having enough light. I guess it's better but still needs some work. Also added a piece of glass for the
table instead of the press board I was using. Other than that, no difference from my previous attempts.
Ok ... I know this doesn't really count as a bottle, but I think I finally got my lighting to work a little better so posted it anyway. Same set up, just moved the lighting and the gobo around to
get the diffuser lit better. I actually took this one before the previous post but didn't mess around with it till later. I must have moved my lighting around before shooting the bottle.
Nice onces, Dave!
First one a bit uneven on the bg, but nice anyway!
"May the f/stop be with you!"
Nice onces, Dave!
First one a bit uneven on the bg, but nice anyway!
Thanks Nik. I'm definitely learning a lot here. | {"url":"https://dgrin.com/discussion/68491/monthly-assignment-1-bottles/p4","timestamp":"2024-11-02T12:28:52Z","content_type":"text/html","content_length":"390604","record_id":"<urn:uuid:f0e1e5b8-3bc5-430b-a4ee-1b0a2e6fbd9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00822.warc.gz"} |
Real homography
A simple example of a Cayley transform can be done on the real projective line. The Cayley transform here will permute the elements of {1, 0, −1, ∞} in sequence. For example, it maps the positive
real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions.
As a real homography, points are described with projective coordinates, and the mapping is
${\displaystyle [y,\ 1]=\left[{\frac {x-1}{x+1}},\ 1\right]\thicksim [x-1,\ x+1]=[x,\ 1]{\begin{pmatrix}1&1\\-1&1\end{pmatrix}}.}$
Complex homography
Cayley transform of upper complex half-plane to unit disk
On the upper half of the complex plane, the Cayley transform is:^[1]^[2]
${\displaystyle f(z)={\frac {z-i}{z+i}}.}$
Since ${\displaystyle \{\infty ,1,-1\}}$ is mapped to ${\displaystyle \{1,-i,i\}}$ , and Möbius transformations permute the generalised circles in the complex plane, ${\displaystyle f}$ maps the real
line to the unit circle. Furthermore, since ${\displaystyle f}$ is a homeomorphism and ${\displaystyle i}$ is taken to 0 by ${\displaystyle f}$ , the upper half-plane is mapped to the unit disk.
In terms of the models of hyperbolic geometry, this Cayley transform relates the Poincaré half-plane model to the Poincaré disk model.
In electrical engineering the Cayley transform has been used to map a reactance half-plane to the Smith chart used for impedance matching of transmission lines.
Quaternion homography
In the four-dimensional space of quaternions ${\displaystyle a+b{\vec {i}}+c{\vec {j}}+d{\vec {k}}}$ , the versors
${\displaystyle u(\theta ,r)=\cos \theta +r\sin \theta }$ form the unit 3-sphere.
Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written ${\displaystyle U[a,b]}$ to indicate that the homogeneous factor multiplies on the left.
The quaternion transform is
${\displaystyle f(u,q)=U[q,1]{\begin{pmatrix}1&1\\-u&u\end{pmatrix}}=U[q-u,\ q+u]\sim U[(q+u)^{-1}(q-u),\ 1].}$
The real and complex homographies described above are instances of the quaternion homography where ${\displaystyle \theta }$ is zero or ${\displaystyle \pi /2}$ , respectively. Evidently the
transform takes ${\displaystyle u\to 0\to -1}$ and takes ${\displaystyle -u\to \infty \to 1}$ .
Evaluating this homography at ${\displaystyle q=1}$ maps the versor ${\displaystyle u}$ into its axis:
${\displaystyle f(u,1)=(1+u)^{-1}(1-u)=(1+u)^{*}(1-u)/|1+u|^{2}.}$
But ${\displaystyle |1+u|^{2}=(1+u)(1+u^{*})=2+2\cos \theta ,\quad {\text{and}}\quad (1+u^{*})(1-u)=-2r\sin \theta .}$
Thus ${\displaystyle f(u,1)=-r{\frac {\sin \theta }{1+\cos \theta }}=-r\tan {\frac {\theta }{2}}.}$
In this form the Cayley transform has been described as a rational parametrization of rotation: Let ${\displaystyle t=\tan \phi /2}$ in the complex number identity^[3]
${\displaystyle e^{-i\varphi }={\frac {1-ti}{1+ti}}}$
where the right hand side is the transform of ${\displaystyle ti}$ and the left hand side represents the rotation of the plane by negative ${\displaystyle \phi }$ radians.
Let ${\displaystyle u^{*}=\cos \theta -r\sin \theta =u^{-1}.}$ Since
${\displaystyle {\begin{pmatrix}1&1\\-u&u\end{pmatrix}}\ {\begin{pmatrix}1&-u^{*}\\1&u^{*}\end{pmatrix}}\ =\ {\begin{pmatrix}2&0\\0&2\end{pmatrix}}\ \sim \ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}\
where the equivalence is in the projective linear group over quaternions, the inverse of ${\displaystyle f(u,1)}$ is
${\displaystyle U[p,1]{\begin{pmatrix}1&-u^{*}\\1&u^{*}\end{pmatrix}}\ =\ U[p+1,\ (1-p)u^{*}]\sim U[u(1-p)^{-1}(p+1),\ 1].}$
Since homographies are bijections, ${\displaystyle f^{-1}(u,1)}$ maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography ${\displaystyle f^
{-1}}$ produces rotations from the ball in ${\displaystyle \mathbb {R} ^{3}}$ .
Matrix map
Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that A^T = −A).
Then I + A is invertible, and the Cayley transform
${\displaystyle Q=(I-A)(I+A)^{-1}\,\!}$
produces an orthogonal matrix, Q (so that Q^TQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as ${\displaystyle Q=(I+A)^{-1}(I-A)}$ .
In fact, Q must have determinant +1, so is special orthogonal.
Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then
${\displaystyle A=(I-Q)(I+Q)^{-1}\,\!}$
is a skew-symmetric matrix. (See also: Involution.) The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.
However, any rotation (special orthogonal) matrix Q can be written as
${\displaystyle Q={\bigl (}(I-A)(I+A)^{-1}{\bigr )}^{2}}$
for some skew-symmetric matrix A; more generally any orthogonal matrix Q can be written as
${\displaystyle Q=E(I-A)(I+A)^{-1}}$
for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries.^[4]
A slightly different form is also seen,^[5]^[6] requiring different mappings in each direction,
{\displaystyle {\begin{aligned}Q&=(I-A)^{-1}(I+A),\\[5mu]A&=(Q-I)(Q+I)^{-1}.\end{aligned}}}
The mappings may also be written with the order of the factors reversed;^[7]^[8] however, A always commutes with (μI ± A)^−1, so the reordering does not affect the definition.
In the 2×2 case, we have
${\displaystyle {\begin{bmatrix}0&\tan {\frac {\theta }{2}}\\-\tan {\frac {\theta }{2}}&0\end{bmatrix}}\leftrightarrow {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end
The 180° rotation matrix, −I, is excluded, though it is the limit as tan ^θ⁄[2] goes to infinity.
In the 3×3 case, we have
${\displaystyle {\begin{bmatrix}0&z&-y\\-z&0&x\\y&-x&0\end{bmatrix}}\leftrightarrow {\frac {1}{K}}{\begin{bmatrix}w^{2}+x^{2}-y^{2}-z^{2}&2(xy-wz)&2(wy+xz)\\2(xy+wz)&w^{2}-x^{2}+y^{2}-z^{2}&2
where K = w^2 + x^2 + y^2 + z^2, and where w = 1. This we recognize as the rotation matrix corresponding to quaternion
${\displaystyle w+\mathbf {i} x+\mathbf {j} y+\mathbf {k} z\,\!}$
(by a formula Cayley had published the year before), except scaled so that w = 1 instead of the usual scaling so that w^2 + x^2 + y^2 + z^2 = 1. Thus vector (x,y,z) is the unit axis of rotation
scaled by tan ^θ⁄[2]. Again excluded are 180° rotations, which in this case are all Q which are symmetric (so that Q^T = Q).
Other matrices
One can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·^T) is replaced by the
conjugate transpose (·^H). This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, one may extend the definition further with choices of
adjoint other than transpose or conjugate transpose.
Formally, the definition only requires some invertibility, so one can substitute for Q any matrix M whose eigenvalues do not include −1. For example,
${\displaystyle {\begin{bmatrix}0&-a&ab-c\\0&0&-b\\0&0&0\end{bmatrix}}\leftrightarrow {\begin{bmatrix}1&2a&2c\\0&1&2b\\0&0&1\end{bmatrix}}.}$
Note that A is skew-symmetric (respectively, skew-Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1.
Operator map
An infinite-dimensional version of an inner product space is a Hilbert space, and one can no longer speak of matrices. However, matrices are merely representations of linear operators, and these can
be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators.
{\displaystyle {\begin{aligned}U&{}=(A-\mathbf {i} I)(A+\mathbf {i} I)^{-1}\\A&{}=\mathbf {i} (I+U)(I-U)^{-1}\end{aligned}}}
Here the domain of U, dom U, is (A+iI) dom A. See self-adjoint operator for further details.
See also
1. ^ Robert Everist Green & Steven G. Krantz (2006) Function Theory of One Complex Variable, page 189, Graduate Studies in Mathematics #40, American Mathematical Society ISBN 9780821839621
2. ^ Erwin Kreyszig (1983) Advanced Engineering Mathematics, 5th edition, page 611, Wiley ISBN 0471862517
3. ^ Gallier, Jean (2006). "Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible". arXiv:math/0606320. As described by
Gallier, the first of these results is a sharpened variant of Weyl, Hermann (1946). The Classical Groups (2nd ed.). Princeton University Press. Lemma 2.10.D, p. 60.
The second appeared as an exercise in Bellman, Richard (1960). Introduction to Matrix Analysis. SIAM Publications. §6.4 exercise 11, p. 91–92.
4. ^ Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins University Press, ISBN 978-0-8018-5414-9
5. ^ F. Chong (1971) "A Geometric Note on the Cayley Transform", pages 84,5 in A Spectrum of Mathematics: Essays Presented to H. G. Forder, John C. Butcher editor, Auckland University Press
6. ^ Courant, Richard; Hilbert, David (1989), Methods of Mathematical Physics, vol. 1 (1st English ed.), New York: Wiley-Interscience, pp. 536, 7, ISBN 978-0-471-50447-4 Ch.VII, §7.2
7. ^ Howard Eves (1966) Elementary Matrix Theory, § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, Allyn & Bacon | {"url":"https://www.knowpia.com/knowpedia/Cayley_transform","timestamp":"2024-11-03T13:46:09Z","content_type":"text/html","content_length":"183879","record_id":"<urn:uuid:05176929-6f4e-44c7-9e0e-072dfaa7c5f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00415.warc.gz"} |
Mathematics Archives - CPL
Catherine Myson-Foehner provides a guide to the new Mathematics Syllabus K-6…
The price of doing the same old thing is far higher than the price of change
Education is evolving rapidly, driven by a strong faith in the ‘magic’ of research. Inside the classroom we keenly feel this maelstrom, with seemingly constant changes to not only what students
learn, but how, why and where they learn. In 2023, all NSW primary teachers are either trialing, or implementing, the reformed mathematics syllabus. Being given a new road map to your job in any
field of work is a stressful and confusing time. It takes energy because things that ran on automatic pilot now demand attention to detail and thoughtful interaction. And change takes from teachers
that one resource which is always in shortest supply – time. As teachers, we have a broad set of mandated goals. We must improve student achievement, but also take the lead in tackling social
problems such as poverty, inequality, complex fast-paced change and fragile mental health. The new tasks thrust upon us require new approaches, new understandings, and above all, a closer relation
between practice, research and theory.
And therein lies the story of our new syllabus.
Why change the syllabus?
There are three main drivers of syllabus reform.
Firstly, we are failing to meet our own national education goals. In 2019, Education ministers agreed on a vision of education for all young Australians under the Alice Springs (Mparntwe) Education
Declaration (Education Council, 2019). The first goal is: ‘The Australian education system promotes excellence and equity.’ This means a commitment to ‘provide all young Australians with access to
high-quality education that is inclusive and free from any form of discrimination’ (Education Council, 2019). And yet in our schools and classrooms, academic achievement is still tied to wealth, to
gender, to indigeneity. By Year 9, students from the lowest quartile of socioeconomic advantage are roughly 3 years behind students from the highest quartile. And startlingly, for each 25% of wealth
and social capital, you lose a year in mathematical achievement. Moreover, how is it possible that in 2021 the gender gap in Year 3 NAPLAN numeracy was the widest yet of any test in favour of boys at
2.52 months? (Thomas, 2021)
Secondly, the NSW Curriculum review (Department of Education NSW, 2022) voiced concern that Australian students’ level of mathematical achievement appears to be in decline. Analysis of PISA data
suggested that Australian students have slipped from being some of the highest performers in mathematics to being near the OECD average. Reforming the curriculum was seen as essential step in
ensuring all students are challenged and engaged to maximise their individual capabilities and potential.
Thirdly, and crucially for us as teachers, a major driver of syllabus change was our own feedback to the NSW Curriculum Review – that the curriculum ‘contains too much clutter, with not enough time
to focus on deep learning’
These three factors, contextualized by the fast pace of educational and social change, brought about an inevitable reform to our curriculum. It is essential we reflect on the ‘why’ of the reform as
we implement these changes because, as teachers, its success rests in our hands. The syllabus always was, and always will be, the basis for all teaching and learning programs. Until it is enacted in
our classrooms, the attempts to support higher achievement, to untie educational destiny from socio-economic status, gender and indigeneity, and to (eventually) reduce our workload, will fail. We are
mandated to carry out the reform and we can use it to illuminate possibilities for the way mathematics is taught and learnt in our classrooms.
The key changes
The structure and content of the syllabus was adapted to reflect current evidence on what makes good teaching and learning. The key changes are:
• clearer, more explicit outcomes for what students are to know, understand and do,
• more deliberate and careful sequencing of content K -6 with a reduction of content and repetition, and a focus on connecting knowledge
• greater emphasis on mathematical reasoning, with one overarching Working mathematically outcome K-6
• increased opportunity for students to apply their knowledge.
The syllabus now sits in a purpose-built, digital portal. Online links provide continuously updated resources such as teaching advice, vocabulary guides, assessment resources, and content examples.
This framework of support is essential viewing because it provides context and support for teaching and learning. A resource tab provides tailored support such as work samples, professional learning
opportunities, and parent/carer guides.
Explicit goals for what students are to know, understand and do.
To make it easier to identify what students need to know, the Mathematics K-10 syllabus has been streamlined and the content described in simpler, more precise language. Stage statements have been
removed, reflecting the fact that content within a stage of learning represents what students ‘typically’ know, do and understand. The change acknowledges that students can have different learning
trajectories and teachers are best placed to make decisions on student learning goals.
Syllabus content remains organised into three conceptual areas: ‘Number and algebra’, ‘Measurement and geometry’ (previously ‘Measurement and space’) and ‘Statistics and probability’. There are
clearer expectations for students’ developmental progression in relation to foundational concepts such as place value, additive and multiplicative relations, and fractions. Focus areas have been
renamed to make the learning content more explicit. For example, in K-2, Addition and Subtraction has been replaced with ‘Combining and separating quantities’, moving to ‘Additive relations’ in Years
3-6. This shifts the focus from treating addition and subtraction as two separate mathematical processes to examining the relationship between them. Similarly, Multiplication and Division is now
‘Forming groups’ in K-2, moving to ‘Multiplicative relations’ in Years 3-6.
Each focus area is also accompanied by teaching advice to assist with programming and lesson design. The advice covers aspects such as possible misconceptions, developmental progression, and
interrelationships with other mathematical concepts. To clarify teaching and learning goals, appropriate content points have drop downs which provide unambiguous examples.
Student goal setting is supported through a tight integration of assessment resources. The K-2 syllabus has key progression point tasks in Representing number, Combining and separating quantities and
Forming groups. To provide a direct link between observable behaviours and syllabus outcomes, National Numeracy Learning Progression V3 (ACARA, 2020) are tagged to syllabus content from K-10.
Teaching advice supports the development of assessment tasks by helping teachers understand where students are on the trajectory of learning. Linked assessment resources provide a range of strategies
to monitor student progress and identify areas where additional support may be needed. Underlining a strong focus on equitable outcomes, sample access points are integrated for students with complex
disabilities who are working towards Early Stage 1 outcomes.
The content is more deliberately sequenced and connected.
The new syllabus draws on contemporary research to redesign the way we identify, introduce and progress key concepts. Content within, and across, focus areas has been realigned and sequenced to
improve the progression of learning and build stronger schemas of understanding. Purposeful connections have replaced isolated repetition. For example, in Measurement and geometry, content relating
to time and mass now fall together under Non-spatial measure, emphasising the different conceptual approach required to measure things we can’t see or touch. Volume now falls under Three-dimensional
spatial structure as a natural connection to how we describe and quantify objects.
Many of the changes reflect that ‘skills and knowledge for focus areas often develop in an interrelated manner and can be addressed in parallel’(NESA, 2022). For example, patterning is a basic
mathematical skill that enables students to sequence, see order and make predictions. It underpins all mathematical relationships from the memorisation of the counting sequence to spatial thinking
and geometry. Being able to identify the repetition of a unit is the basis of multiplicative thinking. Therefore, it just makes sense to liberate ‘Patterns and Algebra’ from its isolated outcome in
the 2012 syllabus and entwine it in all focus areas.
Fractions are another (particularly striking and important) example of how making connections explicit can drive changes to the way we teach, and students learn. We know that almost all students find
fractions challenging, and almost all teachers find fractions challenging to teach. Research suggests ‘a student’s proficiency with fractions is directly related to the conceptual and procedural
interweaving they make over a long period of time’ (Australian Government Department of Education, 2022).
To support this, the ‘Fractions and Decimals’ outcome from the previous syllabus has gone from K-2, and fractional understanding is woven into Forming groups and Geometric measure. The emphasis is on
conceptual understanding of the whole, and its relationship to the parts, rather than on fractions as a number. Of the three different fraction models – linear (partitioning a length or line), area
(partitioning whole shapes or areas) and discreet (partitioning a collection) – the area model is the most challenging. The parts must be equal (‘exactly equal’), and students must understand that a
shape or object has many different attributes and that only some of them contribute to the measurement of area (for instance – not colour, not orientation, not position). K-2 students are slowly
building their ability to estimate and compare area by superimposing shapes, using indirect comparison and, finally, by using grid overlays. Introducing fractions through halves and quarters of
shapes assumes students already have a deep understanding of this challenging concept. Indeed, the typical objects we halve (apples, pizzas, leaves, playdough) are often not halves in terms of their
mass or volume, and only ‘about half’ in terms of their size. We are inadvertently contributing to the misconception that one out of two pieces is a half, rather than focusing on the equality of the
parts and their relationship to the whole.
The new syllabus, therefore, introduces halves through collections when forming groups, and half (and about half) of lengths in Geometric measure. Students are then introduced to the focus area
Partitioned fractions (or fractions as parts of things) in Stage 2, in preparation for representing quantity fractions (or fractions as numbers) in Stage 3.
All of these changes are framed by an increased focus on reasoning. Opportunities for students to reason are tagged to relevant content, and teachers are supported to engage students in mathematical
reasoning activities through linked teaching advice. Research suggests that ‘children’s mathematical reasoning might be the mediator between social background and children’s mathematics’ (Nunes et
al, 2009). If we are serious about closing the equity gaps in mathematical achievement, this is the place to start.
The focus on reasoning informs the move to a single, overarching Working mathematically outcome. It emphasises the interrelationship of the processes that make up working mathematically –
understanding and fluency, problem solving, reasoning, and communicating through mathematical language and models. When teachers feel pressured, they often revert to more traditional teaching methods
which don’t address mathematical reasoning. It seems like a more efficient way of getting through the content. Yet having students listen to, share, and make sense of their classmates’ reasoning is
vital to building and maintaining a focus on mathematical understanding. For example, learning multiplication facts by rote can be helpful but many students are never able to recall them all
accurately. In a classroom where reasoning and communicating is expected, students have to clarify and organise their thinking about the multiplicative relations underlying fact families. This helps
them identify important mathematical connections and build fluency through understanding. That most incorrectly remembered multiplication fact, 6 x 8, can then be accessed or checked through more
familiar facts such as (6 x 4) + (6 x 4), or (7 x 6) + (1 x 6).
Implementation Support
There is no shortage of support for teachers to engage with the new syllabus. NESA has an online learning portal, NESA Learning, which deals with all aspects of the new curriculum. The NSW Department
of Education has a wide range of professional development opportunities, as well as on-demand support through Statewide staffrooms and Curriculum networks. A complete set of sample units for
Mathematics K-2 Syllabus can be downloaded from the Universal Resource Hub. A selection of sample units for Stage 2 and Stage 3 are also available, with the rest being released in a phased manner
into 2024.
However, after all the professional learning is done and all the resources are downloaded, the most important thing will be those discussions in stage meetings, in the staffroom, in classroom
doorways and at student desks. It is here that we, as teachers, will really begin to ‘work mathematically’, exploring, connecting, choosing, applying, reasoning, and communicating newly acquired
syllabus content knowledge, reflecting on our beliefs about successful teaching practices and illuminating our own way forward. The best advice I have? A cut and paste from Jenny Williams’ and
Mary-Ellen Betts’ words of wisdom for teacher approaching a previous ‘new syllabus’ in 2014: “Open the syllabus and read it.”
Australian Curriculum, Assessment and Reporting Authority [ACARA] (2020). National Numeracy Learning Progression Version 3. https://www.australiancurriculum.edu.au/resources/
national-literacy-and-numeracy-learning-progressions/version-3-of-national-literacy-and-numeracy-learning-progressions/ accessed 20 May 2023
Clinton, W. J. (1994). Public papers of the Presidents of the United States, William J. Clinton. Washington, DC :Office of the Federal Register, National Archives and Records
Department of Education NSW (2022). About the Reform. NSW Government https://education.nsw.gov.au/teaching-and-learning/curriculum/nsw-curriculum-reform/about-the-reform accessed 20 August 2023
Education Council (2019). Alice Springs (Mparntwe) Education Declaration. https://www.education.gov.au/resources/alice-springs-mparntwe-education-declaration
NSW Education Standards Authority (2022) Mathematics K-10 Syllabus https://curriculum.nsw.edu.au/syllabuses/mathematics-k-10-2022?tab=course-overview
Nunes, T, Bryant P, Sylva K and Barros, R (2009) Development of Maths Capabilities and Confidence in Primary School. University of Oxford.
OECD (2019). Programme for International Student Assessment Results PISA 2018: Australia, accessed 28 February 2023
Thomas, D. (2021). NAPLAN 2021: Making sense of the reading, numeracy, and writing results https://readwritethinklearn.com/blog/naplan-2021-results/ [website], accessed 14 June 2023.
Australian Government Department of Education (2022) reSolve: Maths by Inquiry [website], accessed 12 August 2023.
Williams, J and Betts, M (2015) How Goes the New K-6 English Syllabus? Journal of Professional Learning, Centre for Professional Learning, S1, 2015
Catherine Myson-Foehner has held classroom teacher and executive roles in NSW schools. She is currently employed by the NSW Department of Education as a Teaching and Learning Officer within the
Educational Standards Directorate. She assists in the development, implementation and evaluation of innovative approaches to planning, programming and assessment for primary mathematics teachers.
Catherine has a strong educational interest in curriculum development and its impact on student equity. She worked on the K-2 and 3-6 writing teams for the K-10 Mathematics syllabus and delivers
professional learning for teachers on syllabus implementation, including workshops at the Centre for Professional learning.
Move and Improve Mathematics: Middle Years
Martin Ommundsen finds that incorporating movement activities into Mathematics can contribute to positive impacts on learning and class dynamics…
If asked ‘What is good for the body, mind and spirit?’ a typical Mathematics teacher might answer ‘Maths!’
This author agrees and presents empirical research that targeted and regular inclusion of movement-based Mathematics pedagogy has beneficial learning outcomes for all students, including those who
might typically be thought to be high achieving and coping well in more traditional, seated, classroom settings. The article sets out background research and context and goes on to detail findings
for learning and social impacts, drawing on student voice and primary research, before presenting a series of examples for effective activities connected to the NSW syllabuses.
The positive connection between physical activity and cognition has been understood since Sibley and Etnier’s meta-study from 2003. They concluded that the academic level of achievement in a range of
different subjects, including mathematics, does not decrease when students spend increased time on physical activities in these subjects (Sibley & Etnier, 2003).
Five years later, Tomporowski and colleagues, in a further meta-study, stated that positive changes in children’s mental functions caused by physical education lessons are primarily seen in the
executive functions. In other words, increased movement in physical education will be followed by increased self-control, short-term memory and cognitive flexibility (Tomporowski et al., 2008).
However, movement during school time can be many other things than those in dedicated subjects, such as NSW’s sport or Personal Development, Health and Physical Education (PDHPE). In my home country
of Denmark, since 2014, it has been a requirement that students should do some kind of movement averaging 45 minutes per day. The aim of the new law was to bring a lot of movement activities or games
into subjects like Mathematics, Danish, English and so on, building on an understanding that physical activity contributes positively to the learning output of students.
The findings below come from empirical research, including my own, which included students interviews with my Class 9 (aged around 15 years old), who frequently received Mathematics instruction
involving planned moving games and activities. Their answers are used to understand how these ‘games’ are experienced from a student’s perspective, and in this article, their responses will be
referred to as “Year 9 Student”.
Impact on learning
There is evidence that physical activity improves student learning. Danish studies report that in Mathematics, students in the youngest school classes can increase their mathematical skills by 35 per
cent compared to students who do not have physical activities incorporated into their Mathematics lessons. An Australian study of NSW students at the older end of the age scale advocated that
implementation of activities like high intensity interval training programs increased students’ fitness and improved their well-being with potential subsequent benefits to academic performance.
(Lubens et al., 2019). However, according to brain researcher Jesper Lundbye-Jensen (Sederberg et al., 2017), for targeted impact on learning outcomes, it is necessary that the physical activities
are connected to the subject itself and that they are not “just” a run around. In Mathematics, several cognitive dimensions can be improved, for example, problem solving, logical thinking, spatial
perception, short-term memory and awareness.
Student experience and brain function
So, what about the Year 9 class? How did they experience these movement games and did they find the strategies helpful in their lessons? One of the students gives the following description:
You are really using your brain a lot while sitting and working, and you can get quite tired. It’s nice with a break halfway through, but you still stay in Mathematics. While moving you refresh what
you’ve learned about prime numbers, square numbers or similar, and after that you are ready to work again.
Another student explains it like this:
It makes things more interesting. You get some oxygen to your brain. You don’t feel so tired. Some things you remember better this way, instead of writing them down.
There are several interesting key points to draw out from this. Firstly, the use of Mathematics when moving instead of sitting seem to freshen up the students. It is also important to remember that
students are usually sitting down in all other subjects except for physical education, so moving activities can really bring a welcome change. Secondly, it apparently helped the students with the
learning afterwards as well. Last but not least, it appears that certain concepts can be easier to remember, perhaps as the body embraces the knowledge in more than one way while moving. This last
insight is also supported by other empirical research arguing that the brain develops when practising motor skills (Sederberg et al., 2017). Furthermore, the role of novelty is important here, as in
an educational world where most learning is expected to occur whilst sitting at similar looking tables, in similar looking classrooms, the body and brain will more quickly remember the exact learning
that happened when it was connected to distinct or novel movement activities in a different setting in the school yard or likewise.
Not only for disengaged
Some might think that movement activities are only or mainly for those students who struggle to sit in their seats doing usual work at tables. However, one of the Year 9 students explained how that
was not correct.
Even though I would categorise myself as a student who is fully capable of sitting down for one hour and listening, I also find it good and welcome to do something where the academic work and
physical activities are blended together.
From my own master thesis research, it was evident that movement activities are absolutely not only for students who are struggling with the normal classroom setting, but indeed for anyone in the
classroom, regardless of academic level and gender. Interestingly, these findings have been very similar across very different countries, such as Denmark, Tanzania and Nepal (Ommundsen 2016). My own
interpretation is that there could be a universal appreciation for such an incorporated model.
Careful selection is required
At this point, though, it is important to clarify that not all parts of the Mathematics syllabus can be met through such activities. In the first instance, it would obviously be too monotone and not
very interesting. The Year 9 students identify certain areas where they find the physical activities fruitful. As example,
Repetition, very clearly! It could be something with questions and answers that are matching. For instance, equations, geometry or mental arithmetic tasks.
Repetition. Like calculation rules, prime numbers or square roots.
This suggests that it is important not to implement new, difficult, or complex Mathematics topics in the movement activity itself. Rather, aim to repeat some previously introduced concepts or areas
and then allow the students to practice on that through the movement strategies. As a Mathematics teacher, that is a welcome opportunity to review some of those things you might not otherwise find
the time for. Also, it can become easier to relate future discussions to something concrete, as for instance, when using prime numbers again, you can refer back to the physical experiences and say
“remember, those were the ones we practiced when we did that activity outside…”.
Impact on social life
In addition to the positive impacts on learning outlined above, there can also be advantages for the social life and dynamics of the class as a group. Year 9 student responses on this specific matter
illustrate a range of different, important points:
That’s a really important part (the social life) and one of the most important reasons to do it. When you’re together in groups, everybody is moving, and everyone is passionate. That creates a better
teamwork. You build much more on your teamwork when there is some movement in the teaching.
You are talking and communicating more with people. You easily come around to each other. Maybe you come to chat to someone you usually aren’t talking to.
Maybe you get some fun memories with each other, you get a little closer with each other, because you have something you can look back on.
Other students interviewed over the years have expressed similar views; that movement activities provide a special mood in the classroom (or outdoor space) when these activities are going on. It
would seem unnecessary here to argue further about why a good mood in the classroom is a positive factor.
Whereas in the normal classroom setting students are not usually meant to talk to very many other people, besides maybe their table partner and the teacher, the students in the movement activities
pass a lot of other students simply because they are moving around to lots of different places. In my master’s thesis, I found that the students actually did not get disturbed by their classmates in
these activities, something that I noticed tended to happen more frequently in the quiet (supposedly) classroom settings. The most likely reason why is that they are so engaged themselves by the
activity (Ommundsen, 2016).
There are obviously different ways of creating fun memories together, but the theory of science of body phenomenology argues that the body plays a distinctly important role. Merleau-Ponty (1994)
argues that bodies have a will for some kind of freedom. Steen Nepper Larsen further explains how the motorical system operates prior to consciousness so that We can, before we know[i]. It has been
my experience that movement activities in and of themselves often help create those important fun memories that build the class up together.
Movement activities linked to the NSW syllabuses
Thus, having argued that there are several benefits, some specific examples of activities to try are included below. All of the following are activities I have had good success with and each is
connected to some relevant syllabus outcomes. The suggestions below are designed to give an indication of starting points and it is anticipated that Mathematics teachers could find many ways to
modify and extend these to suit their students.
Tall, broad, thin, low
• Most likely to be done inside the classroom.
• The teacher writes the following on the board:
□ 76-100: tall
□ 51-75: broad
□ 26-50: thin
□ 1-25: low
• The students are given (the teacher can write on the board) different questions like: 3 x 8 – 2. Students can use lots of different combined calculation methods and indicate their answer by
manipulating their body to reflect the category for the range the answer falls into. In this example, the result is 22, and so all students should make themselves as low to the ground as
• Divide the students into two or three groups and let them compete against each other, or make a class challenge to notice how fast the whole class can do ten questions.
• Raise the difficulty by using percentage. For example, “How much is 25% of 240?” In this case, using the same range above, the answer is 60, and so students make themselves as broad as possible.
NSW syllabus outcomes
• MA3-6NA: selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation
• MA4-5NA: operates with fractions, decimals and percentages
True or false
• Most likely to be done outside or in a big hall.
• The students are divided into two teams. On each team the members stand in a line next to each other, all facing the opponent team. There should be approximately two steps between the two lines,
each student facing a student from the opposite team.
• A “goal line” is marked about 10 m behind each line of the students.
• One of teams is the true team, the other is the false team.
• If the teacher says something which is true, the true team turns around and runs back to their own goal line for safety while the false team tries to catch them. If they teacher says something
which is false, the false team has to turn around and run back to their own goal line for safety, and the true team has to try to catch them. So, if the teacher says, “A triangle consists of 190
degrees”, the false team runs back to their goal line and the true team tries to catch them.
• If someone is caught before the goal line, this student will go on the other team when the students are lining up to the next question. When doing this activity the first time, it can appear a
bit confusing, but the students will soon learn it.
• For making it easier, the teacher can make a break before saying the last part of a sentence, for example “5 x 5 x 2 equals… [pause]… 50”. Then students are given a better chance to calculate.
• Play for a set period of time, until one side has caught 10 players from the other team, or until one team is fully captured.
NSW syllabus outcomes
• MA3-6NA: describes and compares length and distances using everyday language
• MA3-7NA: compares, orders and calculate with fractions, decimals and percentages
• MA4-6N4: solves financial problems involving purchasing goods
One in the middle
• Most likely to be done outside or in a big hall.
• The whole class stand in a big circle, each student standing at a spot (marked by a cone/textbook/chair), with one student in the centre of the circle.
• Every participant is given one of four different numbers (for example 6, 7, 8, 9).
• The teacher calls out a larger number which is a product of one or more of the four selected numbers (for example 21, 24, 72). The students who have the relevant number(s) run from their place to
another vacated place around the circle (several will always be free at the same time as there are only four numbers allocated).
• Importantly, the student in the centre also has to find a free spot each time, and so every time there will be one student who does not find a spot, and this person will be the new person in the
middle for the next round.
• Change the numbers after 5-10 minutes and remember, even though it is a very fun game it is supposed to be a ‘brain break’ and so not consume the whole lesson.
• Note. This game works well for other subjects such as English or languages, with four words allocated and teachers calling out categories, so please consider sharing with your colleagues.
NSW syllabus outcomes
• MA2-6NA: uses mental and informal written strategies for multiplications and division
• MA3-6NA: selects and applies appropriate strategies for multiplication and division, and applies the order of operations to calculations involving more than one operation
Over to you!
The benefits of physical activity are well understood in health fields, and finding ways to incorporate this knowledge into the school experience is a challenge which, if achieved, could have
significant advantages. The strategies outlined in this article suggest that thoughtful integration of subject-specific movement activities into teaching programs can bring improvement in
mathematical understanding as well as potential gains in student wellbeing and class cohesion for the full range of students in the middle years.
Lubens, D., Leahy, A., Smith, J., & Eather N. (2019). Why exercise for cognitive and mental health is especially important in the senior years. Journal of Professional Learning, (2). https://
Merleau-Ponty, M. (1994). Kroppens fænomenologi [Phenomenology of perception]. Det lille Forlag.
Ommundsen, M. S. (2016). Når det sociale mønster i skolen bevæger sig [unpublished Master’s thesis]. Aarhus Universitet.
Sederberg, M., Kortbek, K., & Bahrenscheer, A. (2017). Bevægelse, sundhed og trivsel: I skole og fritid. Hans Reitzels Forlag.
Sibley, B. A., & Etnier, J. L. (2003). The relationship between physical activity and cognition in children: A meta-analysis. Pediatric Exercise Science, 15(3), 243-256.
Tomporowski, P. D., Davis, C. L., Miller, P. H., & Naglieri, J. A. (2008). Exercise and children’s intelligence, cognition, and academic achievement. Educational Psychology Review, 20(2), 111-131.
Martin Ommundsen is a Mathematics, History, Social Science and Physical Education teacher educated in Denmark. He now lives in Australia and has experience teaching students from Year 1 to Year 9 in
a range of cultures and countries including Tanzania and Nepal. Martin completed a Masters Degree at Aarhus University in 2016 and his research interests and final thesis include the sociological
aspects of movement activities in the classroom.
[i] This observation was made by Professor Steen Nepper Larsen during a lecture at Aarhus University, Denmark, in 2015.
Uncertainty, Error and Confidence in Data
Jim Sturgiss provides a straightforward guide to teaching some scientific concepts that are now part of the new Science syllabuses…
Uncertainty is a statistical concept found in the Assessing data and information outcome of the new Science syllabuses:.
WS 5.2 assess error, uncertainty and limitations in data (ACSBL004, ACSBL005, ACSBL033, ACSBL099)
This concept is not found in the previous syllabuses.
This paper addresses uncertainty as a means of describing the accuracy of a series of measurements or as a means of comparing two sets of data. Uncertainty, or confidence, is described in terms of
mean and standard deviation of a dataset. Standard deviation is a concept encountered by students in Stage 5.3 Mathematics and Stage 6 Standard 2 Mathematics.
Not explored in this paper is the use of Microsoft Excel or Google Sheets which can calculate uncertainty of datasets with ease (=STDEV.S(number1, number2,…).
Figure 1 Karl Pearson
Karl Pearson (Figure 1), the great 19th-century biostatistician and eugenist, first described mathematical methods for determining the probability distributions of scientific measurements, and these
methods form the basis of statistical applications in scientific research. Statistical techniques allow us to estimate uncertainty and report the error surrounding a value after repeated measurement
of that value.
1. Accuracy, Precision and Error
Accuracy is how close a measurement is to the correct value for that measurement. The precision of a measurement system refers to how close the agreement is between repeated measurements (which are
repeated under the same conditions). Measurements can be both accurate and precise, accurate but not precise, precise but not accurate, or neither.
Precision and Imprecision
Precision (see Figure 2) refers to how well measurements agree with each other in multiple tests. Random error, or Imprecision, is usually quantified by calculating the coefficient of variation from
the results of a set of duplicate measurements.
Figure 2 Accuracy and precision
The accuracy of a measurement is how close a result comes to the true value.
When randomness is attributed to errors, they are “errors” in the sense in which that term is used in statistics.
• Systematic error (bias) occurs with the same value, when we use the instrument in the same way (eg calibration error) and in the same case. This is sometimes called statistical bias.
It may often be reduced with standardized procedures. Part of the learning process in the various sciences is learning how to use standard instruments and protocols so as to minimize systematic
• Random error, which may vary from one observation to another. Random error (or random variation) is due to factors which cannot, or will not, be controlled. Random error often occurs when
instruments are pushed to the extremes of their operating limits. For example, it is common for digital balances to exhibit random error in their least significant digit. Three measurements of a
single object might read something like 0.9111g, 0.9110g, and 0.9112g.
Systematic error or Inaccuracy (see Figure 3) is quantified by the average difference (bias) between a set of measurements obtained with the test method with a reference value or values obtained with
a reference method.
Figure 3 Imprecision and in accuracy
2. Uncertainty
There is uncertainty in all scientific data. Uncertainty is reported in terms of confidence.
• Uncertainty is the quantitative estimation of error present in data; all measurements contain some uncertainty generated through systematic error and/or random error.
• Acknowledging the uncertainty of data is an important component of reporting the results of scientific investigation.
• Careful methodology can reduce uncertainty by correcting for systematic error and minimizing random error. However, uncertainty can never be reduced to zero.
Estimating the Experimental Uncertainty For a Single Measurement
Any measurement made will have some uncertainty associated with it, no matter the precision of the measuring tool. So how is this uncertainty determined and reported?
The uncertainty of a single measurement is limited by the precision and accuracy of the measuring instrument, along with any other factors that might affect the ability of the experimenter to make
the measurement.
For example, if you are trying to use a ruler to measure the diameter of a tennis ball, the uncertainty might be ± 5 mm, but if you used a Vernier caliper, the uncertainty could be reduced to maybe ±
2 mm. The limiting factor with the ruler is parallax, while the second case is limited by ambiguity in the definition of the tennis ball’s diameter (it’s fuzzy!). In both of these cases, the
uncertainty is greater than the smallest divisions marked on the measuring tool (likely 1 mm and 0.05 mm respectively).
Unfortunately, there is no general rule for determining the uncertainty in all measurements. The experimenter is the one who can best evaluate and quantify the uncertainty of a measurement based on
all the possible factors that affect the result. Therefore, the person making the measurement has the obligation to make the best judgment possible and to report the uncertainty in a way that clearly
explains what the uncertainty represents:
Measurement = (measured value ± standard uncertainty) unit of measurement.
For example, where the ± standard uncertainty indicates approximately a 68% confidence interval, the diameter of the tennis ball may be written as 6.7 ± 0.2 cm.
Alternatively, where the ± standard uncertainty indicates approximately a 95% confidence interval, the diameter of the tennis ball may be written as 6.7 ± 0.4 cm.
Estimating the Experimental Uncertainty For a Repeated Measure (Standard Deviation).
Suppose you time the period of oscillation of a pendulum using a digital instrument (that you assume is measuring accurately) and find: T = 0.44 seconds. This single measurement of the period
suggests a precision of ±0.005 s, but this instrument precision may not give a complete sense of the uncertainty. If you repeat the measurement several times and examine the variation among the
measured values, you can get a better idea of the uncertainty in the period.
For example, here are the results of 5 measurements, in seconds: 0.46, 0.44, 0.45, 0.44, 0.41.
For this situation, the best estimate of the period is the average, or mean.
Whenever possible, repeat a measurement several times and average the results. This average is generally the best estimate of the “true” value (unless the data set is skewed by one or more outliers).
These outliers should be examined to determine if they are bad data points, which should be omitted from the average, or valid measurements that require further investigation.
Generally, the more repetitions you make of a measurement, the better this estimate will be, but be careful to avoid wasting time taking more measurements than is necessary for the precision
Consider, as another example, the measurement of the thickness of a piece of paper using a micrometer. The thickness of the paper is measured at a number of points on the sheet, and the values
obtained are entered in a data table.
This average is the best available estimate of the thickness of the piece of paper, but it is certainly not exact. We would have to average an infinite number of measurements to approach the true
mean value, and even then, we are not guaranteed that the mean value is accurate because there is still some systematic error from the measuring tool, which can never be calibrated perfectly. So how
do we express the uncertainty in our average value?
The most common way to describe the spread or uncertainty of the data is the standard deviation
Figure 5 Standard deviations of a normal distribution
The significance of the standard deviation is this:
if you now make one more measurement using the same micrometer, you can reasonably expect (with about 68% confidence) that the new measurement will be within 0.002 mm of the estimated average of
0.065 mm. In fact, it is reasonable to use the standard deviation as the uncertainty associated with this single new measurement.
This is written:
The thickness of 80 gsm paper (n=5) averaged 0.065 (s = 0.002mm)
s = standard deviation
The thickness of 80 gsm paper (n=5) averaged 0.065 ± 0.004 mm to a 95% confidence level.
(0.004 mm represents 2 standard deviations, 2s)
Standard Deviation of the Means (Standard Error of Mean (SEM))
The standard error is a measure of the accuracy of the estimate of the mean from the true or reference value. The main use of the standard error of the mean is to give confidence intervals around the
estimated means for normally distributed data, not for the data itself but for the mean.
If measured values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of
the number of measurements.
Standard error is often us
For example, two populations of salmon fed on two different diets may be considered significantly different if the 95% confidence intervals (two std errors) around the estimated fish sizes under Diet
A do not cross the estimated mean fish size under Diet B.
Note that the standard error of the mean depends on the sample size, as the standard error of the mean shrinks to 0 as sample size increases to infinity.
Figure 7 Salmon
Standard Error of Mean (SEM) Versus Standard Deviation
In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to
confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process.
The standard deviation of the sample data is a description of the variation in measurements, whereas, the standard error of the mean is a probabilistic statement about how the sample size will
provide a better bound on estimates of the population mean, in light of the central limit theorem.
Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to
which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample
size. This is because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size
Confidence Levels
The confidence level represents the frequency (i.e. the proportion) of possible confidence intervals that contain the true value of the unknown population parameter. Most commonly, the 95.4% (“two
sigma”) confidence level is used. However, other confidence levels can be used, for example, 68.3% (“one sigma”) and 99.7% (“three sigma”).
Knowledge of normally distributed data and standard deviation are key to understanding the notions of statistical uncercertainty and confidence. These concepts are extended to the standard error of
mean so that the significance of differences between two related datasets can be determined.
Absolute error The absolute error of a measurement is half of the smallest unit on the measuring device. The smallest unit is called the precision of the device.
Array An array is an ordered collection of objects or numbers arranged in rows and columns.
Bias This generally refers to a systematic favouring of certain outcomes more than others, due to unfair influence (knowingly or otherwise).
Confidence level The probability that the value of a parameter falls within a specified range of values. For example 2s = 95% confidence level.
Data cleansing Detecting and removing errors and inconsistencies from data in order to improve the quality of data (also known as data scrubbing).
Data set An organised collection of data.
Descriptive statistics These are statistics that quantitatively describe or summarise features of a collection of information.
Large data sets Data sets that must be of a size to be statistically reliable and require computational analysis to reveal patterns, trends and associations.
Limits of accuracy The limits of accuracy for a recorded measurement are the possible upper and lower bounds for the actual measurement.
Measures of central tendency Measures of central tendency are the values about which the set of data values for a particular variable are scattered. They are a measure of the centre or location of
the data. The two most common measures of central tendency are the mean and the median.
Measures of spread Measures of spread describe how similar or varied the set of data values are for a particular variable. Common measures of spread include the range, combinations of quantiles
(deciles, quartiles, percentiles), the interquartile range, variance and standard deviation.
Normal distribution The normal distribution is a type of continuous distribution whose graph looks like this:
The mean, median and mode are equal and the scores are symmetrically arranged either side of the mean.
The graph of a normal distribution is often called a ‘bell curve’ due to its shape.
Reliability An extent to which repeated observations and/or measurements taken under identical circumstances will yield similar results.
Sampling This is the selection of a subset of data from a statistical population. Methods of sampling include:
• systematic sampling – sample data is selected from a random starting point, using a fixed periodic interval
• self-selecting sampling – non-probability sampling where individuals volunteer themselves to be part of a sample
• simple random sampling – sample data is chosen at random; each member has an equal probability of being chosen
• stratified sampling – after dividing the population into separate groups or strata, a random sample is then taken from each group/strata in an equivalent proportion to the size of that group/
strata in the population
• A sample can be used to estimate the characteristics of the statistical population.
Standard deviation This is a measure of the spread of a data set. It gives an indication of how far, on average, individual data values are spread from the mean.
Standard error The standard error of the mean (SEM) is the standard deviation of the sampling distribution of the mean.
Uncertainty Any single value has an uncertainty equal to the standard deviation. However, if the
values are averaged, then the mean measurement value has a much smaller uncertainty, equal to the standard error of the mean, which is the standard deviation divided by the square root of the number
of measurements.
Works Cited
Measurements and Error Analysis, www.webassign.net/question_assets/unccolphysmechl1/measurements/manual.html.
Altman, Douglas G, and J Martin Bland. “Standard Deviations and Standard Errors.” BMJ (Clinical Research Ed.), BMJ Publishing Group Ltd., 15 Oct. 2005, www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/.
Hertzog, Lionel. “Standard Deviation vs Standard Error.” DataScience , 28 Apr. 2017, https://datascienceplus.com/standard-deviation-vs-standard-error/
Mott, Vallerie. “Introduction to Chemistry.”
Schoonjans, Frank. “Definition of Accuracy and Precision.” MedCalc, MedCalc Software, 9 Nov. 2018, www.medcalc.org/manual/accuracy_precision.php.
“Standard Error.” Wikipedia, Wikimedia Foundation, 7 Mar. 2019,
2336 | NSW Education Standards,
1319 | NSW Education Standards, https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/stage-6-learning-areas/stage-6-mathematics/mathematics-standard-2017/content/1319
Jim is an educational researcher and independent educational consultant. His M.Ed (Hons) thesis used an experimental design to evaluate the effectiveness of a literacy and learning program (1997). A
recipient of the NSW Professional Teaching Council’s Distinguished Service Award for leadership in delivering targeted professional learning to teachers, he works with schools to align assessment,
reporting and learning practice. He has been a head teacher of Science in two large Sydney high schools, as well as HSC Chemistry Senior Marker and Judge. For many years he served as a DoE Senior
Assessment Advisor where he developed many statewide assessments, (ESSA, SNAP, ELLA, BST) and as Coordinator: Analytics where he developed reports to schools for statewide assessments and NAPLAN. He
is a contributing author to the new Pearson Chemistry for NSW and to Macquarie University’s HSC Study Lab for Physics.
Follow Me into the Butterfly Garden
Neil Bramsen explores butterflies while teaching Mathematics and Science…
I am always keen to have my students undertake at least one major project based learning (PBL) experience each year.
In mid-2016 I had my stage two class work on revitalising an overgrown and neglected garden area into a ‘Butterfly Garden’. I was inspired by my visit to High Tech High in Chula Vista a few years ago
where I saw a comprehensive PBL program in place, with a butterfly component including garden, plant propagation, egg collection and breeding, all supported by student-generated text and a website.
Talk about comprehensive!
Exploring regional butterflies and appropriate feeder plants introduced a strong environmental and biodiversity perspective as students considered the ecology of a butterfly habitat. Over the course
of six months it was rewarding to document and reflect on the process that covered a multitude of learning areas, such as measurement and science and information reports, as well as the physical
tasks of gardening and assembling materials.
Of course, PBL is a terrific way to ‘access’ this type of learning, and each student was able to achieve success through various entry and exit points that they could identify with. Key Learning
Areas (KLA) such as Mathematics, Science, English and PDHPE came into play and offered a broad scope of learning opportunities.
I have found with any PBL that backward mapping to outcomes is the pragmatic and practical approach. I consider the activities that may be undertaken and then explore the relevant KLA scope and cross
reference to the syllabus involved.
Measuring up
There was extensive use of measurement, both through aerial photography via a DJI Phantom Drone and scale and grid tasks that calculated the area of the garden and path. See a photograph below of the
original site taken by the drone.
This measurement work then evolved into a volume activity for more capable students, and the depth of mulch and crushed concrete was calculated. It is important to note that while all students had an
introduction or refresher to area and square metres for example, I then targeted students that were stretching themselves to explore volume and cubic metres.
The students used websites to source local materials, cost the materials and then ring the landscape company to place the order. They actually used the school credit card under my supervision (I had
the CVV number) to ring and talk to the supplier and arrange the delivery. The students mapped access to the area.
Becoming alive
Highlights of the project included in-depth research into local butterflies and suitable host plants. The class explored colour and the types of colour needed to attract butterflies. Interestingly,
while we initially focused on local plant species and native butterflies, the monarch butterfly and the need for the milkweed plant to support it were identified. We sourced milkweed, and this aspect
has been the most successful, albeit with some winter wind damage to the milkweed. Propagating more milkweed plants would become an ongoing focus.
Importantly, as the image above demonstrates, the project all came together as students physically engaged with and enjoyed the gardening, from clearing weeds and moving barrow loads of mulch to
pouring crushed aggregate to make the path. The area came to life as the seedlings and young plants began to mature.
A little organisation
Students also followed a product procedure to assemble timber benches so that the area was a welcoming learning space. A daily watering regime was added to the class task list, and deep saucers were
added for birds and to provide water for butterflies.
The photograph above shows that, as the area established, it was then used for nature sketching, quiet time, reading and sensory awareness activities by the class.
Rewards worth working for
By late summer and autumn, we began to see monarch butterflies in the garden, just like the one in the photograph below. With some of the students that participated in the PBL project, we carefully
examined the milkweed plants, which act as a host for egg-laying and monarch caterpillars. Not only did we find quite a few eggs on the leaf tips but also fifteen or so caterpillars in varying stages
of maturity.
The kids were totally over the moon with the evidence of success and at seeing a natural life cycle occurring in the habitat that they had helped create. We are looking forward to monitoring the
health of the garden and the number of monarch butterflies that mature. The garden has continued to be popular with my classes for nature sketching and quiet time and has now been dedicated as a
special Year 6 Quiet Area during breaks.
Now, back to the syllabus
The project was an engaging opportunity to introduce teaching points from both the Mathematics and Science syllabuses. Some relevant outcomes are listed below.
Mathematics Stage 2 and Stage 3 outcomes
• selects and uses the appropriate unit and device to measure lengths and distances, calculates perimeters, and converts between units of length MA3-9MG
• measures, records, compares and estimates areas using square centimetres and square metres MA2-10MG
• selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles MA3-10MG
• selects and uses appropriate mental or written strategies, or technology, to solve problems MA2-2WM
• selects and applies appropriate problem-solving strategies, including the use of digital technologies, in undertaking investigations MA3-2WM
• uses simple maps and grids to represent position and follow routes, including using compass directions MA2-17MG
Science Stage 2 and Stage 3 outcomes
• shows interest in and enthusiasm for science and technology, responding to their curiosity, questions and perceived needs, wants and opportunities ST2-1VA
• describes ways that science knowledge helps people understand the effect of their actions on the environment and on the survival of living things ST2-11LW
• investigates their questions and predictions by analysing collected data, suggesting explanations for their findings, and communicating and reflecting on the processes undertaken ST2-4WS
• describes that living things have life cycles, can be distinguished from non-living things and grouped, based on their observable features ST2-10LW
• describes how people interact within built environments and the factors considered in their design and construction ST2-14BE
• describes some physical conditions of the environment and how these affect the growth and survival of living things ST3-11LW
Keys to success
Before attempting your own special learning experience, consider and plan for the following:
• Identify suitable project opportunities in the school grounds or local community;
• Consider the teaching and learning outcomes and prepare to backward map the obvious outcomes while allowing for the unexpected. The opportunities for differentiated learning are extensive and
every student can achieve success and growth in some aspect of learning;
• Allocate sufficient time; PBL takes time, usually more time than you might think!;
• Allocate resources and funding if needed;
• Communicate to other classes, teachers and supervisors the aims and progress of the project to generate community and school ‘buy in’.
We can nurture many positive blooms through our school garden projects. Once your project has concluded, remember to celebrate the successes and share your experiences and new knowledge with your
school and community.
Neil Bramsen is an Assistant Principal at Mount Ousley Public School, Wollongong. He actively engages with ‘the outdoor classroom’ and enjoys citizen science and space science. He is the recipient of
the 2017 Prime Minister’s Prize for Excellence in Primary Science Teaching.
To follow Neil further use: @galaxyinvader and neilbramsen.edublogs.org.au
This is an updated version of the article published in STANSW Science Education News, 2017 Volume 66 Number 4, http://joom.ag/nTUL/p58. Visit STANSW’s website at: www.stansw.asn.au
An Introduction to the New Stage 6 Mathematics Advanced and Extension Syllabuses
Terry Moriarty introduces the new calculus-based courses to be implemented from 2019…
The new NSW Stage 6 Mathematics Advance and Extension Syllabuses were endorsed in 2017. 2018 is a planning year with implementation for Year 11 in 2019 and Year 12 in 2020. There are support
materials, such as sample scope and sequence and assessment tasks, available through the NSW Education Standards Authority (NESA) website.
Due to the online nature of the syllabus documents, teachers are encouraged to download and review each section, including the aim and rationale before moving to the course content. New features of
the Stage 6 syllabuses and common material include:
• Australian curriculum content identified by codes;
• Learning across the curriculum content, including cross-curriculum priorities, general capabilities and other learning across curriculum areas, are incorporated and identified by icons;
• An interactive glossary.
Additionally, the Mathematics syllabuses include coding of applications and modelling as integral parts of each strand. Some strands are now merged together and the Mathematics Advanced and
Mathematics Standard syllabuses contain common material which is identified by a ‘paperclip’ icon.
Mathematics Advanced
Mathematics Advanced replaces the previous Mathematics 2 Unit syllabus. There is a new organisational structure as well as updates to content.
The Year 11 organisational structure
The Advanced course is organised into Strands, with the strands divided into Topics and Sub-topics. Topics within the strands have been updated, including some content from different topics in the
current course, such as Functions, which includes Linear and Trigonometric Functions, as well as new topics.
What to look out for
Some of the topics below have not been included in the new courses:
• Plane Geometry;
• Coordinate Methods in Geometry;
• Harder Applications as a topic;
• Conics.
Some of the topics below have been updated, including some units from different topics:
• Working with Functions includes Linear, Quadratic and Cubic Functions;
• Trigonometry and Measure of Angles, includes the use of two and three dimensions as well as new topics;
• Velocity and acceleration are included in Introduction to Differentiation;
• Financial Mathematics involves sequences and series and their application to financial situations.
Mathematics Advanced: Content
The table below demonstrates the changes between the previous and new syllabus.
│ 2 Unit Preliminary │ │
│ │ New Mathematics Advanced Year 11 Course – Topics and Sub-topics (to be implemented in 2019) │
│ (current in 2018) │ │
│ │ Functions │
│ │ │
│ │ MA-F1 Working with Functions │
│ │ │
│ • Basic Arithmetic and Algebra │ Trigonometric Functions │
│ │ │
│ • Real functions │ MA-T1 Trigonometry and Measure of Angles │
│ │ │
│ • Trigonometric ratios │ MA-T2 Trigonometric Functions and Identities │
│ │ │
│ • Linear functions │ Calculus │
│ │ │
│ • The quadratic polynomial and the parabola │ MA-C1 Introduction to Differentiation │
│ │ │
│ • Plane geometry – geometrical properties │ Exponential and Logarithmic Functions │
│ │ │
│ • Tangent to a curve and derivative of a function │ MA-E1 Logarithms and Exponentials │
│ │ │
│ │ Statistical Analysis │
│ │ │
│ │ MA-S1 Probability and Discrete Probability Distributions │
│ 2 Unit HSC Course │ │
│ │ New Mathematics Advanced Year 12 Course – Topics and Sub-topics (to be implemented in 2020) │
│ (Current until 2019) │ │
│ │ Functions │
│ │ │
│ │ MA-F2 Graphing Techniques │
│ │ │
│ │ Trigonometric Functions │
│ │ │
│ │ MA-T3 Trigonometric Functions and Graphs │
│ │ │
│ • Coordinate methods in geometry │ Calculus │
│ • Applications of geometrical properties │ │
│ • Geometrical applications of differentiation │ MA-C2 Differential Calculus │
│ • Integration │ │
│ • Trigonometric functions │ MA-C3 The Second Derivative │
│ • Logarithmic and exponential functions │ │
│ • Applications of calculus to the physical world │ MA-C4 Integral Calculus │
│ • Probability │ │
│ • Series and series applications │ Financial Mathematics │
│ │ │
│ │ MA-M1 Modelling Financial Situations │
│ │ │
│ │ Statistical Analysis │
│ │ │
│ │ MA-S2 Descriptive Statistics and Bivariate Data Analysis │
│ │ │
│ │ MA-S3 Random Variables │
Mathematics Extension 1: Content
The table below demonstrates the changes between the previous and new syllabus.
│ 3 Unit Preliminary Course │ │
│ │ New Mathematics Extension 1 Year 11 Course – Topics and Sub-topics (to be implemented in 2019) │
│ (current in 2018) │ │
│ │ Functions │
│ │ │
│ │ ME-F1 Further Work with Functions │
│ │ │
│ │ ME-F2 Polynomials │
│ • Other inequalities │ │
│ • Circle geometry │ Trigonometric Functions │
│ • Further trigonometry │ │
│ • Angles between two lines │ ME-T1 Inverse Trigonometric Functions │
│ • Internal & external division of lines into given ratios │ │
│ • Parametric representation │ ME-T2 further Trigonometric Identities │
│ • Permutations combinations │ │
│ • Polynomials │ Calculus │
│ │ │
│ │ ME-C2 Rates of Change │
│ │ │
│ │ Combinatorics │
│ │ │
│ │ ME-A1 Working with Combinatorics │
│ 3 Unit HSC Course │ │
│ │ New Mathematics Extension 1 Year 12 Course – Topics and Sub-topics (to be implemented in 2020) │
│ (current in 2019) │ │
│ │ Functions │
│ │ │
│ │ ME-F1 Further Work with Functions │
│ • Methods of integration │ │
│ • Primitive of sin2x and cos2x │ ME-F2 Polynomials │
│ • Equation dN/dt= k(N-P) │ │
│ • Velocity and acceleration as a function of x │ Trigonometric Functions │
│ • Projectile motion │ │
│ • Simple harmonic motion │ ME-T1 Inverse Trigonometric Functions │
│ • Inverse functions & inverse trigonometric functions │ │
│ • Induction │ ME-T2 Further Trigonometric Identities │
│ • Binomial theorem │ │
│ • Further probability │ Calculus │
│ • Iterative methods for numerical estimation of the roots of a polynomial equation │ │
│ • Harder applications of HSC 2 Unit topics │ ME-C2 Rates of Change │
│ │ │
│ │ Combinatorics │
│ │ │
│ │ ME-A1 Working with Combinatorics │
Mathematics Extension 2: Content
The table below demonstrates the changes between the previous and new syllabus.
│ 4 Unit Course │ │
│ │ New Mathematics Extension 2 Course – Topics and Sub-topics (to be implemented in 2020) │
│ (current until 2019) │ │
│ │ Proof │
│ │ │
│ │ MEX-P1 The Nature of Proof │
│ │ │
│ │ MEX-P2 Further Proof by Mathematical Induction │
│ │ │
│ │ Vectors │
│ • Graphs │ │
│ • Complex numbers │ MEX-V1 Further Work with Vectors │
│ • Conics │ │
│ • Integration │ Complex Numbers │
│ • Volumes │ │
│ • Mechanics │ MEX-N1 Introduction to Complex Numbers │
│ • Polynomials │ │
│ • Harder 3 Unit topics │ MEX-N2 Using Complex Numbers │
│ │ │
│ │ Calculus │
│ │ │
│ │ MEX-C1 Further Integration │
│ │ │
│ │ Mechanics │
│ │ │
│ │ MEX-M1 Applications of Calculus to Mechanics │
Assessment and examination
Advice regarding assessment and examination has been published on the NESA website and teachers should refer to the site regularly for updates. The most significant change is the approach to the
formal school-based assessment program for Year 11 and Year 12.
School-based assessment requirements
Teachers should refer to the NESA Assessment and Reporting in Mathematics Stage 6 document. Some features of the new syllabuses include:
The Year 11 formal school-based assessment program is to reflect the following requirements:
• three assessment tasks
• the minimum weighting for an individual task is 20%
• the maximum weighting for an individual task is 40%
• one task must be an assignment or investigation-style with a weighting of 20–30%.
The Year 12 formal school-based assessment program is to reflect the following requirements:
• a maximum of four assessment tasks
• the minimum weighting for an individual task is 10%
• the maximum weighting for an individual task is 40%
• only one task may be a formal written examination with a maximum weighting of 30%
• one task must be an assignment or investigation-style with a weighting of 15–30%.
NESA has provided the following examples of some approaches to task types for the assignment or investigation-style task:
• an investigative project or assignment involving presentation of work in class;
• an independently chosen project or investigation;
• scaffolded learning tasks culminating in an open-ended or modelling style problem;
• a guided investigation or research task involving collection of data and analysis.
Teachers can benefit from working collaboratively to plan for these new syllabuses. Access to professional learning time and resources will be essential and courses offered by the Centre for
Professional Leaning are an ideal place to begin.
Terry Moriarty has been a Mathematics teacher and Head Teacher in South and South Western Sydney for forty years. He has been involved in curriculum development processes throughout his career.
A Very Useful Aspirin: Networks and the New Stage 6 Mathematics Standard Syllabus
David Watson reflects on why the new Mathematics Standard course is useful for students and explains how to teach the new Networks topic…
A problem
The problem presented by the new Mathematics Standard syllabus did not reveal itself straight away.
In preparation for the new Networks topic, I reviewed everything I could. I searched key words such as Kruskal’s Algorithm and Prim’s Algorithm in Google and reviewed the resources provided by NESA
to support our programming and assessment.
In doing this work I was quickly reminded of my over-confidence while studying Network Theory at university. It was the beginning of this millennium and I was much younger and, perhaps, less wise. I
was twenty-two years old and in my final year and I was amazed at how simple I found the concepts. I even remember thinking that, “I could score 100 in this course!”
Score 100, I did not.
Upon exploring these Networks concepts again now, I enjoyed feeling good at it. It was fun to experience success. Then, while exploring examples online and reviewing the syllabus further, I found the
problem that I now consider the biggest danger in my programming for 2018…
It was all a very nice experience for me to return to my university days, to rediscover learning and knowledge I had thought lost or, at least, forgotten. Yet, in amongst the many applications listed
in the syllabus, including travel times, power cabling and garbage bin routes, all of which made sense to me, I realised I needed to think on how to help Networks make sense for my students.
Not just make sense, but actually be useful!
To steal a metaphor from Dan Meyer, if Network Diagrams, Shortest Paths, Minimum Spanning Trees and Critical Paths are the aspirin, how do we create the headache?
So, what are we doing with Networks?
In this section, I will present some examples of approaches to introducing the new concepts and reflect on some teaching challenges that I encountered while learning about this content. At the end of
the article, I will consider possible solutions to these teaching challenges.
Before you read any further, this article assumes the reader is comfortable to convert an image or table of a real world situation into a network diagram and to understand the language of Network
Theory. If you need help at this level you might visit the Mathspace Essentials free, online textbook for a simple and concise explanation, as this is the first section for both the Mathematics
Standard 1 (MS1) and Mathematics Standard 2 (MS2) pathways.
Konigsberg Bridge
In Mathematics Standard 2, one additional example is the Konigsberg Bridge problem. Images such as the one below are easily found via an internet search. The map of the city of Konigsberg in Prussia
illustrates that the city, either side of the Pregel River and including two islands, includes seven bridges. The problem posed is whether a path can be drawn, with any starting point, so that all
bridges are crossed exactly once.
Source https://en.wikipedia.org/wiki/Graph_theory#/media/File:Konigsberg_bridges.png
This bridge town and problem has many interesting elements, and essentially serves as an opportunity for students to investigate networks and practise their skills in modelling a real world
situation. A possible diagram that models this situation is below. Click here to view image
The seven bridges are represented by edges and the four separate land sections are represented by vertices. The problem now becomes: “Is there a path that travels along each edge exactly once?” The
answer becomes apparent after a few attempts.
It is interesting to note that Konigsberg is now called Kaliningrad and only five of the seven bridges still exist (only two in their original form). This can give rise to discussions about what this
new situation does to the problem, and does it matter which of the five bridges are still in existence?
Konigsberg Bridge teaching challenge
My first teaching challenge with this new topic arose when I found it easier to ‘play’ with this problem using the original image than when I attempted to use the new diagram above. I was fortunate
enough to have stumbled across the same point of view held by many of the students I have encountered in the current Mathematics General 2 course, seeing the creation of this diagram as a needless
extra step.
So why draw a network?
Shortest Path
I will return to the question of drawing a network later. For now, we continue exploring, and look into the concept of Shortest Path and Minimum Spanning Trees. These are concepts that are required
in both the MS1 and MS2 pathways.
The Shortest Path between two points is a fairly obvious concept if we consider the diagram below. We want to find the shortest path from vertex A to vertex B. This image is partway through the
algorithm, and the numbers ‘12’, ‘15’ and ‘14’ in vertices ‘C’, ‘D’ and ‘E’ respectively represent the minimum distance to get to the first three vertices. The shortest path to ‘D’ is through ‘C’.
Click here to view image.
From here we would write ‘27’ in vertex ‘F’, as the shortest distance to ‘F’ is through ‘D’. We would then write ‘31’ in vertex ‘B’, making the shortest distance from A to B equal to 31, with the
shortest path being A>C>D>F>B.
Shortest Path teaching challenge
Similarly to the Konigsberg Bridge problem, I encountered my second teaching challenge here. This algorithm was effective; however, I wondered if it was particularly different to what students would
do anyway? It was, in essence, an exhaustive method of solving the problem and I wondered if it was still a useful tool for students?
Minimum Spanning Trees
Once again, for now we push on and investigate Minimum Spanning Trees.
I discovered the definition: a set of edges with the minimum cost that connect all vertices together. This concept is, obviously, for weighted edges and also for undirected networks. Yet, the
application felt a bit less apparent to me, and so I went searching. The syllabus provided a good recommendation of connecting towns, places or locations to a power station or phone network.
In an online search, the problem that arises most is the ‘Muddy City Problem’. This problem involves a city where the mayor has decided to pave some of the pathways between houses to allow driving
access. The mayor hopes to allow for all houses to be accessed from any other house; however, the major also wants the minimum possible cost. Therefore, only the minimum spanning tree in the network
will be paved. To view a diagram and free lessons for the Muddy City Problem click here.
The number of pavers in the image displays the cost of paving each road (this could be price, resources or time required, and so on). Prim’s Algorithm suggests we first select the shortest edge, and
then continue by selecting the shortest ATTACHED edge. This continues until all vertices are included, and, of course, we avoid all loops. You may have already identified that there are many possible
beginnings, as the more edges with equal costs in a network, the more likely we are to find equal solutions.
Kruskal’s Algorithm requires us to start with the smallest edge, and then select the next smallest edge, regardless of whether it is attached to the existing tree or not. Again we must avoid any
loops. Regardless of where you begin, by the end of the process all distinct sections will link to make a tree.
A breakthrough
It was at this point that I began to see a solution to the teaching challenges outlined above. Not only were these algorithms both immediately helpful and relatively easy to follow, which was
encouraging, but I noticed a key point that I thought I might be able to use. All three problems introduced above can be investigated without the use of Network Theory. They may require scaffolding
for your class, but I found I could successfully introduce these problems to Stage 5 students, and all were intrigued and keen to “play” with the problem.
Critical Path Analysis
Now we move on to Critical Path Analysis, the first of two major skills required only by students following the MS2 pathway. When presented with a list of related tasks to complete a job, Critical
Path Analysis supports us to analyse the situation, identify the shortest possible time taken to complete the list as well as the latest start time for certain steps without delaying the overall
This tool has a variety of applications. A simple one with an example I have created is baking some biscuits for afternoon tea. I enjoy this example because it could be just about any recipe, so
students can create and analyse their own situation. The table below describes the steps involved, the prerequisites and the time for each step, as well as labels.
We are looking for the critical path, so we draw a network diagram, where the vertices represent a moment in time where you are available to start a new task (or tasks), and the edges represent the
tasks themselves. Below is an analysis of the above table.
In the analysis, it is evident that making a cup of tea (Task G) could be started after 21 minutes, and still not delay the entire task. Mixing in eggs, flour and choc chips (Task D) could not begin
until after 10 minutes.
The vertices are split in half and down the centre in my diagram (above), with the number on the left indicating the earliest time that jobs that begin from this vertex could begin. The space on the
right of each vertex is reserved for the latest time that a task beginning at this vertex could begin without delaying the overall job. How to communicate this latest start time varies depending on
the source you are reviewing, and by looking through a variety of textbooks as well as online industry explanations, I have seen a number of different forms of these vertices. These include circles
being divided with a horizontal line, or even vertices divided into three parts.
Critical Path Analysis inspired me with applications relevant to students’ future areas of employment, as well as to their present daily lives. All we really need to consider are tasks that are
dependent upon one another, and contribute to the completion of an overall job. Finally, and sometimes most challengingly, we are asking students to look for tasks that in some instances could be
completed at the same time.
Maximum-Flow, Minimum-Cut Theorem
The final skill included in the new syllabus is the use of Maximum-Flow, Minimum-Cut Theorem. This is used to determine the maximum flow of something through a network. Considering the network from
above from A to B, where A would be considered a source (where the flow originates from) and B considered a sink (where the flow ends). The question is what is the maximum flow that can get from A to
B? The lines cutting though the diagram represent “cuts”, because they completely separate the source and the sink. Click here to view image.
The blue, curved line is the minimum cut, as it severs the connection between A and B and it cuts through a total of 19. If the numbers in this diagram represent the number of litres of water that
can flow from one vertex to the next per minute, then this ‘19’ is the maximum flow per minute from A to B. The most that can flow into ‘B’ is clearly 24, and while we can easily ‘fill’ vertex ‘F’
with 4 litres per minute (min) and therefore maximise this edge (FB), there are only 15L/min worth of edges approaching ‘C’ and therefore we can only fill this with 15L per minute. This means that
while CB is able to allow 20L/min to flow through, only 15L/min is available, giving us a total flow of 15 + 4 = 19.
Similar to the Critical Path Analysis, this strategy has some obvious applications, such as in the area of traffic flow, water and power. In addition, both problems are available to students to
investigate without first being given the algorithms to solve. And I can feel a really pleasant headache.
So what to do about my challenges?
The question I was trying to solve while working through Network Theory was, breaking it right down, “Why?”
Not necessarily “Why is it in the course?”, although this is a question that would be answered as a result, but rather, why is it useful, and would I be able to help my students to see this
usefulness? Again, if these tools are the aspirin, how could I give my students the headache?
The value of the Konigsberg Bridge problem is not discovering whether or not the bridges can be traversed without repetition, but rather, how can we prove and communicate that a solution does not
exist, and why it does not exist. While ‘playing’ with the image might be more natural to students, investigating, discussing and communicating why there is no solution is best supported by the
network diagram. The proof relates to the odd degree of each vertex, which is difficult to examine without first defining the vertices.
The students I have shared Shortest Path problems with have been able to investigate the problem, and generally find the solution. When subsequently shown the algorithm, the room filled with
“ohhhhh”’s of realisation.
They were able to engage with the Muddy City Problem, order events in a critical path scenario and consider the maximum flow through a network. They often found solutions and could explain how they
found them, yet had difficulty convincing me or themselves that this was definitely the maximum, shortest or best solution. Most importantly, their confused looks and questions of one another turned
to smiles and satisfaction that there indeed was an easier and more effective way. Their headache had been relieved.
Final thoughts
Not only does allowing your students to investigate these problems first without the algorithm support them to discover the need for one, it provides a fantastic opportunity to apply problem-solving
skills and communicate and justify their solutions. When an algorithm is introduced, these skills are able to be revisited and enhanced with a deep understanding of useful tools.
And that is a very useful aspirin.
David Watson is a Mathematics Head Teacher in a Sydney High School, experienced in leading teachers from all stages of their careers in syllabus analysis and program development as well as
modernising and engaging the Mathematics classroom. He is a graduate of the University of Technology, Sydney and has worked in a variety of school settings, supporting students from a range of
different socio-economic and cultural backgrounds. Since 2015, David has been a working party member for Lachlan Macquarie College, providing professional learning and networking opportunities for
teachers as well as enrichment days for highly engaged students of Mathematics and Science.
The Making of a Teacher: Why NAPLAN is not Good Enough for Us
Richard Gill has directed the finest Australian Operas. He looks back on his time as a teacher and considers NAPLAN’s place in education today…
It might be expected that I will write about the efficacy of Music Education in the lives of children. I have written thousands and thousands of words on this subject and am always happy to do so.
However, I am now at that stage of my life where I think we have to see things as they really are.
Getting real
Quality Schools, the title of the Review to Achieve Educational Excellence in Australian Schools, or, “Gonski 2.0”, contains a sentence which says:
Australia has an excellent education system but our plateauing or declining results highlight that while strong levels of investment are important, it’s more important to ensure that funding is being
used on evidence-based initiatives that are proven to boost student results.
First, we do not have an excellent education system. If we did, we would not be plateauing (such a politically correct euphemism for failing).
Why do we need to boost results? Is schooling about results? It is hard to believe that in 2018, in a world so rich with wonderful things, all we can think about in a school is results.
How insulting this is to teachers. Is that why you teach? To get results which can be measured by others?
We cannot love what we do not know
As I work to improve our music education system, I am only too well aware of forces that seemingly conspire at every turn to frustrate the creative teacher and reward narrow ‘results’.
I was drawn to teaching because I loved reading novels, poetry and plays and loved music. I still do love all these things. I am also aware that I owe debts to people who helped me directly or
We cannot care about those things we do not love or know, and so we need, in this country, to let our teachers know that there are some of us out there who do care about you, who do share the concept
of a love of learning for its own sake, who don’t give a damn about a NAPLAN score, and who will go to the barricades for you and fight for the right for you to teach children properly.
Section 582, 1958
Allow me to introduce Mrs. Holder…
Mrs. Holder, a Lecturer in English, stood at the front of our section, Section 582 at the then Alexander Mackie Teachers’ College on a frosty September morning in 1958 and uttered the immortal words
which I have never forgotten:
Plan, teach, test.
Section 582, listen to me very carefully. If you don’t plan you can’t teach and if you can’t teach you can’t test and if you can’t test you have no idea what the children know. Remember – plan, teach
Plan, teach, test
At the age of sixteen, I was the youngest member of my section, having passed the Leaving Certificate in 1957 with Bs in English, Ancient History and Modern History and an A in French. Notice the
lack of Maths and Science!
I had applied to go to the then New South Wales State Conservatorium of Music to train as a High School Music Teacher, but my complete lack of Theory and Harmony led the examining panel to suggest to
me that were I to complete Sixth Grade Theory and Sixth Grade piano in that year I would be awarded a scholarship in 1959. They were as good as their word and in 1959 I made it to The Con.
In between times I had accepted a Teachers’ College Scholarship to Mackie as one couldn’t be certain of anything, and failure at tertiary level was real. No appeals, no show cause, no “I had a
nose-bleed in the exam”; just fail!
So it was that as the appointments to primary schools for practise teaching period were posted, I was sent to Eastwood Primary School.
It was decided that I should be given a very difficult class of all boys, a 5D, and from day one with this class and their brilliant teacher, Mr. Peter Black, my life changed. Mrs. Holder was my
supervisor so I planned, taught and tested to insanity.
I still have my three exercise books of lesson preparations with a comment given to me by Mr. Black on every lesson.
I could hardly wait to get to school each day and every day was a joy.
NAPLAN? … Anyone? … No?
So what has all of this to do with the iniquitous NAPLAN?
Even as a very young student teacher it was patently clear to me that the individual classroom teachers had an amazingly detailed knowledge of their pupils.
Morning teas and lunches in the staff room, apart from the usual banter, were often detailed discussions about children and their progress, or lack of it, indicating to me that a classroom teacher
was, in fact, constantly assessing and evaluating her or his students indirectly without having to write reams of pointless information.
If a parent wanted to know something about a child, an interview was arranged with the teacher.
While there were often fireworks, some parents believing that their children were direct descendants of Einstein and the Virgin Mary, with all the attendant virtues, most parents were content with
the reports which the classroom teacher could provide orally with an astonishing level of detail and depth of knowledge of the child in question.
Know thy students
On the matter of syllabus and curriculum, there were documents available for teachers to use which most of the teachers with whom I worked chose to consult rarely or chiefly ignore.
I think this was because they knew what levels their children were attaining in all areas and had realistic expectations of what they could do. In short, they created their own curricula.
The bright classes were well above average in every discipline, and a class such as mine, the fabulous 5D, was working at its own level. There was no point in doing activities or teaching concepts
out of reach of the children.
On one memorable afternoon, I was given a spectacular lesson in over-reach by Mr Black.
I had spent the entire one-hour lunch break creating a solar system on the blackboard, labelling the planets, tables of figures and the like. It was a visual triumph. There was more coloured chalk in
this masterpiece than Leonardo had ever dreamed of.
I gave the lesson during which I had the feeling that the kids couldn’t have cared less. At the end of the lesson Mr. Black asked me to wait behind after school to discuss what I had done.
During the discussion he said:
“These kids don’t have a concept of 10, let alone a concept of millions. The figures you gave them were meaningless to them. They have nothing to relate to and you gave them no real insights.
The use of coloured chalk, however, was very effective. See you tomorrow.”
I was shattered but knew that I had given a really dud lesson. At the same time, I was really grateful for the frank advice. Mrs. Holder, who had also sat in on the lesson, agreed and rammed home the
“You planned nicely but irrelevantly. You taught nothing, they learned nothing and therefore you couldn’t test them. Better luck next time.”
I still have those comments in my practice teaching exercise book lesson plans.
What worked about these times?
The points I am making are:
1. these teachers were fundamentally autonomous;
2. they devised their own curricula and syllabuses to suit their classes;
3. they collaborated with each other and shared ideas;
4. teaching was not competitive and there was no Federal interference;
5. they enjoyed their work, in the main, and the word ‘stress’ was unknown;
6. they knew the strengths, weaknesses and potentials of their charges;
7. they tested officially only twice a year;
8. a school report was a short one-page affair;
9. no one, and in some cases not even parents, knew their charges better than the teacher.
While many of these points would still apply today, NAPLAN has destroyed collegiality, created competition, created stressed-out parents, teachers, Principals and students and, above all, has
promoted a continually sliding scale of under-achievement nationally.
NAPLAN is not diagnostic. Never has been and never will be.
If the robots are permitted to take over marking students’ writing, the next idea will probably be to hire a robot to teach our children too. Creepy!
Looking to our future
No one, but no one, knows Primary school children better than the classroom teachers. Parents who think that a NAPLAN result is an indicator of a child’s abilities, capacities or potential are
seriously deluded. All a parent has to do is make an appointment to see a teacher, who can give the best diagnostic information about the child.
As I travel the country teaching classes and doing workshops, I always ask teachers and Principals what they think of NAPLAN. I haven’t yet met a Primary school teacher who has a good word to say
about NAPLAN. Some Principals tell me they are frightened to speak about NAPLAN because they feel they have to toe a party line.
Recently, I was giving a workshop in which my ten minute attack on NAPLAN was greeted with enthusiastic applause from the assembled teachers. At the end of the workshop a very timid teacher came up
to me, looked around the room several times before whispering “Thank you for that. We are not allowed say anything about NAPLAN to anyone or we will get into trouble.”
She looked once more around the room and then fled.
I hadn’t realised until that moment that we were living in a totalitarian state.
NAPLAN is not good enough for us
Surely teachers should be encouraged to have all sorts of views about all sorts of subjects without imposing any views on their students, but encouraging them also to have views and ideas and to have
all of these views without fear.
It seems to me we go to school for two reasons and two reasons only: to learn how to learn together, and to learn how to think for ourselves. NAPLAN encourages neither of those precepts. The
stranglehold of literacy and numeracy has hijacked all serious learning and enquiry.
Literacy and numeracy are NOT disciplines or subjects. They are states or conditions at which one arrives as a result of being well educated.
Schools which abandon their Arts disciplines in favour of more time given to literacy and numeracy are betraying their children, insulting their teachers, depriving their children’s minds of genuine
creative growth, limiting their imaginations and training them to be all the same.
Music, Art, Dance, Drama and so on are essential in the life of a child. It is through endless hours of play, fantasy, imaginative games, songs, dances, painting and the like that children begin to
make sense of the world. Stories, nursery rhymes, nonsense rhymes, acting out little scenes, together with the other activities already mentioned, are the stuff and lifeblood of education. Children
engaged in these activities learn to love learning.
This attitude to education is recognised in countries which seem to perform consistently well in all areas of learning. Have we anything to learn from them? Or are we too busy testing First Grade
Why are we so obsessed with assessment? Why the absence of commensurate treatment following this relentless ‘diagnosis’?
Why aren’t we as a nation totally devoted in our education programs to those disciplines which promote creative and imaginative thinking, and lead children down the genuine path to literacy and
I’ve seen in this country some brilliant creative teaching which fired up the minds of the children in an extraordinary way. It was inspiring at every level and something every teacher could do.
Teachers need to stand up and be counted and we need to rid this country of an iniquitous and destructive assessment system. I am not suggesting for one minute that children shouldn’t be tested;
remember Mrs. Holder’s wise advice: plan, teach, test. Simply that, in very early education testing is the job of the teacher, not some outside authority who has no real idea of your classroom.
Recently, I attended a Kindergarten assembly at which each child had a specific sentence to read. What was brilliant was that the teacher had devised the sentences according to each child’s ability
so that each child was successful in the eyes of the school community.
Why is this brilliant? Because it meant that the teacher was very well aware of what his children could do and he didn’t need an outside authority to help him.
Let’s all aim for a NAPLAN-free future and a return to teacher autonomy accompanied by appropriate fiscal remuneration for all good teachers.
Life is short and art is long. The minds, souls, hearts and imaginations of our children are immeasurable, priceless, invaluable and bursting with ideas. I want to hear those ideas so I can learn
something too.
Richard Gill AO, founding Music Director and Conductor Emeritus of Victorian Opera, is one of Australia’s most admired conductors and music educators. He has been Artistic Director of the Education
Program for the Sydney Symphony Orchestra, Artistic Director of OzOpera, Artistic Director and Chief Conductor of the Canberra Symphony Orchestra, and Artistic Advisor for the Musica Viva Education
program. He is the Founder and Director of the National Music Teacher Mentoring Program, Music Director of the Sydney Chamber Choir and the inaugural King & Wood Mallesons Conservatorium Chair in
Music Education, at the Conservatorium High School, Sydney.
“Perhaps it’s just as well that Leonard Bernstein is dead. Otherwise he’d probably have to relinquish his great reputation as a musical educator – or at least share it with Sydney’s Richard Gill.”
John Carmody, The Sun Herald
An Introduction to the New Mathematics Standard and Life Skills Syllabuses
Terry Moriarty introduces the new Stage 6 Mathematics syllabuses which are implemented for Year 11 in 2018…
The new NSW Stage 6 Mathematics Standard and Life Skills Syllabuses were endorsed in 2016. 2017 is a planning year with implementation for Year 11 in 2018 and Year 12 in 2019. The new Mathematics
Advanced, Extension 1 and Extension 2 syllabuses will be released following an additional period of consultation and the JPL will provide a guide in the Semester 2, 2017 edition.
Due to the online nature of the syllabus documents, teachers are encouraged to download and review each section, including the aim and rationale before moving to the course content.
New features of Stage 6 syllabuses include:
• Australian Curriculum content identified by codes;
• Learning Across the Curriculum content, including cross-curriculum priorities and general capabilities;
• publication in an interactive online format;
• an interactive glossary.
Initial information regarding assessment has been published by the NSW Education Standards Authority (NESA). The most significant change is the approach to the formal school-based assessment program
for Year 11 and Year 12. Examination specifications are expected to be available in Term 3, 2017.
Mathematics Standard
The Year 11 courses
Organisational structure
Mathematics Standard replaces the previous General Mathematics syllabus. There is a new organisational structure as well as updates to content.
The course is organised into topics with the topics divided into subtopics. Students can complete common content in Year 11 and then move into either Year 12 Mathematics Standard 1 or Year 12
Mathematics Standard 2.
Alternatively, teachers have flexibility within the common Year 11 content to address material that is essential for Mathematics Standard 1 in Year 12. This content is clearly indicated with a
diamond symbol
The content
The Year 11 content is common and there are no longer focus studies. Some of the topics from the previous focus studies have been retained within the topics, such as Plan for the Running and
Maintenance of a Car within the subtopic Money Matters and so existing resources may still be of use.
Modelling and applications are now an integral part of each strand and also merge strands together. The table below demonstrates the changes between the previous and new syllabus structures:
│ General Preliminary Course │ │
│ │ New Standard Year 11 Course (to be implemented in 2018) Topics and Subtopics │
│ (current in 2017) │ │
│ │ Algebra │
│ │ │
│ Financial Mathematics │ MS-A1 Formulae and Equations │
│ │ MS-A2 Linear Relationships │
│ Data and Statistics │ │
│ │ Measurement │
│ Measurement │ │
│ │ MS-M1 Applications of Measurement │
│ Probability │ MS-M2 Working with Time │
│ │ │
│ Algebra and Modelling │ Financial Mathematics │
│ │ │
│ (FS) Communication │ MS-F1 Money Matters │
│ │ │
│ (FS) Driving │ Statistical Analysis │
│ │ │
│ │ MS-S1 Data Analysis │
│ │ MS-S2 Relative Frequency and Probability │
School-based assessment requirements
Teachers should refer to the NESA Assessment and Reporting in Mathematics Standard Stage 6 document at: http://syllabus.bostes.nsw.edu.au/mathematics-standard-stage6/ . Teachers are encouraged to
refer to the relevant NESA documents for updates. Some features for the new syllabuses include:
The Year 11 formal school-based assessment program is to reflect the following requirements:
• three assessment tasks
• the minimum weighting for an individual task is 20%
• the maximum weighting for an individual task is 40%
• one task must be an assignment or investigation-style with a weighting of 20–30%.
NESA has provided the following examples of some approaches to task types for the assignment or investigation-style task:
• an investigative project or assignment involving presentation of work in class
• an independently chosen project or investigation
• scaffolded learning tasks culminating in an open-ended or modelling style problem
• a guided investigation or research task involving collection of data and analysis.
The Year 12 courses
The Mathematics Standard courses are Board Developed Courses and so students can achieve an HSC if they complete the course.
The content
Mathematics Standard 1
The table below demonstrates the changes between the previous and new syllabus structures:
│ General HSC Course │ │
│ │ New Standard 1 Year 12 Course (to be implemented in 2019) Topics and Subtopics │
│ (Current until 2018) │ │
│ │ Algebra │
│ │ │
│ Financial Mathematics │ MS-A3 Types of Relationships │
│ │ │
│ Data and Statistics │ Measurement │
│ │ │
│ Measurement │ MS-M3 Right-angled Triangles │
│ │ MS-M4 Rates │
│ Probability │ MS-M5 Scale Drawings │
│ │ │
│ Algebra and Modelling │ Financial Mathematics │
│ │ │
│ (FS) Design │ MS-F2 Investment │
│ │ MS-F3 Depreciation and Loans │
│ (FS) Household Finance │ │
│ │ Statistical Analysis │
│ (FS) The Human Body │ │
│ │ MS-S3 Further Statistical Analysis │
│ (FS) Personal Resources Usage │ │
│ │ Networks │
│ │ │
│ │ MS-N1 Networks and Paths │
Mathematics Standard 2
The table below demonstrates the changes between the previous and new syllabus structures:
│ General HSC Course │ New Standard 2 Year 12 Course (to be implemented in 2019) │
│ │ │
│ (Current until 2018) │ Topics and Subtopics │
│ │ Algebra │
│ │ │
│ │ MS-A4 Types of Relationships │
│ │ │
│ │ Measurement │
│ Financial Mathematics │ │
│ │ MS-M6 Non-right-angled Trigonometry │
│ Data and Statistics │ MS-M7 Rates and Ratios │
│ │ │
│ Measurement │ Financial Mathematics │
│ │ │
│ Probability │ MS-F4 Investments and Loans │
│ │ MS-F5 Annuities │
│ Algebra and Modelling │ │
│ │ Statistical Analysis │
│ (FS) Health │ │
│ │ MS-S4 Bivariate Data Analysis │
│ (FS) Resources │ MS-S5 The Normal Distribution │
│ │ │
│ │ Networks │
│ │ │
│ │ MS-N2 Network Concepts │
│ │ MS-N3 Critical Path Analysis │
School-based assessment requirements
Teachers should refer to the NESA Assessment and Reporting in Mathematics Standard Stage 6 document for updates. Some features for the new syllabuses include:
The Year 12 formal school-based assessment program is to reflect the following requirements:
• a maximum of four assessment tasks
• the minimum weighting for an individual task is 10%
• the maximum weighting for an individual task is 40%
• one task may be a formal written examination with a maximum weighting of 30%
• one task must be an assignment or investigation-style with a weighting of 15–30%.
Life Skills
The Life Skills course has been re-written to align with the new topics in Standard Mathematics: Measurement, Algebra, Financial Mathematics, Statistical Analysis, and Networks.
Teachers may choose the most relevant aspects of the content to meet the particular needs of individual students and identify the most appropriate contexts for the student to engage with the
outcomes, for example, school, community or workplace. Students will not be required to complete all of the content to demonstrate achievement of an outcome.
In implementing the new syllabuses for Stage 6 Mathematics, the importance of collaboration of teachers between schools and within faculties will be essential. Professional learning opportunities
such as those conducted by the Centre for Professional Learning will also be useful in supporting these processes. For more information visit: http://cpl.asn.au/
Terry Moriarty has been a Mathematics teacher and Head Teacher in South and South Western Sydney for forty years. He has been involved in curriculum development processes throughout his career.
Engagement and Mathematics: What does it look like in your classroom?
Catherine Attard continues her guidance about making Maths come alive in your primary classroom…
What does it look like, feel like and sound like when your students are deeply engaged in a mathematics task? What is it like when they are disengaged? In my previous article for the JPL I provided a
definition of engagement as a multidimensional construct, consisting of three domains: operative, cognitive and affective. The coming together of the three domains leads to students feeling good,
thinking hard, and actively participating in their Mathematics learning (Fair Go Team NSW Department of Education and Training, 2006; Fredericks, Blumenfeld & Paris, 2004).
I also provided a discussion on the importance of establishing positive pedagogical relationships as a foundation for student engagement in Mathematics. In this paper I will move beyond pedagogical
relationships to discuss what happens in practice – the pedagogical repertoires that promote positive student engagement.
The following figure (Figure 1) is an excerpt from the Framework for Engagement (FEM), (Attard, 2014), which provides a summary of the critical elements of engaging pedagogies.
│ In an engaging Mathematics classroom pedagogical repertoires mean: │
│ │
│ │
│ │
│ • there is substantive conversation about mathematical concepts and their applications to life; │
│ • tasks are positive, provide opportunity for all students to achieve a level of success and are challenging for all; │
│ • students are provided an element of choice; │
│ • technology is embedded and used when appropriate to enhance mathematical understanding through a student-centred approach to learning; │
│ • the relevance of the mathematics curriculum is explicitly linked to students’ lives outside the classroom and empowers students with the capacity to transform and reform their lives.│
│ │
│ Mathematics lessons regularly include a variety of tasks that cater to the diverse needs of learners │
Figure 1: Engaging Repertoires (Attard, 2014)
What do these elements look like in practice? I will expand on each of the points illustrated in Figure 1, and provide some practical advice on how the pedagogies can be applied.
Firstly, how do we provide opportunities for substantive conversations between students and the teacher, and amongst students? If you consider a traditional approach to teaching where the Mathematics
lessons are based on a drill and practice approach, it is difficult to see where important mathematical conversations can take place. However, consider an approach where collaboration is encouraged
through problem solving and investigation, and where student reflection is an integral aspect of every Mathematics lesson, regardless of the types of tasks and activities implemented.
We must also consider the Working Mathematically components of our K-10 Mathematics Syllabus (Board of Studies New South Wales, 2012). Promoting substantive conversation allows students access to
each of the five components: Reasoning, Communicating, Understanding, Fluency and Problem Solving, and provides teachers with opportunities to assess them.
The provision of tasks that provide opportunity for all students to succeed can be a challenge for teachers. It is often difficult to differentiate activities to ensure the diversity of academic
ability is not only addressed, but provides sufficient challenge. Learners need to experience success and a sense of achievement if they are to develop a positive attitude towards Mathematics. One
way of ensuring all learners are challenged is to provide open-ended, rich tasks rather than closed problems that only have one correct answer or limited opportunities to apply a range of strategies.
Allowing student choice in the Mathematics classroom is an important element of engagement and sends important messages relating to power and control. You can provide choice by having alternative
activities within a specific mathematical content area, or you can have students choose how they present their work. Perhaps students may choose to work with concrete materials or interact with
appropriate technology. This does not have to occur in every lesson, but allowing students the freedom to make choices every now and then can contribute to their overall engagement.
Technology has become an integral part of contemporary life, and as such, our curriculum requires us to use it meaningfully to enhance the teaching and learning of Mathematics. The challenge with
using technology in Mathematics lessons, however, is to ensure that we promote a student-centred approach. If you take for example, the interactive whiteboard, consider how it positions the teacher.
The whiteboard is fixed and usually located at the front of the classroom. Any interactivity usually occurs between one person (often the teacher), and the whiteboard. The teacher has control and
students are generally passive (Attard & Orlando, 2014). How can this engage all learners?
Many schools have introduced 1:1 laptop or tablet programs, however there is a danger that the devices may be used simply as a replacement for a traditional textbook or as a word-processing device to
replace pen and paper. Online Mathematics programs provide some functional improvement to textbooks, however the opportunities for students to collaborate and become involved in substantive
mathematical conversations is limited.
Fortunately, the introduction of mobile technologies such as tablets has now provided us with rich opportunities to develop highly engaging, student-centred mathematical activities and tasks.
The use of contemporary technologies in Mathematics lessons provides opportunities to illustrate the relevance of Mathematics and bridge the digital divide between the school and students’ lives
outside school. However, it does not necessarily mean students will be engaged. Caution must be taken to ensure the use of technology is driven by good pedagogy, rather than the technology becoming
the focus of the lesson. Other ways to illustrate the relevance of Mathematics is to, where possible, embed mathematical concepts into real-life contexts and allow opportunities for students to apply
Mathematics in meaningful and purposeful ways. This not only deepens mathematical understanding but will enhance engagement. Of course, as mathematical concepts become more abstract in the senior
years it is not always possible or practical to apply all concepts to real-life contexts, however if students have developed a love of Mathematics through quality practices, their engagement will be
The final aspect of the FEM relating to pedagogical repertoires refers to the provision of variety within Mathematics lessons. Although young students do require some structure, variety can be
provided within that structure. For example, in the primary classroom children can be presented with a range of tasks that use a range of resources. Sometimes Mathematics lessons can be conducted
outside the classroom – consider running a maths trail at your school where students can participate in interesting mathematical investigations based upon their physical surroundings. Explore the
use of tools such as Thinkers’ Keys (Attard, 2013) to provide Mathematics tasks that are open-ended and creative, and set homework that takes advantage of the Mathematics in students’ lives, rather
than drill and practice activities.
I have provided a brief exploration of engaging pedagogies that are listed in the Framework for Engagement with Mathematics (FEM), (Attard, 2014). Engagement with Mathematics during the compulsory
years of schooling is critical if students are to develop an appreciation for and understanding of the value of Mathematics learning. Students who are engaged are more likely to learn, find the
experience of schooling more rewarding, and more likely to continue with higher education. How can you adapt your practices so that your students value the Mathematics they are learning and see
connections between the Mathematics they do at school and their own lives beyond the classroom now and in the future?
Attard, C. (2014). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives, 34(3), 1-14
Attard, C. (2013). Engaging maths: Higher order thinking with thinkers’ keys. Modern Teaching Aids: Brookvale
Attard C, & Orlando J, 2014, Early career teachers, mathematics and technology: device conflict and emerging mathematical knowledge. In J. Anderson, M. Cavanagh, & A. Prescott, Curriculum in Focus:
Research Guided Practice, proceedings of the Mathematics Education Research Group of Australasia annual conference, pp 71-78. MERGA: Sydney
Board of Studies New South Wales. (2012). Mathematics K-10 syllabus. Retrieved from http://syllabus.bos.nsw.edu.au/
Fair Go Team NSW Department of Education and Training. (2006). School is for me: pathways to student engagement. Sydney: NSW Department of Education and Training, Sydney, Australia.
Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59 -110
Dr Catherine Attard worked as a school teacher and proceeded to complete a PhD on student engagement. She has been a part of the Fair Go Project Team at the University of Western Sydney. She is
also editor of the journal Australian Primary Mathematics Classroom.
Catherine Attard conducts a weekly blog at http://engagingmaths.co/about/?blogsub=confirming#blog_subscription-3 that has a number of resources that teachers are able to access and use.”
Getting Passionate About Maths
Catherine Attard explores some strategies to increase student engagement in Maths …
“I like having a teacher who is really passionate about maths”: Getting students to engage with mathematics through positive pedagogical relationships
How often do teachers of Mathematics hear the phrase “why do I need to learn this?” or “I’m no good at Maths”? Many people attribute anxiety or a dislike of Mathematics to their experiences during
the middle years of schooling (Years 5 to 8) and although students are influenced to some degree by parents and peers, it is the teacher who has the most influence on students’ engagement with
mathematics. This article explores the construct of engagement as it relates to Mathematics, and suggests that for deep and sustained engagement to occur, positive pedagogical relationships, the
interpersonal relationships between teachers and students that optimise engagement, must first be established.
Defining engagement
As teachers, we use the term ‘engagement’ often, but do we really understand what real engagement looks like? When we see students who are ‘on task’, are they engaged, or are they just involved in
busy work, and in getting the task done? Consider the difference between students who are ‘on task’, and students who ‘in task’. When students are ‘in task’, their minds and bodies are focused on
what they are doing. They might be participating in substantive dialogue about the topic, or they might be working in silence, thinking deeply about Mathematics they are involved in – either way,
they are engaged.
Many definitions of engagement are found in education literature. Some provide a narrow view that relates only to behaviour and participation. Others provide a deeper understanding that is
multi-dimensional. Fredricks, Blumenfeld and Paris (2004), define engagement as a deeper student relationship with classroom work, multi-faceted and operating at cognitive, emotional, and behavioural
levels. In this paper, I draw on work of the Fair Go Project (Fair Go Team NSW Department of Education and Training, 2006) and define engagement as the coming together of three facets – cognitive,
operative, and affective, which leads to children valuing and enjoying, and actively involved with school mathematics, and seeing connections between the Mathematics they do at school, and their own
lives beyond the classroom now and in the future.
Pedagogical relationships and mathematics
This paper is informed by a longitudinal study on the influences on engagement (for a more in depth description see Attard, 2011, 2013, in print). In the study, data were collected from a group of 20
children across three years of their schooling from Year 6 to Year 8. The major selection criterion for participation in this project was that the students had to identify themselves as being engaged
with Mathematics (through the use of a Motivation and Engagement Scale (Martin, 2008). Data were collected through individual student and teacher interviews, student focus groups, and classroom
During the first phase of the study when the students were still attending primary school, they identified their current teacher as someone they perceived to be a good Mathematics teacher. They
articulated several attributes directly relating to the pedagogical relationships the teacher had formed with her students, such as her ability to cater to individual needs through the
differentiation of tasks, and her modeling of enthusiasm and passion towards Mathematics. Comments such as these were typical: “I like having a teacher who is really passionate about Maths” (Alison,
Year 6), and “…while you’re doing the work she also has fun teaching the Maths as well” (Tenille, Year 6).
In the second phase of the study, things changed for this group of students. They began their secondary education, at a new school that was significantly different at the time from traditional
secondary schools. At the time the school identified itself as a ‘ground breaking’ learning community in relation to its multi-disciplinary approach to curriculum, large open teaching spaces and a
teaching structure that saw a group of Mathematics teachers rotate amongst classes, which meant each class group did not have one allocated teacher and saw each teacher every fourth lesson. These
structures were not conducive to building relationships – the teachers had very limited opportunities to identify student needs and abilities, and as a result, students became disengaged: “everyone’s
excited when there’s no Maths. I think it’s because, not having someone explain it to you and you don’t get it. If you don’t get it that means you don’t like it” (Kristy, Year 7).
Fortunately circumstances improved for the students in Year 8. Teachers were allocated a class group and the students were back on the path to engagement. They felt that they were now seen as
individuals rather than a collective, and teachers cared more about their learning. They also felt that if they required assistance from their teachers, they felt safe in asking for help and felt the
teachers now wanted to help them. The increased opportunity to develop pedagogical relationships also improved the level of feedback students received, which began to re-build their confidence as
well as their engagement.
During the course of the study the students experienced a wide range of teaching and learning situations that resulted in significant fluctuations of their engagement levels. Although the data
overwhelmingly confirmed the teacher was the strongest influence on these students’ engagement, this influence appeared to be complex, consisting of two separate yet inter-related elements:
pedagogical relationships and pedagogical repertoires. Pedagogical repertoires refer to the day-to-day teaching practices employed by the teacher.
Results of this study suggest that it is difficult for students to engage with Mathematics without a foundation of strong pedagogical relationships. Positive pedagogical relationships exist when:
• students’ backgrounds and pre-existing knowledge are acknowledged and contribute to the learning of others;
• interaction among students and between teacher and students is continuous;
• the teacher models enthusiasm and an enjoyment of Mathematics and has a strong Pedagogical Content Knowledge;
• the teacher is aware of each student’s abilities and learning needs; and
• feedback to students is constructive, purposeful and timely.
It can also be argued that it is through engaging pedagogies that positive pedagogical relationships are developed, highlighting the connections between relationships and engaging repertoires. So
what are considered engaging pedagogies in the Mathematics classroom? These will be explored in the next issue of The Journal of Professional Learning.
Catherine Attard worked as a school teacher and proceeded to complete a PhD on student engagement. She has been a part of the Fair Go Project Team at the University of Western Sydney. She is also
editor of the journal Australian Primary Mathematics Classroom.
Catherine Attard, University of Western Sydney
Attard, C. (2011). “My favourite subject is maths. For some reason no-one really agrees with me”: Student perspectives of mathematics teaching and learning in the upper primary classroom. Mathematics
Education Research Journal, 23(3), 363-377.
Attard, C. (2013). “If I had to pick any subject, it wouldn’t be maths”: Foundations for engagement with mathematics during the middle years. Mathematics Education Research Journal, 25(4), 569-587.
Attard, C. (in print). “I don’t like it, I don’t love it, but I do it and I don’t mind”: Introducing a framework for engagement with mathematics. Curriculum Perspectives.
Fair Go Team NSW Department of Education and Training. (2006). School is for me: pathways to student engagement. Sydney: NSW Department of Education and Training, Sydney, Australia.
Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence. Review of Educational Research, 74(1), 59 -110.
Martin, A. J. (2008). Motivation and engagement Scale: High school (MES-HS) test user manual. Sydney: Lifelong Achievement Group. | {"url":"https://cpl.nswtf.org.au/subject/mathematics/","timestamp":"2024-11-09T11:05:49Z","content_type":"text/html","content_length":"202248","record_id":"<urn:uuid:dcaba3a3-92d6-4004-9487-f31ddd4f377d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00004.warc.gz"} |
Source: “Questions of Philosophy”, No. 6, 1951, pp. 143-149
To the results of the discussion of questions of logic
In recent years, many unclear and controversial issues have emerged in the teaching of logic and in published logical works.
By raising issues of logic for discussion and organizing a wide exchange of opinions on them, the editors of the journal considered it necessary to identify different opinions in the interpretation
of a number of issues of logic and put an end to the confusion and confusion that exists in the views of many logic specialists. The insufficient mastery of the fundamentals of Marxism-Leninism by
some of them has led to the fact that this confusion, as the materials of the discussion show, exists not only on issues that have not yet been sufficiently developed, but also on issues that were
resolved long ago by the classics of Marxism-Leninism and, therefore, are completely indisputable.
It is well known that thinking, the laws and forms of which constitute the subject of logic, is inextricably linked with language. J.V. Stalin teaches: “Whatever thoughts arise in a person’s head and
whenever they arise, they can arise and exist only on the basis of linguistic material, on the basis of linguistic terms and phrases. Bare thoughts, free from linguistic material, free from
linguistic “natural matter”, do not exist. “Language is the immediate reality of thought” (Marx). The reality of thought is manifested in language. Only idealists can talk about thinking that is not
connected with the “natural matter” of language, about thinking without language”[1]. The confusion and direct vulgarization of Marxism in the field of logic basically followed the same line as the
confusion and vulgarization in linguistics. The vulgarizers of Marxism—Marr and his followers—considered class language and attributed it to the superstructure. Similar attitudes existed in logic.
And here the vulgarizers of Marxism considered the laws and forms of thinking studied by formal logic to be superstructural, class, and in accordance with this, at one time they declared formal logic
to be a weapon of the class enemy, the basis of a religious worldview, and on this basis they expelled it from high school. As a result of this, Soviet youth did not receive knowledge of basic rules
and techniques of logical thinking in high school. V.I. Lenin, back in 1921, pointed out the need to study formal logic (with amendments) in the lower classes of the Soviet school; similar
instructions were repeatedly given by J.V. Stalin.
In 1946, the Central Committee of the All-Union Communist Party of Bolsheviks, on the initiative of Comrade Stalin, ordered the introduction of the teaching of logic in secondary schools. However,
the vicious, anti-Marxist concept of the class character of logic continued to enjoy support from some workers of the Ministry of Higher Education of the USSR, the Institute of Philosophy of the USSR
Academy of Sciences and other leading philosophical institutions of the country, which found expression in programs on logic, in books on logic prepared for publication, and even more so in oral
Opinions were expressed that since in an exploitative society logic always served the ruling classes in order to strengthen their class dominance, then by its very essence it always had a
superstructural character. For example, in order No. 361 of March 23, 1948, the former Minister of Higher Education of the USSR S.V. Kaftanov, who assessed the work of the Department of Logic of
Moscow State University, stated that “formal logic in ancient times defended the ideology of slave owners, in the Middle Ages it was the handmaiden of theology, and in capitalist society adapts to
the bourgeoisie in order to keep the oppressed classes captive to bourgeois ideology.”
In accordance with this, the idea of creating a special, “Soviet” logic was put forward, which should be the opposite of the old, supposedly entirely bourgeois, formal logic. In the “Program on Logic
for Departments of Logic and Departments of Logic and Psychology of Pedagogical Institutes and Universities,” approved by the Department of Teaching Social Sciences of the Ministry of Higher
Education of the USSR in July 1949, it is printed: “Partyism in the science of logic. The auxiliary role of logic in relation to ideology.” “Soviet logic is a sharply honed ideological weapon of the
Soviet people in the fight against the remnants of the past in the minds of people, in the fight against bourgeois ideology.”
Non-Marxist attitudes that the logic of thinking is superstructural, class in nature, that each socio-economic system has its own logic and that therefore it is necessary to create some kind of
special, “Soviet” logic, are like two peas in a pod, similar to the views of Marr and his students on the tongue.
A direct consequence of this attitude was the assertion of supporters of the classism of logic that the logic taught in our schools should be considered not as the old, formal logic, freed from
idealism, scholasticism, metaphysics, but as a kind of “dialectized” formal logic. This was clearly a vulgarizing, alien to Marxism, approach to mixing formal logic with dialectics, to replacing
Marxist dialectics with formal logic, which V. I. Lenin and I. V. Stalin spoke out against with all harshness. Calling for the creation of a new, “Soviet” logic, as a single one, in which formal
logic and dialectics are inseparably fused (more precisely, mixed), the vulgarizers of Marxism essentially rejected both formal logic and Marxist dialectical logic. By this, without any
justification, that is, violating the most elementary requirements of logic, they rejected the direct instructions of Engels and Lenin regarding the main features and characteristics of Marxist
dialectical logic, distinguishing it from elementary, school logic, usually called formal logic.
The discussion on linguistics and the work of Comrade Stalin “Marxism and Questions of Linguistics” forced the vulgarizers of Marxism to change their minds somewhat in matters of logic. They had to
abandon the original concept of “class logic” as too obviously un-Marxist. But with all the greater tenacity they began to defend the idea of a “single” logic, which was entirely derived from this
vicious concept, and in fact a formal logic mixed with Marxist dialectics, with the worldview of the Bolshevik party. But they forgot - or pretended to forget - that such a mixture of formal logic
with dialectics was proclaimed by them in their time precisely in order to distinguish the “new”, “Soviet”, supposedly class logic from the old, bourgeois logic. class formal logic. Being forced to
abandon the initial position about the class nature of logic, they want to necessarily preserve the consequence that follows from it, thereby revealing their inability to be in harmony with logic, to
be logically consistent. In some published articles, the nihilistic attitude towards formal logic is motivated by the fact that logic, as a science about forms of thinking, is class-based,
party-based, although the object of its study is forms of thinking that are universal to mankind, and that therefore “bourgeois” formal logic should give way to “Soviet” , “dialectized” logic. The
authors of these articles vulgarize the Leninist principle of the partisanship of science, lumping together theoretical sciences about society (political economy, sociology, etc.), the entire essence
of which is class, with sciences that study non-class phenomena (for example, grammar, formal logic), which, Of course, like any other sciences, they are used by different classes, but the main
content of which cannot be considered class-specific.
There is no doubt that such confusion and vulgarization have a detrimental effect on the activities of researchers and teachers of logic, graduate students and students, disorienting them. Decisive
and swift action is needed to stop this confusion and vulgarization.
As the discussion showed, Soviet logicians, relying on the Stalinist doctrine of language and its organic connection with thinking, made correct conclusions regarding the logical forms and laws of
thinking studied by formal logic. These conclusions can be formulated in the following main points:
a) Logical forms and laws of thinking are not a superstructure over the basis, just as language, which is closely connected with thinking, is not a superstructure over the basis. Thinking does not
disappear with the disappearance of one or another basis and its corresponding superstructure, it only changes. Consequently, the laws and forms of thinking also do not disappear, but only develop.
b) Not being a superstructure over the base, the forms and laws of thinking are not of a class, but of a universal human nature. The logical apparatus of thinking, its forms (concept, judgment,
inference) and the laws of their functioning among representatives of different classes are absolutely the same, just as they are absolutely the same among representatives of different nations. The
forms and laws of thinking are a reflection of one and the same objective reality, the result of billions of times repeated practical activities of people.
c) Like language, thinking, in contrast to the superstructure, is directly related to production and any other human activity. Any significant change in human activity is reflected in thinking in the
form of the emergence of new concepts, judgments, conclusions, without waiting for changes in the basis to occur.
d) The logical system of thinking, its laws and, to an even greater extent, their theories are constantly changing and developing. However, as in the development of language, there are no explosions.
The forms and laws of thinking develop slowly, through the gradual death of elements of the old quality and the accumulation of elements of a new quality.
The discussion further showed that the majority of Soviet logicians and philosophers adhere to the correct, Marxist point of view on formal logic and its relationship to dialectical logic. This
Marxist point of view boils down to the following: formal logic is the science of elementary laws and forms of correct thinking. It is a collection of elementary rules on how to use concepts,
judgments, and inferences so that our thinking is definite, coherent, consistent, demonstrative, and consistent. Formal logic is elementary. According to Lenin’s characterization, it “takes formal
definitions, guided by what is most common or what most often catches the eye, and limits itself to this”[2].
Formal logic, being absolutely necessary, although not sufficient for complete knowledge of the subject, is by no means metaphysics, since it is not absolutized, it is not recognized as the only
possible one.
There are not two formal logics: the old, metaphysical, and the new, dialectical, just as there are not two - metaphysical and dialectical - arithmetic, grammar. There is one formal logic, universal;
it is a set of elementary rules of thinking, it is the simplest teaching about these rules.
The need for formal logic is due to the fact that it provides rules for logical thinking, which are mandatory for all people and non-compliance with which leads to the destruction of thinking, to
chaos, confusion in thinking. These rules must be followed in order to think harmoniously and consistently. Those who violate these rules have no order in their thoughts, and therefore cannot have
order in their actions, since a person’s actions must always be meaningful.
Knowledge and observance of the elementary rules of formal logic are necessary not only for schoolchildren, but also for every adult. They are necessary for party and Soviet workers, engineers,
teachers, doctors, agronomists, lawyers, etc. Without the ability to think consistently and definitely, one cannot manage any area of work. Chatterboxes and muddle heads are distinguished, in
particular, by the fact that in their reasoning, which violates the elementary rules of logic, they drown the living matter, introducing chaos and confusion. If a person does not know and does not
follow the rules of logic, he cannot be understood; Such a person's thinking is unsystematic, and the results of thinking are incorrect. Just as a person who does not know the rules of arithmetic and
grammar cannot count and write correctly, so a person who does not know the rules of logic cannot reason and act correctly. This is the meaning of formal logic.
Marxist dialectical logic coincides with the dialectics and theory of knowledge of Marxism; it, in essence, represents an identity with them.
Dialectical logic “is a teaching not about external forms of thinking, but about the laws of development... of the entire concrete content of the world and its knowledge, that is, the result, sum,
conclusion of the history of knowledge of the world”[3]. Dialectical logic is applied both to the study of laws and forms of thinking, and to the study of the laws of reality. It reveals the organic
connection between the forms and laws of thinking and the laws of the objective world, showing that they are nothing more than a reflection of the laws of the objective world.
Compared to formal logic, dialectical logic is a qualitatively new, higher stage in the development of thinking. Its relationship to formal logic, according to Engels' deep comparison, is similar to
the relationship of higher mathematics to lower mathematics.
“Dialectical logic, in contrast to the old, purely formal logic, is not content with listing and without any connection placing next to each other the forms of the movement of thinking, that is,
various forms of judgments and inferences. On the contrary, it derives these forms from one another, establishes between them a relationship of subordination, not coordination, it develops higher
forms from lower ones”[4]. Dialectical logic, being the highest logic, does not eliminate the lower, formal logic, but shows its limitations. Dialectical logic is an integral part of Marxism, but
formal logic is not an integral part of Marxism.
This is the Marxist point of view on formal logic and its relationship to dialectical logic. This point of view is clearly stated in the works of the classics of Marxism-Leninism.
From this Marxist point of view, “projects” for the creation of some kind of “new”, “special”, “dialectical formal logic”, or, as some put it, “formal logic of the dialectical method” are refuted as
worthless and obviously harmful. Such a “dialectization” of formal logic leads, on the one hand, to the vulgarization of Marxism, and on the other hand, undermining the very foundations of the very
existence of formal logic, to its complete elimination, because it is impossible to “dialectize” formal logic without thereby destroying it as logic.
“Dialectical formal logic” is complete nonsense.
The “dialectization” of formal logic in scientific and educational work has always been and will remain an eclectic mixture of formal logic with dialectical materialism. An eclectic confusion of
formal logic with dialectical logic occurs whenever facts or results obtained through the application of dialectical logic are attempted to be explained by means of formal logic, or when formal logic
is attempted to be presented “dialectically.”
The line of mixing formal logic with dialectical materialism, of incorporating formal logic into Marxism, is currently the most confused, erroneous and harmful line in logic, distorting the
principles of Marxism.
Therefore, the task of Soviet logicians is to wage the most decisive struggle against this line.
Another distortion of the principles of Marxism is the attempt revealed during the discussion by some logicians to present formal logic as the science of such laws of thinking that supposedly do not
reflect any aspects of objective reality, but are only specific laws of thinking itself. To prove this, the argument is usually given that if in nature and society everything develops and changes,
then in thinking, on the contrary, the law of identity operates as the fundamental law, erroneously interpreted as the law of constancy and immutability. It is absolutely clear that this idea of
formal logical laws (the law of identity, contradiction, excluded third and sufficient reason) is a Kantian, idealistic idea. It must be remembered that formal logic, like any field of knowledge,
throughout its history has also been the arena of a fierce struggle between materialism and idealism. The separation of the forms and laws of thinking from reality, the denial that they are a
reflection of objective connections and laws, inevitably leads to a separation of subject and object. The slightest concession to such views, and even more so their defense, means a betrayal of the
basic principle of materialism. Soviet logicians must wage the most implacable struggle against this kind of idealistic perversion.
Finally, some logicians have developed a tendency to believe that formal logic is the only science about the laws and forms of thinking, whereas in fact, in addition to formal logic, dialectical
logic is also concerned with the laws and forms of thinking. The discussion of questions of logic had a number of shortcomings. Some participants in the discussion took an unprincipled position of
the “golden mean”, recognizing, on the one hand, that formal logic is inferior logic in relation to dialectical logic, and on the other, seeing it as a necessary component of dialectical logic. Some
comrades, instead of considering the issue on its merits, limited themselves to simply quoting the classics of Marxism-Leninism, taking individual statements out of context, without revealing their
deep meaning, and sometimes arbitrarily interpreting them.
One cannot but admit that a disadvantage of the discussion is the fact that many professional philosophers did not take part in it.
The discussion revealed among logicians a misunderstanding of quite clear and long-solved questions in Marxism. These erroneous, non-Marxist views have greatly prevented logicians from creating a
full-fledged textbook of formal logic and are hindering the development of problems of logic; they indicate that not all Soviet logicians fully mastered the fundamentals of Marxist-Leninist theory.
The path that should be followed in order to overcome the major errors and deviations from Marxism that have emerged in some of our logicians is the path of a serious, in-depth study of the works of
the classics of Marxism-Leninism.
Soviet logicians are faced with tasks of enormous importance and, above all, tasks arising from the work of J.V. Stalin “Marxism and Questions of Linguistics.”
1) Soviet logicians must persistently and purposefully cultivate the skills of thinking accurately and consistently. They must mercilessly combat all violations of logical rules, regardless of
whether these violations occur in students or adults.
2) It is necessary to develop, using specific material, the question of the unity of language and thinking, logical and grammatical forms, and the relationship between logic and grammar.
J.V. Stalin teaches: “A distinctive feature of grammar is that it gives rules for changing words, meaning not specific words, but words in general without any specificity; it gives rules for
composing sentences, meaning not any specific sentences, say, a specific subject, a specific predicate, etc., but all kinds of sentences in general, regardless of the specific form of a particular
sentence. Consequently, abstracting from the particular and concrete, both in words and in sentences, grammar takes that general thing that underlies changes in words and combinations of words in
sentences, and builds from it grammatical rules, grammatical laws. Grammar is the result of long, abstract work of human thinking, an indicator of the enormous success of thinking.
In this respect, grammar resembles geometry, which gives its laws by abstracting from specific objects, considering objects as bodies devoid of concreteness, and defining the relations between them
not as specific relations of such and such specific objects, but as relations of bodies in general, devoid of any concreteness "[5].
Formal logic, examining its subject, deals with it in the same way as grammar deals with its subject. When studying the forms and laws of thinking, she preserved the general and abstracted from the
individual, the specific. Considering a concept, judgment, inference, she formulates rules that relate not to certain specific concepts, judgments, inferences, but to concepts in general, judgments
in general, inferences in general; consequently, it is abstracted from the specific content of concepts, judgments, and conclusions. Studying the connection between logic and grammar from this side
is an extremely rewarding task.
3) It is necessary to continue the development of questions of formal logic, in particular, questions about the definition and division of concepts, about judgment and its relationship with a
sentence, about inference, evidence, etc.
Of exceptionally great interest is the specific study and demonstration of how the classics of Marxism-Leninism expose their opponents for ignoring and violating the elementary rules of logic in
order to push through logical tricks and tricks hostile to Marxism.
4) V.I. Lenin’s instructions on making amendments to the formal logic have not yet been fulfilled and require fulfillment. These amendments should go in the direction of completely purifying formal
logic from medieval scholasticism, eliminating the separation of formal logic from life, from practice. It is necessary to banish from formal logic scholasticism and idealism in the interpretation of
the forms and laws of thinking, in particular, in the interpretation of the essence of syllogism, inductive methods of research, etc. Without making these amendments, it is impossible to create a
full-fledged textbook on formal logic for the Soviet school.
5) In the fight against the Kantian distortion of the principles of formal logic, it is necessary to show on concrete material that the elementary rules and axioms of logical thinking are a product
of socio-historical practice, generalized and fixed in the human mind. Thus, formal logic receives a materialistic justification. It is necessary to take into account that relative stability is
inherent in the things and phenomena themselves, and this side of objective reality is reflected in logical laws (identities, contradictions, etc.).
6) Huge tasks face Soviet logicians in exposing anti-scientific, reactionary trends in foreign logic - intuitionism, alogism, etc. - criticism and exposure of sophistry and metaphysics in the logical
constructions of the enemies of Marxism. In separate articles and books, it is necessary to reveal the entire inconsistency of fashionable logical “schools” and trends in bourgeois science, such as
Carnap’s logical positivism, the symbolic logistics of Russell and Whitehead, etc., etc.
The classic works of Marx and Engels, Lenin and Stalin, the brilliant work of J.V. Stalin “Marxism and Questions of Linguistics” give Soviet logicians everything they need to successfully solve the
problems they face."
Source: “Questions of Philosophy”, No. 6, 1951, pp. 143-149
[1] I. Stalin. Marxism and issues of linguistics, p. 39. 1951. [2] V.I. Lenin. Op. T. 32, p. 72.
[3] V.I. Lenin. Philosophical Notebooks, p. 66. 1947.
[4] F. Engels. Dialectics of Nature, p. 177. 1950.
[5] I. Stalin. Marxism and issues of linguistics, p. 24. | {"url":"http://directdemocracy4u.uk/scientific-marxist-philosophy/formal-logic-and-dialectical-logic","timestamp":"2024-11-06T20:33:25Z","content_type":"text/html","content_length":"40117","record_id":"<urn:uuid:97009b2a-3920-4fdc-8d27-23ac9d3aa608>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00261.warc.gz"} |
Covid-19 - How is the outbreak growing? A deep dive analysis with Power BI - Ben's Blog
Covid-19 – How is the outbreak growing? A deep dive analysis with Power BI
Covid-19 – How is the outbreak growing? A deep dive analysis with Power BI
With the rapid spread of the novel coronavirus Covid-19 across the globe, a massive amount of data is generated every day.
Many organizations such as the WHO or the CDC have publicly shared datasets on the worldwide impact of COVID-19.
By now, we’ve probably seen hundreds of graphs and charts across the internet or on the TV depicting the new confirmed cases or cumulative cases around the world.
Although those charts are highlighting important daily statistics, I still feel that the data is not analyzed in an efficient way to provide insights and can sometimes be misleading.
So using Power BI I will attempt to provide a more in-depth analysis of the outbreak and share the insights I found.
So what are the actual insights about the outbreak?
What do we really want to know?
• Where does the virus spread faster?
• Which countries are the most affected?
• Which countries have the most severe cases?
• Which countries better handle the outbreak?
• Is the epidemic slowing down? And where?
• Has the curve flattened?
• When the peak will be reached? Or has it been already reached?
Data sources
The dataset contains time series data of the number of cases, deaths and recoveries across each country on a daily basis.
Terms of Use:
This GitHub repo copyright 2020 Johns Hopkins University, all rights reserved, is provided to the public strictly for educational and academic research purposes. Reliance on the Website for medical
guidance or use of the Website in commerce is strictly prohibited.
Flatten the curve
Countries around the world are working on slowing the spread of the infection. “Flattening the curve” is a strategy to reduce the number of new cases from one day to the next to prevent healthcare
from being overwhelmed.
Most of the charts shown on the news represent the new daily cases or the total number of cases/fatalities over the past few weeks by country. These statistics are good at making the headlines but
what does that tell us? What are the actual insights that we can take from it? How do we know if the curve is flattening?
Daily figures
The above chart shows the daily cases over time in Italy, it seems that the numbers of new confirmed cases have begun to plateau or even fall but this is still no really obvious if Italy has started
to flatten the curve of cases.
However, if instead of looking at the new cases we look at the progression change rate of today’s data versus yesterday’s data or the last 7 days average data we then start to get a sense of where
the outbreak progression rate is heading.
Figure 1: Daily Cases in Italy – Data as of Tuesday 14th April 2020
So how can we spot a flatten curve in this chart?
A flattened curve will show a downward trend in the last 7 days avg whereas an upward trend will indicate that the virus is still spreading rapidly.
We clearly observe a downward curve in the “Last 7 Days Avg” trend so this a good sign that Italy has managed to flatten the curve. Adding the last 7 days avg trend on top of the daily cases provides
a much clearer view on whether the infection rate is slowing down or still rapidly growing.
Analysis by country
Total cases, Mortality rate, Recovery rate
Figure 2: Total Cases, Deaths and Recovered by Country- Data as of Tuesday 14th April 2020
What does this chart tell us? Does it provide any insights?
Yes and no…
It says where are things right now? How does a country compare to other countries?
The US seems to be the most impacted country as it has far more cases than any other countries as well as more fatalities.
Italy has the highest mortality rate.
Russia has the lowest mortality rate.
China has the highest recovery rate.
I could go on and on to list the insights given by this chart but hang on how do we compare the US with Switzerland?
They have very close mortality rate US 4.25% and Switzerland 4.52% but the US has 24 times more case than Switzerland and the US population is about 38 times bigger than Switzerland population.
So what can we infer about it now?
If instead of looking at the number of cases we look at the number of cases per million inhabitants disparities across each country will become much clearer.
Which countries seem to better handle the outbreak?
Now let’s visualize Covid-19 cases and fatalities per million inhabitants:
Sorted by Cases for 1M:
Figure 3: Cases per 1M Inhabitants by Country – Data as of Tuesday 14th April 2020
Sorted by Deaths for 1M:
Figure 4: Deaths per 1M Inhabitants by Country – Data as of Tuesday 14th April 2020
Now, this chart provides a lot more insights we clearly observe a significant difference between each country. And we can get a better intuition on which country better handle or has more resources
to handle the pandemic.
Spain has the most severe cases rate with 3,676 cases per 1M, followed by Switzerland and Belgium. The United States, the country with by far the most cases, still has a relatively low rate in
comparison, 1,844.
Germany has more than 130k cases which is nearly 5 times more than Belgium but its fatality rate stands at 2.5% only compared to 13% in Belgium. Germany has a deaths For 1M rate of 39.6 whereas Spain
has a rate of 384.7 which is 10 times more.
Is that because Germany has been testing far more people than other countries? At the time of writing this post, I haven’t gathered any data about the number of tests by country. I’d be tempted to
say yes but as I can’t back it up I won’t say it!
How can we better visualize disparities across countries?
We’ve seen above that using a ratio per 1M inhabitants gives a clearer view of the disparities between countries but raw data still not provide an easy way to visualize it.
One chart I like to use when I want to compare two ratios is the scatter plot.
Figure 5: Top 15 Countries Ratio Cases/Deaths per 1M Inhbaitants (14.04.2020)
How to read this chart:
• Circle size represents the number of cases
• The dotted line represents the ratio between Cases per 1M over Deaths per 1M
• On the lower side of the chart (right symmetry), we assume that the longer the distance is between a country and the dotted line the better the country handles the outbreak
• On the upper side (left symmetry), we assume that the longer the distance is between a country and the dotted line the worse the country handles the outbreak
Why do I use ratio and scatter plot? Well does the number of cases on its own tell which country is the most impacted?
No, If we assume that the population size of a country is associated with the ICU beds capacity and medical equipments like ventilators. (I’m not saying it’s true)
So since we know how to read this chart and we suppose that in theory population size is associated hospital beds capacity. Let’s deep dive into this chart again.
Figure 6: Top 15 Countries Ratio Cases/Deaths per 1M Inhbaitants (14.04.2020)
Which countries seem to do better and worse than others?
From the previous visuals, we started to get an intuition on which country was doing better than an another but it was still not obvious to see how Switzerland and the US were different.
To compare the ratio of each country side by side scatter plot is in my opinion by far the most appropriate visual to go with.
So far we’ve had identified that Germany was doing far better than other countries but we’ve had no clue that Switzerland was also standing out.
So among the most impacted countries based on the ratio (Deaths/Cases per 1M inhabitants) Germany, Switzerland and the US seem to be better handling the outbreak than other countries while Belgium,
Italy and the UK have a higher Deaths/Cases ratio than other countries.
Now we’ve seen that depicting the relationship between the two ratio Deaths/Cases per 1M inhabitants gave us a clear picture of which countries are the most gravely affected. But how do we know where
the virus spread faster? What about countries where the infection has just started?
Don’t show the date on the X-axis
The number of cases isn’t going to start on the same day across all countries. Instead, the virus will tend to spread in a specific location then to nearby locations and then gradually all over the
So, in that scenario how does one country compare to another?
Date is not relevant, in fact, in February China was the hotspot now they have mostly eradicated the pandemic, then in March Italy was the hotspot, now in April, the US is the hotspot.
If we were to compare these 3 countries using the date scale it would look like this:
Figure 7: Cumulative Cases over time Jan 01 to Marh 25 – China, US, Italy
Figure 8: Daily Cases over time Jan 01 to Marh 25 – China, US, Italy
AS the outbreak did not begin at the same time in these 3 countries. These charts do not provide actual insights and cannot answer the question “Where does the virus spread faster”
Instead of using the date on the X-axis, we will use the number of days since 50 cases were first recorded or since 10 deaths were first recorded, thus, we bring all the countries at the same
starting point.
Where does the virus spread faster?
The two charts below allow us to compare how fast the number of confirmed cases increased after the outbreak has reached a similar stage in each country.
The first chart represents the cumulative number of cases across the top 10 most affected countries, by number of days since the 50th case was recorded (over 20 days).
Figure 9: Total cases by number of days since 50th case recorded (0-20 days)
We can see how robust the spread is over Turkey which has around 3 times more cases than France and Italy since the infection begun to spread within a period of 20 days.
As the virus begun to spread later in Turkey we wouldn’t have been able to visualize it using the date scale.
So using the “number of days since 50th case recorded” gives a much accurate view on how rapidly the virus spread across each country.
This second chart represents the cumulative number of cases across the top 10 most affected countries, by number of days since the 50th case was recorded (over 40 days).
Figure 10: Total cases by number of days since 50th case recorded (0-40 days)
Now this time we see something interesting the US was not among the 10 most-affected countries when looking at a period of 20 days since the infection started but at 40 days its number of cases is
far higher than any other countries.
It looks like it’s only after 25 days that its number of cases started to grow exponentially.
Another interesting point is that at 20 days the virus seemed to spread much faster in Turkey than anywhere else in the world but 5 days later Spain took over.
So did Turkey manage to slow down the infection or perhaps Spain had a sudden increase in cases?
How can we effectively compare the growth in cases of different countries?
Logarithmic scale
The log scale will help better visualize early exponential growth.
So now we get even more insight on when and where the virus spread faster.
Turkey had an early exponential growth in cases at days Turkey had times more cases than Spain, times more than France and UK and 50 times more than the US!
The exponential growth for France, Spain and the Uk started at around the 10th day of the outbreak (since 50th case recorded) and it started around the 15th day for the US.
(Note: I use the term exponential growth to mean “really fast” not to mean cases double every day)
We’ve now seen where the virus is spreading faster.
And most of the countries impacted by the rapid spread of the virus have ordered lockdown in order to slow the epidemic
So how can we track the effectiveness of the lockdown?
Tracking the effectiveness of lock-down period on the spread of the virus is an important indication of how well government responses around the world worked.
How long does it take to the curve to flatten since lockdown started?
The number of cases or fatalities in a country isn’t going to start flattening overnight it might take up to two weeks to have the symptoms and even more to go from being infected to unfortunately
passing away.
Most estimates of the incubation period for COVID-19 range from 1-14 days, most commonly around five days.
Let’s visualize the effectiveness of lockdown on the new daily cases in a few different countries:
Lockdown seems to be quite effective it takes on average 20 days to see the new daily cases slowing down after the lockdown started as we can see from the recap table below.
Contry Lockdown Curve starts to flatten Duration
Italy 09 March 27 MArch 18 days
Spain 14 March 01April 18 days
US 19 March 11 April 23 days
UK 24 March 15 April 22 days
Now let’s visualize the effectiveness of lockdown on the new deaths cases in a few different countries:
Again lockdown seems to be quite effective it takes on average 22 days to see the new daily deaths slowing down after the lockdown started as we can see from the recap table below.
Contry Lockdown Curve starts to flatten Duration
Italy 09 March 03 April 24 days
Spain 14 March 04 April 21 days
US 19 March – ? – 25 days+
UK 24 March 15 April ? 22 days
However, for the US there’s seem to be an issue with the data that twists the actual trend, it could indicate either a time of explosive growth of fatalities and thus that the lockdown is not
effective or just a change in how deaths are counted like in France where fatalities in nursing homes were excluded from official numbers until the beginning of April.
A sudden extreme growth or shrinking of the number of cases or fatalities is what we call in statistics an outlier. Outliers affect the mean value of the data and can make trend harder to forecast.
(We’ll see that in the next part)
When the peak will be reached?
Just to be clear here.
The model “linear regression” that I will use is very basic and can by no mea
Just to be clear here.
The model “linear regression” that I will use is very basic and can by no mean accurately predict the future outcome of the pandemic.
Even for experts, the future of the pandemic is still hard to predict and as experts said no matter how much data we gather, models can’t predict human behaviour.
First let me explain how I’ll try to predict when the peak will be reached.
To predict when the peak will be reached I look at the daily rate of change in the number of cases. So when a country has fewer new daily cases or fatalities than the previous day the change rate
will be negative and if there are more cases today than yesterday then the change rate will be positive.
So if the change rate is positive it means the outbreak is still in a growing phase and not yet under control. If the daily number of cases is still growing, but the change rate is negative it means
that the outbreak is slowing down.
A picture is worth a thousand words so let’s visualize this:
Daily Cases Change Rate in Italy – 01 March – 14 April
How to read this chart:
Here we have the change rate of “Daily Cases Change” (yellow) and its trend “Estimated Cases Growth %” (dotted blue line) using a linear regression.
Here we have the change rate of “Daily Cases Change” (yellow) and its trend “Estimated Cases Growth %” (dotted blue line) using a linear regression.
If the rate is positive, the daily number of deaths/cases is growing, if it is negative, the daily number of deaths/cases is shrinking.
If the linear trend is going up, the overall change rate is growing, if it’s going down the overall change rate is decreasing. When the linear trend crosses 0, the peak has been reached and if the
linear trend approaches 0 in the future we predict that the peak will be reached at this time.
So here in Italy we the linear interpolation of the change rate reaches 0 on So here in Italy, we see that the linear trend of the change rate reaches 0 on April 5th which we were predicted the peak
to be. We already know that the peak was reached a few days earlier.
Now let’s see how outliers can affect visuals as well as the forecast trend. In the below visual in the UK, there was only one confirmed case on the 15th March and 406 confirmed cases on the 16th
March. This is a 40,500% increase in the daily change rate and we see how this outlier is affecting the visual.
It seems that any other values apart from the second outlier are just lying on the X straight line.
And if we were to forecast the daily change rate in the future while keeping the outliers we would predict an increase of 212% for the 16th April so 3 times more cases than the day before…
Fortunately, this is completely wrong my linear regression model has been skewed away from the true underlying relationship due to the few outliers.
So what can we do about outliers?
In that scenario outliers are likely to be associated with a change in
In that scenario, outliers are likely to be associated with a change in reporting methods by public health or government so experts will know what to do with it.
In my case I will just drop them since missing value won’t impact my model, however, in other scenarios, we would probably have to cap them or assign them another value such as the mean or a
Anyway after dropping the few outliers this how the UK Daily Cases Change rate looks like:
Now after droping the outliers we predict a change rate of 0.04% which is
Now after dropping the outliers we predict a change rate of 0.04% which is much closer to the reality.
This last part was more to showcase what we can do in Power BI by using only DAX (no R or Python involved) rather than providing a real forecast.
I appreciate that it is so hard at that time to see into the future of Covid-19. However, I do believe that short term estimation helps countries best prepare to combat the virus.
In this post, I’ve shown how we could predict when the peak will be reached but many other things could also be predicted such as estimating the peak duration or predicting the time when COVID-19
infections will fall below a certain threshold.
Final Thoughts
Thanks for sticking with me until the end!
I hope you feel that we have uncovered some useful insights and that I have demonstrated that using the appropriate visuals help to analyse data more efficiently.
Let’s recap a few of them:
Daily figures:
Comparing Daily figures with rolling avg or with last 7 days avg helps to
Comparing Daily figures with rolling avg or with last 7 days avg helps to visualize when the curve has flattened.
By now most countries from the West seemed to have flattened the curve especially those that have ordered a lockdown
Using different scales tell different stories:
Logarithmic scale helps to visualize exponential growth. Turkey had the earliest exponential growth in cases.
Tracking the effectiveness of lockdown:
It is crucial for public healts and governements to track how effective lockdowns are in stopping the spread of the virus.
Lockdown seems to have worked in most countries but there are still a few countries where a lockdown seems to be a bit less effective like in the US or even Belgium.
Using Number of days since the 50th case instead of the date helps visualize how rapidly increases the infection across each country.
Using deaths/cases per capita:
Knowing how many people have died compared to how many people live in that country is more insightful than showing the raw number of deaths or cases and helps to see which countries are the most
affected and those which better handle the outbreak.
When will the outbreak peak?
As we’ve seen it’s easy to know when we’ve reached the peak.
As we’ve seen it’s easy to know when we’ve reached the peak.
Once we start seeing the downward trend it means the peak has been reached.
Knowing when the pandemic will peak is also useful, in fact, if we know that the virus will peak soon and will start to be contained within a few weeks it will help panic and fear of the unknown to
fade rapidly and lifting restrictions swiftly.
Power Bi Link
You can fully interact with the reports shown in this post here.
I plan to add more features and insightful visuals so stay tune. | {"url":"https://datakuity.com/2020/04/18/covid-19-how-is-the-outbreak-growing-a-deep-dive-analysis-with-power-bi/","timestamp":"2024-11-05T13:42:56Z","content_type":"text/html","content_length":"139710","record_id":"<urn:uuid:ee53992f-45ff-438c-b24d-370e4198363f>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00607.warc.gz"} |
Introduction to MAPpoly
MAPpoly is an R package to construct genetic maps in autopolyploids with even ploidy levels. This quick start guide will present some essential functions to construct a tetraploid potato map. Please
refer to MAPpoly’s Reference Manual and the Extended Tutorial for a comprehensive description of all functions.
Reading data set
There are several functions to read discrete and probabilistic dosage-based genotype data sets in MAPpoly. You can read a data set from TXT, CSV, VCF, fitPoly-generated or import it from the R
packages polyRAD, polymapR, and updog. The data set distributed along with MAPpoly is a subset of markers from (Pereira et al., 2021) in CSV format. Let us read it into MAPpoly
file.name <- system.file("extdata/potato_example.csv", package = "mappoly")
dat <- read_geno_csv(file.in = file.name, ploidy = 4)
print(dat, detailed = T)
The output figure shows a bar plot on the left-hand side with the number of markers in each allele dosage combination in \(P_1\) and \(P_2\), respectively. The upper-right plot contains the \(\log_
{10}(p-value)\) from \(\chi^2\) tests for all markers, considering the expected segregation patterns under Mendelian inheritance.
Data quality control
Quality control (QC) procedures are fundamental to identify:
• Individuals from crosses other than \(P_1 \times P_2\)
• Individuals and markers that exceeds a defined threshold of missing data points
• Markers with distorted segregation
• Markers with the same genotypic information (redundant markers). Removed markers are positioned into the final map.
Depending on the data set, these procedures can be conducted in any order. Let us first remove individuals from crosses other than \(P_1 \times P_2\). When using the interactive function, the user
needs to select a polygon around the individuals to be removed by clicking its vertices and pressing Esc.
Now, let us filter out markers and individuals with more than 5% of missing data. You can update the threshold interactively
dat <- filter_missing(dat, type = "marker", filter.thres = .05)
dat <- filter_missing(dat, type = "individual", filter.thres = .05)
Finally, we can filter out markers with distorted segregation and redundant information. At this point, we do not consider preferential pairing and double reduction.
seq.filt <- filter_segregation(dat, chisq.pval.thres = 0.05/dat$n.mrk)
seq.filt <- make_seq_mappoly(seq.filt)
seq.red <- elim_redundant(seq.filt)
After filtering, 1990 markers were left to be mapped. Now, let us create a sequence of ordered markers to proceed with the analysis. You can also plot the data set for a specific sequence of markers
and check the distribution of the markers in the reference genome, if the information is available
Two-point analysis
The two-point analysis calculates the pairwise recombination fraction in a sequence of markers. At this point of the analysis, where we have many markers, we use the function est_pairwise_rf2, which
has a less detailed output than the original est_pairwise_rf (which will be used later) but can handle tens of thousands of markers, even when using a personal computer. Nevertheless, the analysis
can take a while depending on the number of markers if few cores are available.
ncores <- parallel::detectCores() - 1
tpt <- est_pairwise_rf2(seq.init, ncpus = ncores)
m <- rf_list_to_matrix(tpt) ## converts rec. frac. list into a matrix
sgo <- make_seq_mappoly(go) ## creates a sequence of markers in the genome order
plot(m, ord = sgo, fact = 5) ## plots a rec. frac. matrix using the genome order, averaging neighbor cells in a 5 x 5 grid
We can cluster the markers in linkage groups by using function group_mappoly. The function uses the recombination fraction matrix and UPGMA method to group markers. Use the option comp.mat = TRUE to
compare the linkage-based clustering results with the chromosome information. If your data set does not contain chromosome information, use the option comp.mat = FALSE. You also can use the
interactive version to change the number of expected groups
In the table above, the rows indicate linkage groups obtained using linkage information and, the columns are the chromosomes in the reference genome. Notice the diagonal indicating the concordance
between the two sources of information.
Ordering markers
Markers are ordered within linkage groups. In this tutorial, we will show the step-by-step procedure using Linkage Group 1 (LG1). You can do the same for the remaining linkage groups.
Since we had a good concordance between genome and linkage information, we will use only markers assigned to a particular linkage group using both sources of information. We will do that using
genomic.info = 1 , so the function uses the intersection of the markers assigned using linkage and the chromosome with the highest number of allocated markers. To use only the linkage information, do
not use the argument genomic.info. You also need the recombination fraction matrix for that group.
Let us order the markers in the sequence using the MDS algorithm.
Usually, at this point, the user can use diagnostic plots to remove markers that disturb the ordering procedure. We didn’t use that procedure in this tutorial, but we encourage the user to check
the example in ?mds_mappoly. Now, let us use the reference genome to order the markers.
You also can order the markers the reference genome
For the sake of this short tutorial, let us use the MDS order (s1.mds) to phase the markers. Still, you can also try the genome order (s1.gen) and compare the resulting maps using function
compare_maps, and you will notice that the genomic order yields a better map since its likelihood is higher.
Phasing markers and estimating multilocus recombination fractions.
Estimating the genetic map for a given order involves the computation of recombination fraction between adjacent markers and inferring the linkage phase configuration of those markers in both
parents. The core function to perform these tasks in MAPpoly is est_rf_hmm_sequential. This function uses the pairwise recombination fraction as the first source of information to sequentially
position allelic variants in specific parental homologs. The algorithm relies on the likelihood obtained through a hidden Markov model (HMM) for situations where pairwise analysis has limited power.
Once all markers are positioned, the final map is reconstructed using the HMM multipoint algorithm. For a detailed description of the est_rf_hmm_sequential arguments, please refer to MAPpoly’s
Reference Manual and the Extended Tutorial.
First we need to calculate the pairwise recombination fraction for markers in sequence s1.mds using est_pairwise_rf, which contains the information necessary to the proper working of the phasing
tpt1 <- est_pairwise_rf(s1.mds, ncpus = ncores)
lg1.map <- est_rf_hmm_sequential(input.seq = s1.mds,
start.set = 3,
thres.twopt = 10,
thres.hmm = 20,
extend.tail = 50,
info.tail = TRUE,
twopt = tpt1,
sub.map.size.diff.limit = 5,
phase.number.limit = 20,
reestimate.single.ph.configuration = TRUE,
tol = 10e-3,
tol.final = 10e-4)
Now, use the functions print and plot to view the map results:
Now let us update the recombination fractions by allowing a global error in the HMM recombination fraction re-estimation. Using this approach, the genetic map’s length will be updated by removing
spurious recombination events. This procedure can be applied using either the probability distribution provided by the genotype calling software using function est_full_hmm_with_prior_prob or
assuming a global genotype error like the following example
lg1.map.up <- est_full_hmm_with_global_error(input.map = lg1.map, error = 0.05,
verbose = TRUE)
plot(lg1.map.up, mrk.names = TRUE, cex = 0.7)
We can also use the ordinary least squares (OLS) method and the weighted MDS followed by fitting a one dimensional principal curve (wMDS_to_1D_pc)
lg1.map.ols <- reest_rf(lg1.map, m1, method = "ols")
lg1.map.mds <- reest_rf(lg1.map, m1, method = "wMDS_to_1D_pc", input.mds = mds.o1)
Now let us create a list with the maps and plot the results
Homolog probability and preferential pairing profile
To use the genetic map in conjunction with QTL analysis software, we need to obtain the homolog probability for all linkage groups for all individuals in the full-sib population. In this short guide,
we will proceed only with one linkage group, but this procedure should be applied to all chromosomes in real situations. Let us use the updated map
g1 <- calc_genoprob_error(lg1.map.up, step = 1, error = 0.05)
to.qtlpoly <- export_qtlpoly(g1) #export to QTLpoly
h1 <- calc_homologprob(g1)
plot(h1, lg = 1, ind = 10)
Now let us compute the preferential pairing profile for linkage group 1
Exporting a phased map
It is possible to export a phased map to an external CSV file using
For a script with a complete analysis of the data set presented here, please refer to the Complete script
Utility functions
To use the next functions, let us load the complete genetic map
in.file <- "https://github.com/mmollina/SCRI/raw/main/docs/tetra/maps_updated.rda"
map_file <- tempfile()
download.file(in.file, map_file) | {"url":"https://cran.stat.sfu.ca/web/packages/mappoly/vignettes/mappoly_startguide.html","timestamp":"2024-11-15T03:31:14Z","content_type":"text/html","content_length":"811901","record_id":"<urn:uuid:16874eaa-c9f0-4d71-81b1-c95391d3de80>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00133.warc.gz"} |
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Sum of probabilities of all the events is equal to
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A coin is tossed 3 times, what is the probability that at least one head will occur?
1. 1 ⁄ 8
2. 2 ⁄ 8
3. 7 ⁄ 8
4. 8 ⁄ 8
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Probability of a sample space is always equals to
1. 0
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3. 2
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For two mutually exclusive events say A and B, the (A ∩ B) returns
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<no title> — tsfresh 0.20.2.post0.dev10+g1297c8c documentation
abs_energy(x) Returns the absolute energy of the time series which is the sum over the squared values
absolute_maximum(x) Calculates the highest absolute value of the time series x.
absolute_sum_of_changes(x) Returns the sum over the absolute value of consecutive changes in the series x
agg_autocorrelation(x, param) Descriptive statistics on the autocorrelation of the time series.
agg_linear_trend(x, param) Calculates a linear least-squares regression for values of the time series that were aggregated over chunks versus the sequence from 0 up to the
number of chunks minus one.
approximate_entropy(x, m, r) Implements a vectorized Approximate entropy algorithm.
ar_coefficient(x, param) This feature calculator fits the unconditional maximum likelihood of an autoregressive AR(k) process.
augmented_dickey_fuller(x, param) Does the time series have a unit root?
autocorrelation(x, lag) Calculates the autocorrelation of the specified lag, according to the formula [1]
benford_correlation(x) Useful for anomaly detection applications [1][2]. Returns the correlation from first digit distribution when
binned_entropy(x, max_bins) First bins the values of x into max_bins equidistant bins.
c3(x, lag) Uses c3 statistics to measure non linearity in the time series
change_quantiles(x, ql, qh, isabs, f_agg) First fixes a corridor given by the quantiles ql and qh of the distribution of x.
cid_ce(x, normalize) This function calculator is an estimate for a time series complexity [1] (A more complex time series has more peaks, valleys etc.).
count_above(x, t) Returns the percentage of values in x that are higher than t
count_above_mean(x) Returns the number of values in x that are higher than the mean of x
count_below(x, t) Returns the percentage of values in x that are lower than t
count_below_mean(x) Returns the number of values in x that are lower than the mean of x
cwt_coefficients(x, param) Calculates a Continuous wavelet transform for the Ricker wavelet, also known as the "Mexican hat wavelet" which is defined by
energy_ratio_by_chunks(x, param) Calculates the sum of squares of chunk i out of N chunks expressed as a ratio with the sum of squares over the whole series.
fft_aggregated(x, param) Returns the spectral centroid (mean), variance, skew, and kurtosis of the absolute fourier transform spectrum.
fft_coefficient(x, param) Calculates the fourier coefficients of the one-dimensional discrete Fourier Transform for real input by fast fourier transformation algorithm
first_location_of_maximum(x) Returns the first location of the maximum value of x.
first_location_of_minimum(x) Returns the first location of the minimal value of x.
fourier_entropy(x, bins) Calculate the binned entropy of the power spectral density of the time series (using the welch method).
friedrich_coefficients(x, param) Coefficients of polynomial
has_duplicate(x) Checks if any value in x occurs more than once
has_duplicate_max(x) Checks if the maximum value of x is observed more than once
has_duplicate_min(x) Checks if the minimal value of x is observed more than once
index_mass_quantile(x, param) Calculates the relative index i of time series x where q% of the mass of x lies left of i.
kurtosis(x) Returns the kurtosis of x (calculated with the adjusted Fisher-Pearson standardized moment coefficient G2).
large_standard_deviation(x, r) Does time series have large standard deviation?
last_location_of_maximum(x) Returns the relative last location of the maximum value of x.
last_location_of_minimum(x) Returns the last location of the minimal value of x.
lempel_ziv_complexity(x, bins) Calculate a complexity estimate based on the Lempel-Ziv compression algorithm.
length(x) Returns the length of x
linear_trend(x, param) Calculate a linear least-squares regression for the values of the time series versus the sequence from 0 to length of the time series minus one.
linear_trend_timewise(x, param) Calculate a linear least-squares regression for the values of the time series versus the sequence from 0 to length of the time series minus one.
longest_strike_above_mean(x) Returns the length of the longest consecutive subsequence in x that is bigger than the mean of x
longest_strike_below_mean(x) Returns the length of the longest consecutive subsequence in x that is smaller than the mean of x
matrix_profile(x, param) Calculates the 1-D Matrix Profile[1] and returns Tukey's Five Number Set plus the mean of that Matrix Profile.
max_langevin_fixed_point(x, r, m) Largest fixed point of dynamics :math:argmax_x {h(x)=0}` estimated from polynomial
maximum(x) Calculates the highest value of the time series x.
mean(x) Returns the mean of x
mean_abs_change(x) Average over first differences.
mean_change(x) Average over time series differences.
mean_n_absolute_max(x, number_of_maxima) Calculates the arithmetic mean of the n absolute maximum values of the time series.
mean_second_derivative_central(x) Returns the mean value of a central approximation of the second derivative
median(x) Returns the median of x
minimum(x) Calculates the lowest value of the time series x.
number_crossing_m(x, m) Calculates the number of crossings of x on m.
number_cwt_peaks(x, n) Number of different peaks in x.
number_peaks(x, n) Calculates the number of peaks of at least support n in the time series x.
partial_autocorrelation(x, param) Calculates the value of the partial autocorrelation function at the given lag.
percentage_of_reoccurring_datapoints_to_all_datapoints Returns the percentage of non-unique data points.
percentage_of_reoccurring_values_to_all_values(x) Returns the percentage of values that are present in the time series more than once.
permutation_entropy(x, tau, dimension) Calculate the permutation entropy.
quantile(x, q) Calculates the q quantile of x.
This feature calculator accepts an input query subsequence parameter, compares the query (under z-normalized Euclidean distance) to all
query_similarity_count(x, param) subsequences within the time series, and returns a count of the number of times the query was found in the time series (within some predefined
maximum distance threshold).
range_count(x, min, max) Count observed values within the interval [min, max).
ratio_beyond_r_sigma(x, r) Ratio of values that are more than r * std(x) (so r times sigma) away from the mean of x.
ratio_value_number_to_time_series_length(x) Returns a factor which is 1 if all values in the time series occur only once, and below one if this is not the case.
root_mean_square(x) Returns the root mean square (rms) of the time series.
sample_entropy(x) Calculate and return sample entropy of x.
set_property(key, value) This method returns a decorator that sets the property key of the function to value
skewness(x) Returns the sample skewness of x (calculated with the adjusted Fisher-Pearson standardized moment coefficient G1).
spkt_welch_density(x, param) This feature calculator estimates the cross power spectral density of the time series x at different frequencies.
standard_deviation(x) Returns the standard deviation of x
sum_of_reoccurring_data_points(x) Returns the sum of all data points, that are present in the time series more than once.
sum_of_reoccurring_values(x) Returns the sum of all values, that are present in the time series more than once.
sum_values(x) Calculates the sum over the time series values
symmetry_looking(x, param) Boolean variable denoting if the distribution of x looks symmetric.
time_reversal_asymmetry_statistic(x, lag) Returns the time reversal asymmetry statistic.
value_count(x, value) Count occurrences of value in time series x.
variance(x) Returns the variance of x
variance_larger_than_standard_deviation(x) Is variance higher than the standard deviation?
variation_coefficient(x) Returns the variation coefficient (standard error / mean, give relative value of variation around mean) of x. | {"url":"https://tsfresh.readthedocs.io/en/latest/text/_generated/tsfresh.feature_extraction.feature_calculators.html","timestamp":"2024-11-08T20:29:58Z","content_type":"text/html","content_length":"46551","record_id":"<urn:uuid:a8fb34cd-3047-45a1-a49a-29a4d9d91074>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00683.warc.gz"} |
Welcome to the Department of Mathematics at the University of Central Arkansas. The Department of Mathematics is committed to excellence in education. The primary mission of the department is to
help students obtain a solid foundation of mathematical knowledge and to make skillful use of appropriate technology and apply it to the work force. The department is also committed to scholarly
activity and service to enhance the professional and academic growth of students and faculty while at the same time, serving the university and community by consulting with business and industry
and delivering supplemental mathematics instruction through seminars, workshops, conferences, and in-service teacher training.
There are 16 graduate faculty members with doctoral degrees which include four in applied mathematics, five in mathematics education, four in data science, one in pure mathematics, three lecturers
and two visiting lecturers. The mathematics and applied mathematics faculty are active in symmetry analysis, PDEs, numerical analysis, control theory, mathematics modeling, bio-statistics, queueing
theory and functional analysis. The mathematics education faculty are active in the areas of curriculum development and alignment, professional development, performance evaluation and assessment,
innovative instructional strategies, instructional technology, and math anxiety. Regardless of their expertise, all faculty are dedicated to student success. We maintain a collegial environment
where collaboration within the department and college faculty provides a supportive atmosphere. The University of Central Arkansas (UCA), located in Conway, is only 30 miles from downtown Little
Rock. The Department of Mathematics is located in the Mathematics and Computer Science Building (MCS).
Academic Focus
The focus in the Department of Mathematics is two-fold: Applied Mathematics and Mathematics Education. In our undergraduate program, students learn to apply techniques needed to model and analyze
complex problems and at the same time are exposed to the underlying theories. We have a strong undergraduate research program. Each year, a group of faculty and students present their research
findings at regional and national mathematical meetings. In addition, technology is integrated in the curriculum in the forms of graphics calculators, computer algebra systems, and computer software.
Academic Programs
The department offers an undergraduate degree Baccalaureate of Science (B.S.) in Mathematics. The B.S. program has multiple tracks: Applied Mathematics, Data Science, Pure Mathematic and
Math Education. The Applied Math track offers preparation that integrates technology, critical thinking, and problem solving which culminates in a mastery of mathematical skills needed to succeed in
careers in business, government, industry or in advanced studies. The Mathematics Education track provides excellent preparation for middle or high school teacher licensure in mathematics. The Data
Science track prepares students by equipping them with computing, statistical, analytical, and business skills needed in the field of data science. The pure mathematics track offers excellent
preparation that integrates critical thinking and problem solving and culminates in a mastery of mathematical skills needed for advanced studies. All tracks requires a minor.
In addition, the Math Department offers two graduate degrees. The M.A. Program in Mathematics Education was developed to increase the mathematical knowledge of secondary school teachers, prepare
candidates for teaching in community colleges, and enrich the mathematical background of professionals. The M.S. Program in Applied Mathematics, was developed to train students in mathematical
modeling, prepare students to serve both business and government agencies or for further graduate studies. Please contact me if you would like additional information about our department or to
arrange for a campus visit.
Dr. Loi Booher, Department Chair & Associate Professor
201 Donaghey Avenue
Conway, AR 72035
Phone: (501) 450-3147
e-mail: math@uca.edu | {"url":"https://uca.edu/math/","timestamp":"2024-11-10T02:23:27Z","content_type":"application/xhtml+xml","content_length":"23791","record_id":"<urn:uuid:7196d4fb-e9de-4016-b9aa-0b3205b0fadb>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00299.warc.gz"} |
Enhancing Power System Resilience through Computational Optimization
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Enhancing Power System Resilience through Computational Optimization
In this dissertation we develop models, solution techniques, and derive policy implications for a number of important applications in improving power system resilience and flexibility: black start
allocation and power system restoration, optimal islanding, stochastic unit commitment and flexible wind dispatch. The models we employ are predominantly mixed integer programs (MIPs), i.e.
optimization problems in which some of the variables must take integer values. These problems are NP-hard in general, so there is no guarantee that we will be able to obtain solutions within
acceptable time and accuracy as the scale of the problem increases. For that reason, we also develop or utilize specialized computational techniques, which broadly fall into one of the following
categories: decomposition algorithms that exploit the problem structure (sparsity), reformulations of the problem constraints, and customized heuristics.
We start by exploring the problem of restoring the normal operation of the power system after a blackout. The restoration of the system builds around specific units with the ability to start
autonomously (black start units). We formulate a planning problem for deciding the allocation of these units on the grid - black start allocation (BSA) - in an optimal way, while simultaneously
optimizing over the possible restoration plans. We include, among others, considerations for thermal limits of lines, alleviating overvoltages, and constraints to model the startup curves of
generators. Due to the size and complexity of the resulting MIP, commercial solvers are unable to tackle it directly. We construct a randomized heuristic based on linear programming relaxations of
the optimization problem and an understanding of the underlying physics of the power grid to aid the solvers. The heuristic execution is parallelized and implemented on a high-performance computing
environment. We are able to obtain solutions with optimality guarantees within reasonable times for test power systems with a few hundred buses.
We proceed to extract a substructure of the feasible region from any problem in power systems that employs reconfiguration of the physical topology (i.e. switching on/off of generators, branches, and
buses): each energized island in the power system needs to contain at least one energized generator. We explore reformulations that describe the feasible region corresponding to this requirement. We
employ two families of valid inequalities to strengthen the formulation, both exponential in size, but separable in polynomial time. We study polyhedral properties of the integer hull and the
strength of some of these inequalities under simplifying assumptions. We proceed to conduct computational experiments for two problems in which the substructure appears: the optimal islanding problem
and a simplified version of the BSA problem. We are able to observe significant computational benefits by using suitable reformulations for both problems. Finally, we describe an approach to obtain
solutions with an optimality guarantee for a simplified model of BSA in a synthetic test case with 2000 nodes representative of Texas.
We extend the modeling framework of the BSA problem to accommodate for uncertainty in the power outages. Specifically, we consider optimal allocations of the black start units over a number of
scenarios of partial or total outages. These scenarios may also include irreparable damage caused to components of the system. The resulting stochastic mixed integer program exhibits sparsity, so we
employ a decomposition algorithm by scenario to solve instances of the problem.
We then return to the optimal islanding problem to examine it in more detail. We observe that, despite the formulation improvements, the branch and cut algorithm is still fairly slow for an online
application of that scale, which requires to act within seconds to prevent a cascaded outage of the system. For that reason, we propose a heuristic based on a reformulation of the optimization into a
problem in graph theory. We utilize an algorithm to obtain heuristic solutions with high computational efficiency and good quality compared to state-of-the-art techniques.
To conclude this dissertation, we introduce a framework for evaluating the cost of priority dispatch for wind power. Renewable generation is commonly considered a must-take resource in power systems,
despite the technical capabilities of current wind turbines to dispatch at levels lower than their available output. The cost of that policy compared to one that instead optimizes over the available
wind output is evaluated for a reduced California system, by employing a two-stage stochastic program for stochastic unit commitment. A scenario decomposition algorithm for the resulting large-scale
MIP, parallelized on a high-performance computing environment, enables us to obtain near optimal solutions and calculate the difference in cost between the two policies.
Main Content
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Operation Types
Operation Types From the simplest microprocessor (8-bit) to a large mainframe with an embedded microprocessor, the types of ALU operations range from basic add and subtract
operations to sophisticated trigonometric operations and separate coprocessor and math pacs, which operate independent of the ALU. The types of instructions most ALUs canperform can be divided into
two categories: arithmetic operations and logical operations. The ALU uses the logical products of the logic gates to perform the arithmetic and logical
instructions. Depending on the sophistication of the computer, the logic gates are arranged to perform the instructions included in the computer s set of instructions.
Computers can be designed to have an adder to perform its adding and subtracting or a subtracter to perform its adding and subtracting. Or they can have a combined adder/subtracter system. Because a
computer can really only add or subtract, the add and subtract capabilities allow the computer to perform the more complicated arithmetic operations: multiply,
division, and square root functions. Addition and subtraction functions are embedded in division, square root, and the more complicated arithmetic functions,
such as trigonometric and hyperbolic, to name a couple. The computer can be designed where a single instruction will accomplish the results or a series of
instructions can be written to produce the results. The only drawback to a series of instruction is they consume more time to accomplish the results. The multiply,
divide, square root, and trigonometric instructions are examples. Computers can multiply by repetitive adding or they can use a series of left shift instructions both using
a compare instruction, which may be how a computer with a dedicated multiply function accomplishes the function anyway. The same principle can be applied to
the divide and square root functions. A divide can use repetitive subtractions or a series of right shifts with a comparison function. A square root would use a
combination of additions/subtractions and comparisons for the multiplying and dividing necessary to accomplish a square root function. A trigonometric
function using separate instructions would use logical instructions to accomplish the same results that a single trigonometric instruction would accomplish. ALU operations include signed operations.
Depending on the sophistication of the computer, ALU functions can include the following functions: 5-23 Arithmetic Add, subtract, shift, multiply, divide, negation, absolute value. (The more
sophisticated ALUs can perform square root, trigonometric, hyperbolic, and binary angular movement or motion (BAM) functions.) Logical AND, OR, NOT (complement), and EXCLUSIVE OR (compare).
Also depending on the design, numeric data coprocessor and math pacs are used in some computers in addition to the normal arithmetic instructions
available. They execute the arithmetic instructions the CPU s ALU cannot, and they are still controlled by the CPU s program control. These additional logic circuits
can be used to amplify the capabilities of the ALU and arithmetic section in general. Remember, the ALU is part of a CPU module or a microprocessor chip on a
printed circuit board. The numeric data coprocessor and math pac are separate modules or chips. NUMERIC DATA COPROCESSOR. The numeric data coprocessor is a special-purpose
programmable microprocessor designed to perform up to 68 additional arithmetic, trigonometric, exponential, and logarithmic instructions. The coprocessor
performs numeric applications up to 100 times faster than the CPU alone and provides handling of the following data types: 16-, 32-, and 64-bit integers; 32-,
64-, and 80-bit floating-point real numbers; and up to 18-digit binary coded decimal (BCD) operands. The numeric data coprocessor operates in parallel
with and independent of the CPU using the same data, address, and control buses as the CPU. In effect, the coprocessor executes those arithmetic instructions that
the CPU s ALU cannot. The CPU is held in a wait mode, while the coprocessor is performing an operation. The CPU still controls overall program execution, while the coprocessor recognizes and
executes only its own numeric operations. MATH PAC. Math pac is a module used as a hardware option for some militarized minicomputers. The math pac module provides the hardware capability
to perform square root, trigonometric and hyperbolic functions; floating-point math; double-precision multiply and divide instructions; and algebraic left and right quadruple shifts.
TOPIC 3 COMPUTER INTERNAL BUSES To transfer information internally, computers use buses. Buses are groups of conductors that connect the | {"url":"https://firecontrolman.tpub.com/14100/css/Operation-Types-127.htm","timestamp":"2024-11-12T13:16:32Z","content_type":"text/html","content_length":"34910","record_id":"<urn:uuid:d07fed38-d776-4831-bc86-543e923547f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00122.warc.gz"} |
1028 List Sorting
Source: [PAT 1028]
Excel can sort records according to any column. Now you are supposed to imitate this function.
Input Specification: Each input file contains one test case. For each case, the first line contains two integers N(≤105) and C, where N is the number of records andCis the column that you are
supposed to sort the records with. Then N lines follow, each contains a record of a student. A student’s record consists of his or her distinct ID (a 6-digit number), name (a string with no more than
8 characters without space), and grade (an integer between 0 and 100, inclusive).
Input Specification
For each test case, output the sorting result in N lines. That is, if C = 1 then the records must be sorted in increasing order according to ID’s; if C = 2 then the records must be sorted in
non-decreasing order according to names; and if C = 3 then the records must be sorted in non-decreasing order according to grades. If there are several students who have the same name or grade, they
must be sorted according to their ID’s in increasing order.
Output Specification
For each test case, output the sorting result in N lines. That is, if C = 1 then the records must be sorted in increasing order according to ID’s; if C = 2 then the records must be sorted in
non-decreasing order according to names; and if C = 3 then the records must be sorted in non-decreasing order according to grades. If there are several students who have the same name or grade, they
must be sorted according to their ID’s in increasing order.
Sample Input 1:
000007 James 85
000010 Amy 90
000001 Zoe 60
Sample Output 1:
000001 Zoe 60
000007 James 85
000010 Amy 90
Sample Input 2:
000007 James 85
000010 Amy 90
000001 Zoe 60
000002 James 98
Sample Output 2:
000010 Amy 90
000002 James 98
000007 James 85
000001 Zoe 60
Sample Input 3:
000007 James 85
000010 Amy 90
000001 Zoe 60
000002 James 90
Sample Output 3:
000001 Zoe 60
000007 James 85
000002 James 90
000010 Amy 90
解法： 1.读入所有数据 2.根据c调用不同的排序方法 3.打印 | {"url":"https://blog.alomerry.com/ioi/pat-a/1028","timestamp":"2024-11-14T23:58:06Z","content_type":"text/html","content_length":"23786","record_id":"<urn:uuid:50ae92b5-2f4f-4900-bf7b-7be9ac01cb1c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00420.warc.gz"} |
The S.I. unit of gravitational potential is
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Attacking Lattice-based Cryptography with Martin Albrecht - Security Cryptography Whatever
Security Cryptography Whatever
Some cryptography & security people talk about security, cryptography, and whatever else is happening.
Security Cryptography Whatever
Attacking Lattice-based Cryptography with Martin Albrecht
• Security, Cryptography, Whatever • Season 3 • Episode 4
Returning champion Martin Albrecht joins us to help explain how we measure the security of lattice-based cryptosystems like Kyber and Dilithium against attackers. QRAM, BKZ, LLL, oh my!
Transcript: https://securitycryptographywhatever.com/2023/11/13/lattice-attacks/
- https://pq-crystals.org/kyber/index.shtml
- https://pq-crystals.org/dilithium/index.shtml
- https://eprint.iacr.org/2019/930.pdf
- https://en.wikipedia.org/wiki/Short_integer_solution_problem
- Frodo: https://eprint.iacr.org/2016/659
- https://csrc.nist.gov/CSRC/media/Events/third-pqc-standardization-conference/documents/accepted-papers/ribeiro-saber-pq-key-pqc2021.pdf
- https://en.wikipedia.org/wiki/Hermite_normal_form
- https://en.wikipedia.org/wiki/Wagner%E2%80%93Fischer_algorithm
- https://www.math.auckland.ac.nz/~sgal018/crypto-book/ch18.pdf
- https://eprint.iacr.org/2019/1161
- QRAM: https://arxiv.org/abs/2305.10310
- https://en.wikipedia.org/wiki/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm
- MATZOV improved dual lattice attack: https://zenodo.org/records/6412487
- https://eprint.iacr.org/2008/504.pdf
- https://eprint.iacr.org/2023/302.pdf
"Security Cryptography Whatever" is hosted by Deirdre Connolly (@durumcrustulum), Thomas Ptacek (@tqbf), and David Adrian (@davidcadrian)
Analyzing the Security of Post-Quantum Cryptography
Finding Short Vectors in Lattices
Quantum Speedup in Cryptography
Understanding and Applying BKZ Algorithm
Lattice-Based Cryptanalysis and Improvements
RAM and Storage in Classical Attacks
Discussion on AES Quantum Computing Costs
Hello, welcome to Security Cryptography Whatever. I'm Deirdre,
I'm Thomas.
and we have a returning champion back on the pod today. We have Martin Albrecht back. Hi, Martin. How are you?
happy to be here.
Great! Thank you. We promise to send you your N timers club jacket in the mail. We have another one that we have to mail out in the future. FYI, myself and Martin both work for Sandbox AQ, at least
for part of our time. And we invited Martin back to talk about one of his other areas of expertise, which is cryptography and how to analyze the attack security against them. Because we wanted to
understand more about some of the new NIST schemes that have been selected for post quantum cryptography. And that includes a lattice-based KEM, key encapsulation mechanism called Kyber, and at least
one of the signature schemes called Dilithium. Falcon's also lattice-based, right?
But it's different for reasons and it uses floating point things and I don't like them. Um, but there, there's been some discussion about like, how do we analyze? The security levels of not just
lattice schemes, but kind of like post quantum schemes in general that are supposed to be resilient about, against a classical attacker with a regular computer and theoretically a non noisy, there's
like some There's some acronym about how it's, like, got enough logical qubits and enough error correction and enough, uh, you know, handling of noise that happens with quantum computers in the real
world to be an actual threat against, uh, these cryptographic schemes and actually run things like Shor's algorithms or some of these other attacks efficiently to target, you know, some quantum
scheme, uh, sorry, uh, cryptography scheme. So, we, we're here to pick Martin's brain, and maybe he can help us, and in the, the fun medium of... Not being able to use visual aids to show things like
lattices and vectors and bases. So first off, Martin, can you kind of give us a high level of like, why we kind of like these schemes like Kyber and Dilithium that are based on lattice assumptions
for post quantum cryptography?
Right. So after you do your intro that you don't really like them, let me tell you why you should like them. Okay. So like, I mean, one thing is like, they are fairly efficient, right? For post
quantum schemes. So like, that's one of the reasons is like they are based on some problems that we believe to be hard, even on a quantum computer, as you just said. And like, given that, like their
sizes, uh, at the computation time, um, somewhat good while. convincing at least many people that they are safe. So, and really, uh, we're here to talk about, I guess, is their security, right? So,
um, what grounds this? And there's a, there's the first question of like structurally, why do we think it makes sense to base security on them? And then there's the question of parameters and like,
you know, what is the security level? Which I think is the focus of today's episode, if I gather this correctly. So first, the structural thing is. Roughly speaking is if you can break Kyber, then
you can solve a problem that is called module learning with errors.
We've heard about that
And we believe that this is a hard problem even for a quantum computer. So, let me unpack that, because the module learning with errors problem will also reappear at the lithium, so it makes sense to
spend a bit more time on that. So, it's essentially, so you do linear algebra mod q, so, uh, nothing too fancy there, but you add some small noise. So, instead of having a proper solution to a linear
system, You have a solution and that is offset by some noise and small here means it's not 0, it doesn't hit 0, but it hits 5 or it hits minus 3 or something like that. Like something small relative
to the size of Q. And then the module in there means that we're actually not doing this just over the integers mod Q, but we do this as matrices over polynomial rings mod Q. and that kind of then
gives you module learning with errors, um, and it's, um, assumed we have, uh, some reasons to believe that this problem is hard, also in quantum computer. And then the, the lithium, um, in addition
to being based on this module LWM, LWE problem is also based on the module SIS problem. Which is simply, it's, it's, it's a very simple to stay problem. So I give you a wide matrix with many columns,
but few rows, uh, modQ. It's uniformly random. And all you have to do is find me a short solution that sums it up to zero. So you sum up the columns to, to hit zero everywhere. And again, short is
something, you know, like, you know, maybe the solution is between, uh, 20 and minus 20. Like, these are not dilithium parameters, but to give you a sense. And then Q is a few thousand. Of course,
the problem is really easy if you have something that is quite big, a big solution, but it's considered to be a hard problem if the solution is small. So that's the structural reason. So these two
problems are believed to be out on a quantum computer. If you can break either of those two schemes, you make, you can solve these problems in some parameter selection. So structurally, that's why we
believe this is hard. It's the same thing as if we had a reduction that if you could solve Diffie Hellman, you could compute discrete blocks.
So, like, the, the basic learning with errors problem, I, I imagined it as, like, you know, taking, like, the standard problem of, like, you know, using Gaussian elimination to find a solution for,
you know, a, a system of linear equations, right? Which is, like, first week of first semester of linear algebra, easy problem to do, right? But if you mix a small error term in, With the solution
you're trying to find, it's a hard problem. And that's because when you multiply out that whole matrix times the error, that error term blows up, right? That doesn't involve polynomials, that doesn't
involve, you know, anything really complicated at all, maybe except for doing everything in modq, right? And you can build a cryptosystem off of just normal integer LWE. But nobody does, right? It's
all modular LWE, it's all over polynomials at this point. I have, like, sort of a basic intuition for, like, how that kind of module LWE kind of works. I have no intuition for why we do that. Like,
why do we complexify it in that manner?
Yeah, so first, I mean, there was Frodo that was submitted and some people really like Frodo because they're conservative and that is based just on the plain LWE problem. So, it's not quite right to
say no one does it. But why do people like module LWE is, so, before module LWE came ring LWE, which is essentially you replace the matrix that you do your linear algebra on. You replace that by a
polynomial. And then if you know this trick of that you can, you know, when you do a polynomial multiplication, multiply another polynomial, you can phrase this as a matrix multiplication. And then
you can say like, so instead of really having a uniformly random matrix, what I have is one that is derived from this polynomial. So now instead of having to store n squared coefficients, Or entries,
I now have to only stay a store ahead. So that is the reason that, you know, that immediately kind of, you know, like you get it from a quadratic, uh, size to a linear size or a quasi linear size.
And then, so that's nice. And then there's some omas that are nice, some degrees that are nice and some that are not so nice. And you might also think like, oh, maybe it's a little bit much
structure, but like mostly for performance reasons. What if, I hear you like matrices, I hear you like polynomials, what if I put matrices on top of your polynomials? So what you have is a 3x3
matrix, and in the case of Kyber768, And each of the kind of like, uh, nine blocks that you have is a little polynomial and where the, you know, the, and the matrix there is derived from this
polynomial. So that still gives you a saving and some flexibility of parameter choices.
And so the nice and not so nice, is that both in terms of security or performance or size or all of the above?
So it's, it's very nice in terms of performance and it might actually be, it's faster, uh, more efficient typically to consider this module LWE, at least in the case, uh, here. Because we like powers
of two for, for reasons that are a bit too boring to go into here, uh, for these rings, for these dimensions of these rings, and we really like... 7, 6, 8 as a dimension. It just hits a security
parameter sweet spot. It's not a power of two, is it? And that's why actually you get, you know, nice security versus kind of performance trade off if you just believe parameters. The reason why you
might not like it is, well, I'm adding more structure, right? And, you know, the rules of the game are more or less like the more structure you add to like a cryptographic primitive, the more you
worry that maybe an adversary can exploit the structure. But I should say... When we think about like what, how do we pick parameters, we just ignore the structure because we know of no way of using
the structure to even give us a factor that is linear in n. Like, it's just like, it seems like we don't know how to do much better, even for these more structured instances.
And so this is a bit of a tangent for the parameters are powers of two. They work nicely for these kind of polynomials you use in module LWE. That I, the first thing I hear when like, we like powers
of two, I'm like, because our regular digital computers are all in binary, and they really like computing over powers of two, it seems that it is a quote, a happy accident, not quite an accident,
that the crypto system based over module LWE He also likes those things and maybe that is why, partially why, they're quote so efficient on our modern digital computers.
Not quite, because
Cool, okay.
so the power of two here is a dimension of your matrix, right? So you're, so the 7, 6, 8 is like how big is your matrix? And there, you know. Not talking about bits. So there was a, there was a
finalist for NIST that actually, because there's still what is Q? Everything is mod Q and what's Q? And you would typically choose a prime because, you know,
makes sense. It's easy to reason about things are fields. But Saber was a finalist that said like, no, actually we use powers of two for the reason that you just mentioned.
if you use that, then actually our computers or actually our mathematics doesn't like it so much because if you pick the prime right, then you can use the NTT, the number theoretic transform, to do
your polynomial multiplications. And then that gives you a better performance than just doing general polynomial multiplication over arbitrary modular.
Got it. Okay, so, ignore what I said. But, picking parameters, of course, is like this balance between performance It's maybe a little bit of like how these parameters affect complexity of both the
structure but maybe also implementation complexity and what we know or what the designers know of the best attacks and It generally, if I say best attacks to people, they may think purely in terms of
computational complexity of an attack algorithm parameterized by some security level parameter like lambda or whatever, but There's more dimensions to this, especially when you have a quantum attack
where you're trying to design a scheme that is resilient against a quantum attack, you have to consider adversaries that have classical modern day computing capabilities and quantum adversaries that
have access to a efficient large quantum computer and Apparently, there's more to consider. Well, not apparently, but there is more to consider than just the algorithmic complexity of ATT& CK. So can
you tell us a little bit more about what is sort of in the space that you're considering that influences choosing parameters for these, like, LWE based schema, or module LWE and module SIS, whatever?
Yeah, so that is then kind of completing the circle because we know that if you find a magic way of breaking Kyber, then you can also solve some module LWE instance. So what is the best way that we
know how to break Kyber? Actually, it's attacking the module LWE instance. So you really think of this, so you don't really think about the encryption scheme anymore, you think of this hard
computational problem. Which is essentially, well, it's a noise linear system. Can you find a solution, please? And then you asked a question, um, how would you go about solving multi LWE? And then
that is a thing that proceeds in, you know, you have three levels. One is an overall attack strategy that is known as the primal or the dual attack. Then in both of these, you run a lattice reduction
algorithm, which more or less, uh, means people consider the BAK has Z algorithm. And this algorithm in turn now called the subroutine called, uh, shortest vector Problem Solver and this shortest
vector problem solver, they're the fastest algorithm is a lattice S And then the, the key question is like, well, how expensive is it to run a ladder SIF in a given dimension? So the key question is.
Because we have a pretty good idea of, you know, what parameters we need to run for this BKZ algorithm in order to make progress, you know, to have a solution with either of this primal or dual
attack, um, that gives us something called a block size. And the block size essentially tells us the dimension in which we have to solve the actually hard problem.
And so in the case of Kyber 512, the block size is something in the ballpark of 400. So you have to find shortest vectors in the lattice of dimension 400. And the question is like, how long does this
take with a quantum computer, with considering memory even so. And then that's the key datum. And there you want to hit whatever magical threshold you pick for saying, you know, my, my estimate for
this cost needs to be higher than And that's, and then maybe you take the polynomial overhead of running the BKZ algorithm and so on into account, and then that tells you how small you're allowed to
make it.
cool. I'm writing down each of these steps and it goes deeper and deeper and deeper and deeper. It's like, okay, this is the, this is the hot path. This is the heavy hitter way down here, parameters
by dimensions, approximately 400 and you just crank. Now this is. Is this, like, the best known, uh, attack, classical and quantum? Like, is this, like, where is the advantage for, uh, a quantum
attacker to run, say, uh, lattice sieve versus on classical?
Yeah, so like, maybe it kind of makes sense to, so, to talk a little bit about the algorithm that we use to find short vectors and lattices. So it really is, so you're thinking of, so what is a
lattice in the context here? Think of it really as a matrix, but a matrix over integers, so not mod q. And then you're saying, I'm going to do integer linear combinations, say, consider the rows of
this matrix, and I'm going to add the marks. And what combination of them gives me a vector that is the shortest, which has the shortest, the smallest entries, or Euclidean long, but like, you know,
like more or less, just something that makes me short entries. So that's the key computational task, right? That's all that's required of us. And so how do we go about this? Like, what's the best
known algorithm, the most efficient algorithm that we know of?
Before you, before you say that, just, just so I'm, just so I'm sure of what I think I understand here. On a normal episode of this podcast, I put on a, a delicate balancing act of play acting
ignorance about all of these things. Um, in this case, I won't need to be acting so much. Actually, I'm, I'm, I'm never acting. But, um, we're talking here, when we're talking about finding short,
uh, short vectors in a lattice, this is from a starting point of a random lattice. Where the basis of that lattice is, like, the row space of the matrix of the... It's the vectors that make up the
lattice, right? And if it's a, if it's a random... lattice, then those are all going to be of like weird random big size or whatever. And we're looking for short vectors, which we don't immediately
have in front of us. We have to somehow compute them from this random lattice. How much of that was crazy?
Uh, no, this is fine and like for, uh, so pedantically, it's a bit difficult to define what a random lattice is because it's an infinite space, but like that is maybe kind of not the thing that we
should now kind of dive into. But I think like, so what you also want to distinguish, there's the question of which lattice you pick. So that is essentially the span of all these vectors of what,
what can you produce? And then the question is, what's my input? How do I actually encode it? What's the basis of this lattice? And the key thing is that the input that you get is essentially a
random basis of this lattice and that is a notion that can, it's a bit clearer kind of what we mean by random. So you more or less compute the Hermit normal form of the integer matrix and because
that is canonical, that is the perfectly good input. Well, this is the thing that I had prepared and let's see, let's see if my analogy lands. So how do we go about finding these short vectors? And
it's just Barton's algorithm, and I've just offended, uh, all my friends. So, how do we go about this? We essentially say like, well, let me just, I have my input, right, is some, some basis, and
then I'm going to just kind of, Produce many more of them. I'm going to sample many vectors, kind of, that are just linear combinations of them, right? Pick some random linear combinations and you
probably pick small coefficients of your rows to kind of make it not blow up because you all care about small stuff. But you sample, like, a lot of them. And then you just check them pairwise. Look,
if I subtract these two, do I get something shorter? Oh yes, I do! Let me keep that. no, I don't. Let me not keep that. And you keep doing this pairwise comparison until you're done. And by the end,
you have a new list, but they're all a little bit shorter. You do the same thing again. Oh, yes, these are shorter. Let me keep them. And you keep on going. Until you eventually, and under some
heuristics, you hit the shortest thing you can make this way. Right? It's no different from... How do you kind of eliminate the hamming distance, kind of like, of a bunch of vectors? Well, if I add
these, does it reduce the hamming distance? Oh yeah, let me keep them.
This is amazing. I could code
I'd really love this. Yeah,
The whole thing is, uh, what I just described is, uh, is the Gauss NV CIF, and that is the, kind of like, the, the first, kind of, of the, heuristic CIFs. And by now we have a bit of, and this will
be maybe relevant when we talk about memory, Slightly better. And the, but the slightly better idea also, I think, is fairly quickly explained. And that is, if I have to, and again, let me do Hemming
distance, because I assume that kind of this is what maybe listeners here are familiar with. If I have three strings, and the first string is close in Hemming distance to kind of like some string
that I shall call the center, And the second thing, a string is also close to that string that I call the center. Then probably the two strings that are close to the center are also somewhere close
to each other. If they share many bits in common with the center, they will also share, you know, some number of bits with each other. That is the key trick for the asymptotically faster SIFs that we
use. So we just dedicate some vectors and say, like, yeah, let's just pick some random ones, you know, that are kind of, like, somewhat nicely distributed. And then, what do we mean by somewhat
nicely distributed? Oh, might as well be random. And then we say, like, okay, so now we're very generous. We say, like, oh, is it kind of close? You know, if I subtract these, is this kind of small?
But I got this very loose definition of small. And then we put them in buckets. And then I do this pairwise comparison within these buckets. And then now I have this quadratic search only happens in,
uh, in, you know, the things that are close to the center and, uh, I've improved the asymptotic complexity. So that is, roughly speaking, the BGJ1 SIF, which, uh, is implemented in Jessica, which is
this open source implementation of SIVing.
And then if you want to go to the asymptotically fastest of these algorithms, instead of picking these random centers, You pick some centers that are essentially error correcting codes and you just
use something that you can find, find out very easily which of them you're close to, right? So you specifically chose these centers and then instead of having to check all of them, am I close to
them? There's an algorithm that is somewhat better that tells you if you're close to one of these centers.
Yeah, yeah. It was cool.
That is the BDGL SIDH and that is the state of the art for classical SIDHs. So classical meaning not on a quantum computer.
Right. So do we get any speedups on a quantum computer for any, any of it, part of it? I mean, the whole, like, comparing all these little vectors and seeing which one is shortest with a little bit
of, like, ways of doing bucketing comparisons and stuff like that, it almost smells like you could get a quantum speedup on some of that stuff, but how much do you get?
So the quadratic search that I described, I hold on to one string and I compare it against all the other strings, right? That's an exhaustive search. And in an exhaustive search, there we have an
improvement using Grover's algorithm. And so we can just use Grover's algorithm for this part of the, uh, quadratic search. So, uh, that means you can at best hope for a quadratic speedup, right? So,
or a square root. The speedup is not that impressive because we have just reduced the quadratic part of the SIF dramatically by using these, uh, funky centers and only doing the quadratic pairwise
search kind of in these smaller buckets. And so you get a quadratic speedup in the size of these buckets. Or the running time, which is quadratic in the size of the bucket. And then Grover more or
less turns it down to linear in the size of these buckets. But these buckets now are not the whole dimension. You have like, they're smaller. And so the advantage that you have for kind of a quantum
computer are much less than what you would expect for Grover speedup. Because you always assume like, well, AS128.
would hope that Grover gives you something of 2 64 quantum operations, whatever that means exactly, and then a quantum operation is much more expensive than a classical operation, but yeah, whatever.
And then we are much further away from this kind of square root speedup in lattice setting.
I know Isogeny's the best. Before everything got broken in SIDH land, it was like, the best classical was like, fourth root speed up. And then if you throw, if you throw a Grovers on that, or you
apply Grovers as part of what was called a claw attack, uh, you would get sixth root. And so some of the sizes were, were scaled for that reason. And yeah, it sounds like you wouldn't, you don't even
have to worry about it in that. much in the lattice world, but then it got broken on the, uh, the construction basis of torsion points. And everyone was worried about the torsion points for a long
time. So we don't talk about that anymore. Analysis thrown in the bin. Uh, not really.
another, there's another reason why you select what I've just described in the quantum setting, so this Grover speedup. So, the database that we're searching for in the Grover setting here is an
exponentially large database of actual vectors.
So, like, it's different from an AS key search where, you know, like, it's some uniform space over, like, some bit strings, but, like, no, it's not all vectors that exist in the world, but it's,
like, an exponentially large collection of vectors. So, you actually need to produce these vectors first. And then you need in your Grover algorithm, you need something called a QRAM. So you need a
quantumly accessible classical RAM. So you have RAM, but you now can carry it in superposition.
Do we have that yet? Has anyone built that yet?
No, like, so QRAM is equivalent to a full scale quantum computer. And there is some debate about that I'm not an expert on, about what is the cost actually of running QRAM. roughly speaking, like an
intuition for why this is much more expensive than in a classical setting, you know, if I have a very large amount of data, I could, in principle, if I don't need some data for a while, kind of, I
could, you know, Put it in stone or put it on some magnetic tape and bury that somewhere and then leave it there for 200 years and then come back to it. But I don't need to keep the power on for
this, this thing for that while. But if I want to have superpositions over all my data, like I need to, to have it powered on entire time for my entire computation. Because I'm not varying like this
particular memory cell, but I'm varying some superposition of all the memory cells. Again, quantum computing people are probably a bit upset with me about like how I butcher that analogy. But that is
a reason like the, the question of memory cost is one for in the quantum setting that small speed up that we get from Grover, it would be very surprising if, if that would even hold
you need this QRAM and that seems to be a very expensive resource. So all the costs that we have is that we just assume you get QRAM for free. Because otherwise there just is not, no, no advantage,
Now, this is kind of goes to the point of like, how do you model your adversary? And one kind of nice thing about designing cryptographic schemes that will theoretically be resilient against a
quantum adversary is that. You ideally want to be resilient against a incredibly powerful quantum adversary. You want to give your adversary all the advantages that you can kind of get away with,
because if you can stand up to that, then you're, you've got a lot of margin. But then, on the other side, you have to... If you want your thing to be adopted, you need to be able to make it
performant and usable, and that means picking parameters that are resilient enough against your adversary that you have modeled, but small enough that make it performant and small and, you know, all
that sort of stuff. So, I'm guessing the basic answer is, like, Okay, we need to, my sort of ideal world is like, if we modeled the quantum adversary running these, uh, attacks against these module
LWE as having perfect QRAM, how does that totally bone the parameter sizes?
not at all.
Hey, then why don't we just assume they have, like, perfect tons of QRAM and then just... Just move on with their day.
Uh, yeah, and then this is what we do. I mean, the, we wrote a point where we tried to estimate that, but we had to set that QRAM as unit cost. And even then, I think if you, for the largest Kyber
parameters, the speedup was something that you wouldn't care about. I'm trying to open it live while I'm talking to you, but like it was. It was very, very small in, in large dimensions of like the,
the part that you have. And then, I don't know, maybe I'm leaning out of the window a bit too much, but I, I don't think that's going to be an issue. And then also the shouting matches that I, people
are not shouting at each other about quantum adversaries at the moment. So that's not the issue that's at stake. I think more or less everybody's like at the moment, like, yeah, the, the costs, how
we understand, um, what they can do is so far from something that we would consider a threat. Like quantum adversaries is for parameter selection, you can more or less ignore because, because of this
kind of weird thing, you need to have exponentially large database, you produce that first, and then the, the speed up is in this quadratic part of the SIDH, and we learned how to make that a lot
smaller, uh, using classical techniques.
So when we're like, when we're talking about like the strength of these algorithms, the security levels of these algorithms, we're talking about classical attacks. We're talking about like, if you
swap out curve25519 or the NIST curves or RSA for an LWE problem, are we getting comparable levels of security against kind of classical attackers like The attackers that actually exist in the real
Yeah. So that's, uh, I think it's at stake. Mm
so like, the lowest level of the stack of this attack is the sieving stuff, and it turns out the sieving is both cool and also not as hard to understand as I expected it to be, right? And the sieving
that we're doing, we're using that as kind of like a plug in for the BKZ algorithm. I'm I'm right so far? All right. So I have a sort of intuition for lattice reduction because it comes up in some
basic cryptanalytic attacks for public key stuff and for GCM and stuff, but like lattice reduction, LLL, I sort of get like I get it. Gram Schmidt because, you know, I did an undergrad linear algebra
with my daughter, right? I mentally understand LLL as like, okay, LLL is like Gram Schmidt if you had Gram Schmidt to call as a subroutine, right? But it's comparably as, you know, as complicated as
Gram Schmidt is. BKZ. In my head, I have like, okay, BKZ is like a better version of LLL. And that's as much as I know. So the idea of like a lattice reduction thing that has an SVP Oracle as a
plugin, I have no intuition for. How complicated is the BKZ part of this attack?
I think it's, if you're comfortable with LLL, then it's quite easy. Um, so let me first kind of try to do it through the medium of LLL and then let me try to do it without the medium of LLL. So first
through the medium of LLL. LLL is BKZ with block size 2. So, because what LLL does is, it does the Gauss reduction and so that means it looks at two vectors and then those span a little lattice, a
little projected sub lattice, and you just find the shortest vector there. And so now instead of looking at two vectors and finding the shortest vector in there, you have some parameter beta and you
look at that lattice and there your oracle finds the shortest
It proceeds exactly like the LLL algorithm. Okay, it doesn't quite exactly, but it's more or less the
I'm the one person that's listening to this that got anything out of that probably, but that all made perfect sense to me. That's great.
ha! Yay!
have at least benefited from this. This is awesome.
Let me try to do it without LLL. So LLL is a very famous algorithm for doing lattice reduction. And then You can, um, and that might not be the most intuitive, but you can think of this essentially
as a, as a sand pile. And this is if you know Gram Schmidt Orf organization. You can do that for a matrix, and then you can plot the norms of these Gram Schmidt vectors. You probably want to take the
logs of these and then what you get is essentially after LL reduction, something that is a downward sloping line. So the model that we use to analyze is literally called the sandpile model. So that's
the analogy I'm going to use. So essentially what this algorithm does, you start with like a very steep sandpile. And what you're trying to do is you're trying to flatten this. Ideally at the end of
the day, you would want something that's parallel, that's flat, right? So just a flat line. So, um, how do you do that? It's like, well, you pick a little part of your, your pile and it's like, I
want to flatten that just locally. I'm going to locally kind of like make this flat. So I take the peak and I make it flat. And so that means you push the beginning down and that means because you
know, the sand has to go somewhere. So the, the, the rest is pushed up and then let me move on, you know, next to the peak. Now I try to make that flat and you keep on going until the end. Oops. I
have nothing left to do, you know, I've kind of started from the peak and I'm at the bottom. Let me go back to the peak and see what's happened there, right? And then you kind of flatten that again,
that again pushes some sand out, and then that kind of modifies some stuff later. And you flatten, you keep on doing that. And so the flattening operation is more or less this SVP oracle, right? So
it does that for you, one little block, then you do the flattening, and you just keep on doing that and going round and round and round, and keep flattening it in little, in little steps. And then
suddenly, like, all of these steps together give you something that is a lot flatter. And, but as far, as much progress as you can make is as much, you know, stuff you kind of look at in the, uh, in
each little step. And so the bigger the block size, The more you flatten it. One goal.
Awesome. Ha ha ha ha ha! Thomas, this is, this is
I agree. No, it's not. I agree that that was awesome.
Oh, that's
Okay, so I'm still, I'm back to trying to chart my position in the whole attack as a whole, right? So I have like a, I have an idea of what we've done with BKZ to the original lattice. I have an idea
of how we use the sieving thing as kind of a plug in to BKZ to kind of locally sort out the shortest vector from a block of I've lost track of where we are in the overall attack.
Yeah. So, okay, so now let's say we have ABKE algorithm. So how do I use this kind of in the most straightforward way of solving module WE. And I already mentioned that like we just ignore all the
prolonging structure. We just pretend it's just like some matrix, right? Because that's actually how we cost this algorithms.'cause we don't know how to do better. So then what you do is like, so an
LWE is essentially have a matrix, A as a random matrix, and I multiply it by some vector S and then I add some more error term e. And let's call the result A times S plus E, C. It's some vector. And
so what I'm really like, how would I go about solving it? Well, one thing I can do is I can, you know, try all possible S's to see if I multiply them by A and subtracting them from C, I get something
small. Right. That would be kind of like, you know, take me, give all integer, you know, linear combinations of A and C. And when I subtract them from, from C, give me something small.
We're trying to find, like, the specific error term there, then. And we know the error term is small, because, like, it's a parameter of the algorithm. Like, the rate of the, uh, of whatever the
construction is, is... Like, by design, it's a small band of possible errors.
yeah, and then the parameters are chosen. There is no other S that also gives you something small. So, like, if you found it, then it's unique. And that already should smell a little bit like lapse
reduction because I have some, I have some vectors and I do some integer combination of them. And if I do the correct integer combination of them, then it's very small. Then suddenly everything
collapses and everything is very small. And that is known as the unique shortest vector problem. So it's a, it's a situation where like, Ooh, I have this thing, this matrix here over the integers. I
do some linear, integer linear combination to them. And there's one thing in there that is really quite small. So this BKZ algorithm is really good at finding kind of these short vectors. So I more
or less take my LWB instance and apply my BKZ algorithm to it and I have the primal attack. So the primal attack is in a way a very direct, it's a direct way of you tackling the problem head on.
There exists a linear combination of the rows of this matrix that gives something very small. Well, we know a class of algorithms that do exactly that. So let's, let's call that. That's it.
Got it. Okay. And that's actually, that feels kind of similar to the way, like, LLL gets applied to existing crypto problems, right? It's like, it's set up so, like, the smallest vector is going to
be, is like the likely candidate solution or whatever, right?
So one intuition for, you know, like maybe other people also, uh, you know, heard of Breitenbach's attack where you solve the hidden number problem kind of for poor randomness kind of ECDSA. So the
hidden number problem is one dimensional LWE. So if you understand the hidden number problem and, you know, have an idea how LL solves that, it's literally the same thing as the primal attack on LWE.
Neat. Okay.
I really love this. Just sort of like, Hey, you know, this other attack that, you know, from like 30 years of classical asymmetric cryptography, it is in fact, a short dimension or small dimension
instance of these like bigger ones, that's going to help me a lot to learn these a little bit more in depth.
I'm assuming that the reason everyone talks about the dual attack is because the primal attack is not the way you would really do this.
Yeah, I would say broadly, the prime attack should be the way to do this because it really solved the underlying problem. So like the dual attack, but like, uh, that is the segue, I guess, is you
actually solving the sys problem. So that's what we mentioned the module SIS or module sys problem. And that is finding a short solution kind of to get sums to zero. That is actually how the dual
attack proceeds. So we know that A times S plus E equals C. And now, say, I have a lot of rows of A. So it's not just a square matrix, but I have, like, quite a few rows. So that means there exists a
short linear combination of the rows that sums to zero because these are linearly dependent. And so all I have to do is to find a short linear combination that sums to zero. So now if I apply this
short linear combinations of the rows of A, well, that makes it zero point. And so if I now apply it to C, then this is a short linear combination. I've just killed A, so all that's left to do is a
short linear combination of the error term. But the error term itself is small, so I can, after I've done that, I get an element out that is much smaller than Q. So what I do is I do this many times
and then I say like, well, what I got out is that a short element all the time or is that a uniform element? That's the dual attack. And then how do I find short linear combinations kind of of A?
Well, the BKZ algorithm finds short vectors in a lattice and it turns out the kernel of A is a lattice. So what I do is I compute the kernel of A. And then that gives me an integer matrix. And then I
say, like, BKZ, would you please be so kind and give me some short vectors in this lattice? I would really like to multiply them by my C.
hm, uh, he,
And then you actually do something slightly more complex called the normal form of the dual attack. The reason why I'm mentioning this is the lattice that you fundamentally reduce in the dual and the
primal attack. is one has A in it, which is the LWEA, and the other one has A transpose in it. So, like, more or less the difference in the BKZ tests that you're running is whether you will reduce A
or A transpose, which, reminder, A is a random matrix. The only difference is, in the one case, you'll, at some point, you unlock the secret E, because, like, you know, you do a linear combination
and suddenly everything collapses. In the other case, there is nothing to collapse. You just do that and later on, take your short vector and multiply it by E, and then it says something's wrong. And
then it turns out, so it was, and then the, the dual attack is in a way, it's a quite roundabout way of attacking it, right? So you're saying like, oh, you know, I find something short in A that
doesn't have anything to do with A times S plus E. I just, uh, and then I multiply it by A times S plus E. And suddenly I can distinguish now. It feels like a less direct way. So, like, morally you
would say, like, I don't think this should be better than just, like, tackling the problem head on. You're even running the same BKZ algorithm on it. But it turns out, as far as we understand these
attacks, you can play some tricks there of, uh, how you go about it. And that kind of makes these attacks currently kind of a few, um, a little bit faster than the private attack, at least kind of as
far as our estimates are concerned. Right, so like, it seems like, yeah, indeed, these algorithms perform slightly better. And if that's an artifact of like, we just haven't really... found out how
to analyze either of them really properly or that's actually there's something, something deeper happening with this roundabout way is more efficient. So
Okay. So. That mostly made sense to me, and I won't torture you by, uh, having you explain that further. Um, I'll torture you a different way, which is, so we have, like, for the PQC stuff, we have
target security levels, right, which is, like, roughly, like, matching the strength of the asymmetric algorithm, the key exchange, or whatever, to whatever, um, like, bulk encryption we're doing,
like, 128 bits of security, whatever it is, right? So, like, I guess there was news last year. I think it was 20, it was 2022, right? where, like, the Israeli equivalent of the NSA published a
technical report, right, where they had brought cost of, I think, the dual attack down to below the target security levels. I don't know how meaningful that reduction of security was. It was, like,
on the order of, like, 10 or 15 bits or something like that, which the number of bits we had was already higher than AES or something like that. I don't know, right? But I'm more interested just in
what was the meaning of that technical report. Like, what was the interesting bit of that?
they, they did two things. So let me compare with the bit adaptations that they did. Yeah, so the target security level was meant to be something in the ballpark of 143 and they reduced it to 138 or
mm hm, so five bits.
Uh, something for the smallest permit and it's bigger for 1,024, it went from 2 7 2 to 2 5 7, so that's, that's a bit better. Right? And then what they did is two things. On the one hand, they
improved, so they build on some model of how you cost the sub exponential part of this algorithm. You know, I mentioned this decoding in the sitting earlier, and that is a sub exponential part, so
that's great. But it's still, it's expensive. And then, and so in this paper that I put in the meeting notes, we gave like some costs for that. And they said like, actually, you can do better than
this. And so they improved the sub exponential factor. That is one of the big savings. The other one is they are building, or maybe I think they, it might be independent, might have been independent
work, but there was a paper slightly earlier by Guo and Johansson at AsiaCrypt. And they, uh, essentially introduced an FFT based approach to do a part of guessing in the dual attack. So let me
unpack that. So the way I've described the dual attack now, it was a distinguishing attack. Like, do I get something small or large? But, um, what you can do is you can say like, how about I guess
some components of the secret. And then if I guessed incorrectly, then I assume that the distribution of what I'm left over is uniform. If I guess correctly, I get a smaller dimensional LW instance.
If I can distinguish those two, I can confirm or deny my guess. And, uh, what they did is they, uh, and the, and this is, this is useful because now the last problem that you have to solve is
smaller, right? And the nice thing about this attack from an attacker perspective that the costs are more or less additive. So you can run the lattice part of the attack once and then you do the
guessing stage kind of like on this preprocessed data. And, uh, and they kind of showed a way of, like, making this guessing step cheaper using some FFT based approach by essentially targeting only
kind of some lower order bits. And that allowed them to increase the dimension of the stuff that you guess, which decreased the dimension of the lattice problem, and then all parameters, uh, kind of,
you know, like, attack parameters were a bit smaller and everything was a bit faster. And that's kind of, they have, uh, in this Matzov paper, they have a slightly different approach to kind of like
also using an FFT, which also allows them to generalize this and not just, uh, for bits, but like any, any small prime. And then that flexibility allows them to reduce this further still. This then
also opens the door for like the kind of the back and forth, because there's a paper that is joyously, uh, joyously, uh, titled, does a dual attack even work?
Mm hmm.
Uh, by Leo and Ludo, and the problem that you might run into is at some point you have to distinguish between, is this thing random or is this thing from an LWE distribution?
But now, if we guess a lot of secrets, we actually, the correct secret has to compete with potentially an exponential number of false solutions. And then you have to take into account, like, maybe
some of them randomly behave even better than your correct solution. And then there's this paper by Leon Duda, as I mentioned. And then there's a paper by Yixin and her co author, where they said,
like, okay, where's the regime where we can prove this? But it's essentially the question of... We're doing some statistics. We're more or less making some independence assumptions here that this is
fine, but these things are not necessarily independent is actually sound. So there's a, it's a bit of a back and forth of like, do you actually get the claim complexity or not?
But here we're talking about not whether the particular MatSav attack worked, but whether the dual attack approach works at all.
So the dual attack is, it has been known to work, but like the, so what the question really means, does the attack when you do this key guessing work? So can you, can you do this step? And then, and
what is the regime in which you can do this step? And you still have some guarantees like, no, no, no, my correct solution will be the one that I identify.
And what they kind of, uh, showed in this paper said like, like the analysis is kind of, uh, theoristic, like we have some, some, some counter examples. Yeah. And so I think there's still, if the
attack has this complexity or not, at this point is a bit up in the air is my understanding. I haven't followed the latest on this in detail, but like when we're talking about, yeah, like ballpark of
do you gain five bits or not in the lowest attack, right? So like, we're not talking about like, is the whole principle doomed? What we're talking about. Do you really get this complexity that, that
pushes you below this kind of magical number of one, four,
hmm. Awesome. So, all of these, even including the, the Matzov attack or some of these improved, uh, sieving attacks, these are all classical. So, we talked a little bit about how... In the quantum
attack model, you have to consider how, how good or how much QRAM you have, but it seems that for a lot of schemes, you don't even have to consider that because even if you consider them having all
the QRAM all the time, and it's perfect and never has errors, it doesn't really affect the security margins for those attacks. What about for these, like, best in class, if the dual attack actually
gets the speed up and formats off and all these things like that? What do we have to worry about in terms of RAM and storage and all that sort of stuff? Is that a consideration for classical? It
seems like it's not, not so much a consideration for quantum attacks.
it definitely is a consideration for classical. So the, uh, the sieving algorithm, so the key routine that we kind of argued about here is an algorithm that runs in exponential time and requires
exponential memory.
Oh, okay.
And so, and that
sounded so easy!
heh heh
the key thing is you need exponentially, many of your strings in order to do this Paris comparison, actually get a reduction. Right. That's the reason. So the numbers here, the memory that you need
for sif is roughly oh 0.2 times the dimensional of your sif. So if you have block size beta, it's oh 0.2 times.
Okay. Mm hmm.
So we had block size, roughly 400 for the smallest cargo parameters. So you're looking at a memory of roughly to the 80 vectors. And then I had already mentioned that the way we actually do this
attack is that we actually kind of put them in these small buckets. And it's only in these buckets that you do a pairwise comparison.
hmm. Within each pocket.
Yeah, within each bucket you do this pairwise comparison. And that, uh, a bucket is roughly the square root of your entire database. That's how you parameterize it. So you're looking at something
like 2 to the 40 vectors. And that's where you do the quadratic SIF. And so like some, some work that's been kind of, we have recently done that is kind of hopefully put on Eprint soon is to say
like, okay, can you sift on multiple machines?
and then the, and turns out you can, because you, um, the thing that you really need to be fast, the memory to be kind of, because you, you, you're looping through your database and find something
that is close to your vector that you're currently holding onto, right? And there you need memory to be fast. That is something of 0. 1 of your dimension, right? So like, it's, it's quite a bit
smaller. And so if you have local RAM that is kind of in that ballpark, then you can essentially put one bucket per machine and there enjoy all the benefits of like fast handling, right? So that was
the thing that we're focusing. And there's been some designs of like, you can even, you know, do a range computers in a ring structure and so on, so that you kind of minimize this, like. But like,
more or less, yes, you need exponential memory, but it's not like you're jumping around randomly in this exponential memory all the time, but you can hide a lot of, uh, load latency by essentially
running through this inner circle and just, uh, touching the next vector. So in that sense... There is some cost that is associated with kind of accessing all of these, uh, exponentially many
vectors, but it's not like each call now means like I now have to go out in an exponentially large database and find the random element in there and, you know, I have no, you can, caches and so on
make sense.
This smacks of, not quite map produce, but like you can just break it up into chunks and you can farm it out and pretty much just bucket it into parallel processes, get some results, and compare
them, and then kind of work on it. Work on. I work on it as opposed to every. sieving bucket, having to see all the other data that the other sieving processes are also working on or something like
and there's, of course, some synchronization that you need to do, but like, I mean, in our, you know, like, somewhat limited experiments, but like, you know, like a bit free servers, like we saw like
a pretty tidy linear speedup. So it seems like, yeah, you can scale this across multiple servers. And then of course, like the, really what you want to do is like have networks of FPGAs doing this
sort of thing. Uh, but like, you know, we leave that for future work
So does this affect any of the current estimates of the hardness of the different security parameters? Compared to, because it's supposed to be like, well, level one is approximately AES 128, and
level three is supposed to be AES 256, and so on and so on. Do you think it affects that?
because at the moment, all of these estimates that we're talking about, assume that RAM is free.
Uh huh.
So, so if anything, they push the cost up, not down.
Cool. All right.
And so the dispute is over, does it push it up enough to regain the security level that is, was originally claimed or not? So, if you think that kind of these numbers that are, for example, just
take, you know, say, like, uh, we find out actually Matzov behaves exactly, the Matzov attack behaves exactly as they claimed in their paper, despite problems in the analysis identified, but somehow
it turns out, you know, yeah, yeah, yeah, we made some independence assumptions. It's not true, but it seems to work in practice sort of thing. Let's, let's say that. And if you're happy with that,
like, yeah, it's not quite AES256, it's, it's a little bit lower than that, but like, I accept that then, yeah, the RAM, taking RAM costs into account will only push that up. So, it's, the, the own,
the whole dispute is over if you're uncomfortable with these numbers right now. Does taking memory into account allow you to be comfortable again?
see. Okay.
And you're gonna have to take memory into account, somehow. But like, you might not take the worst case assumption about how much complexity or how much time it's gonna, or effort it's gonna take.
Like having to have every bit be connected to every other bit because of like, we can, if we can bucket things and blah, blah, blah. Okay.
Yes, at the moment we're just assuming this is free, right? So like I, every, every cell I can, I can just address and like immediately, like it costs, it's, it's a no op, right? So like, I want to
compare a vector, then, you know, I can just do that. And that is, of course, unrealistic, but the question is like, how well does different levels of cache, how much do they mask the true cost? And
how much do you, should you actually care? I think is a, is a, is an open research question. So
On the flip side, like, has there anything changed with, like, analysis of how hard AES 128 256 are against quantum attackers with, you know, perfect QRAM or classical attackers with perfect all
access and it's free? Like, has any anything evolved on that? Or is it just purely like, Model groves with perfect QRAM with everything everywhere all at once.
no, there has been, so all the lattice schemes, let me try to phrase this the right way around, more secure
there was a revised cost estimate for breaking AS on a quantum computer.
So there's a paper, I can put it in the meeting notes by Fernando Verde and co authors. I apologize to the co authors, but Fernando was my student, so I know his name best. Meeting notes, and they
kind of revise the costs, uh, for, uh, for AES quantum circuits. still get, like, so the... This target of as hard as AES on a quantum computer was a moving target, like it makes sense why they
picked this, but like we, it's not that it is settled of like how many operations does it take to solve AES on a quantum computer. And even what is the right cost model for a quantum computer isn't
clear. So what are the operations that you should count? Because we don't really know yet which of the architectures will, uh, will be dominant and then what are the relative costs of various kind of
quantum gates.
Thank you because like, I know that there's like discussion when you're looking at specific new crypto schemes about All these attacks and how to measure and that sort of stuff, but like the other
side of it is like the thing you're trying to categorize against, it also may move because it's also subject to analysis and, you know, crypt analysis and modeling of, you know, whatever. And so I
was, I was very curious if like these, if these things were moving at all, or if, if AES 128, you know, complexity of attacks kind of stayed stable, and I'm glad that you pointed out that I haven't
read that paper. So, yeah. Cool! Thomas, do you have anything else?
No, my brain is leaking out of both my ears and my
hee hee
but that was, that was incredible. That was awesome. So I'm, I know a little bit more than I did before, but it's so difficult to get me to learn new things that you should be very, very impressed
with yourself for accomplishing that. That was awesome. Thank you so much.
Okay, I feel honored as an educator.
Yeah! Thank you, Martin, for, uh, meeting the challenge of how do I talk about these things with no visual aids, audio only, and I think you did an excellent job because I understood a lot of it.
Thank you, Martin. Thank you very much. Uh, Security Cryptography Whatever is a side project from Deirdre Connolly, Thomas Ptacek, and David Adrian. Our editor is Nettie Smith. You can find the
podcast on BlueSky and Twitter at SCWpod, and the hosts on BlueSky and Twitter @durumcrustulum,@tqbf, and @davidadrian. You can buy merchandise at merch. securitycryptographywhatever. com. Thank you
for listening! | {"url":"https://securitycryptographywhatever.buzzsprout.com/1822302/episodes/13962690-attacking-lattice-based-cryptography-with-martin-albrecht","timestamp":"2024-11-04T14:32:52Z","content_type":"text/html","content_length":"139237","record_id":"<urn:uuid:c6406579-a9df-447f-9321-dabf5291a9b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00396.warc.gz"} |
Symmetry in Physical Laws – Gaurav Tiwari
Symmetry in Physical Laws
'Symmetry' has a special meaning in physics. A picture is said to be symmetrical if one side is somehow the same as the other side. Precisely, a thing is symmetrical if one can subject it to a
certain operation and it appears exactly the same after the operation.
For example, if we look at a base that is left and right symmetrical, then turn it 180° around the vertical axis it looks the same.
Newton's laws of motion do not alter when the position coordinates are altered, that is, they are moved (linearly) from one place to other. This is equally true for almost all other physical laws.
Therefore, we can say that (almost all) laws of physics are symmetrical for linear displacements. The same is true for rotational displacement.
Not only Newton's law, as said, all the other laws of physics known so far are symmetric under translation and rotation of axes. Using these concepts of symmetry, new mathematical techniques has been
developed for writing and using physical laws, for example Tensor Analysis.
Remark: We use many other different terms for symmetry whenever needed, for example in high-schools we use the term conservation and at grad-level it turns out to be invariance, invariant or
symmetry, itself.
There are following main symmetry (conservation) operations in physical laws:
• Symmetry in Matter and Energy or Conservation of Mass (Matter) and Conservation of Energy
• Conservation of Momentum
• Conservation of Angular Momentum
• Conservation of Electric Charge
• Conservation of Baryon Number
• Conservation of Lepton Number
• Conservation of Strangeness
• Conservation of Hypercharge
• Conservation of Iso-spin
• Conservation of Charge Conjugation
• Conservation of Parity.
Conservation of Mass & Energy
This conservation involves the following two different definitions and one hypothesis by Einstein.
Conservation of Mass / Matter
Matter can never be created or destroyed, but it can convert itself into several other forms of either matter or energy or both.
Definition. 2
Conservation of Energy
Energy can never be created or destroyed, but it can convert itself to other forms of matter & energy.
In practical, we see that if we burn a coal, it emits heat & remains ash. Scientifically, the coal (matter) is converted into heat (energy) and precipitate (matter). This is a balance conversion in
which matter converts into energy. Similarly, we can generate a lot of energy after nuclear fission, in which also matter is converted directly into energy. We have also seen, energies forming
different kind of unstable matters in nature. Physics' famous equation $E=mc^2$ given by Einstein also says the same : $E$ (energy) is directly related to $m$ (mass). Matter (mass) and Energy both
are conserved with their inner-conversions and the total value of mass + energy is a constant, since origin of universe. The complete hypothesis was by Albert Einstein.
After combining the two definitions and the hypothesis we have,
The mass and energy can neither be produced nor destroyed --- but they can be converted from one form to another.
Conservation of Momentum
The linear momentum of a system is constant if there are no external forces acting on the system of physical bodies.
Conservation of Angular Momentum
The angular momentum of a system remains constant if there are no external (angular) torques acting on the system.
Conservation of Electric Charge
The electric charge can neither be created nor destroyed. The net algebraic sum of positive and negative electric charges is constant.
Conservation of Baryon Number
In any nuclear reaction, the number of baryon particles must remain the same, at least ] until the reaction completes.
Conservation of Lepton Number
The lepton number, i.e.; the algebraic sum of number of leptons and anti-leptons, remains constant throughout a nuclear reaction.
Conservation of Strangeness
The algebraic sum of the number of kaons and hyperons , called strangeness, remains constant in electromagnetic and strong interactions.
Conservation of Hypercharge
The flavor of quarks remains the same throughout an internuclear interaction.
Conservation of Isospin
The isospin of hadrons is constant in strong interactions.
Conservation of Charge Conjugation
Remark: Charge conjugation C is the operation of changing a fundamental particle into its anti-particle. It's something like applying inverse function to any value.
For example, let C be the charge conjugation operator
• $C (\pi^+) = \pi^- $ (i.e., $\pi$-mesons being converted into their antiparticles, and;
• $C (x^2-3x+5) = 3x-5-x^2 \Box$
The charge conjugation operator is conserved in strong and electromagnetic interactions.
Conservation of Parity
Remark: The parity operation, $P$ is reflection of all coordinates through the origin. That is, in two dimensions co-ordinate system X-Y, $P(x, y) = (-x, -y)$ or $P(\mathbf{r})=-\mathbf{r}$.
The parity of any wave function describing an elementary particle is conserved. | {"url":"https://gauravtiwari.org/symmetry-in-physical-laws/","timestamp":"2024-11-08T05:19:21Z","content_type":"text/html","content_length":"68804","record_id":"<urn:uuid:dd2c8c03-d052-41b3-b7c2-680fa53b41f1>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00190.warc.gz"} |
Murphy's Mangled Math
In his blog Stranded Resources Tom Murphy argues that space resources will likely remain beyond our reach. He concludes humanity should learn to live within its means and conserve our resources. This
sound advice is the theme for most of his Do The Math blogs.
But the math on which he builds his argument is wrong.
To calculate delta V from earth to Mars he adds 3 quantities:
Earth escape velocity (~11 km/s),
Earth to Mars velocity (~6 km/s)
Mars escape velocity (~5 km/s)
Which totals ~22 km/s.
But you don’t simply add these three quantities. Break the Earth to Mars velocity into two parts. These parts form legs of two right triangles. The other legs being Earth escape velocity and Mars
escape velocity. Add each hypotenuse for the actual delta V.
So the total delta V is around 17 km/s, not 22 km/s.
But wait. Murphy did generously round his 22 km/s to 20 km/s.
And there is also a ~2 km/s gravity loss incurred during vertical ascent. Add this 2 km/s to 17 km/s and you get 19 km/s. Murphy isn't shy about mentioning gravity loss. But he doesn't include it in
his calculations, giving the impression that he's being quite generous to the addled space cadets. Including gravity loss takes the actual delta V to about 19 km/s. This isn't too far off from
Murphy's 20 km/s.
But Murphy neglects the use of aerobraking.
For the Mars orbiter missions, a small burn is done to park the probe in a capture orbit rather than a low circular orbit. This can be done with as little as .7 km/s. The lowest point in these
capture orbits pass through Mars upper atmosphere. Each time the probe passes through Mars' upper atmosphere a little velocity is shed by atmospheric friction. Using aerobraking, a capture orbit can
be reduced to a low circular orbit using virtually zero propellant.
For the Mars landers, aerobraking sheds around 6 km/s.
Including gravity loss and using aerobraking the delta V budget for Earth surface to Mars surface is more like 14 km/s, about the same for delivering a comsat to geosynchronous orbit. So even
Murphy's apparently generous 20 km/s is 6 km/s too much.
Given that the exponent of Tsiolkovsky's rocket equation scales with delta V, 6 km/s is a serious error.
Tsiolkovy's equation:
(start mass) / (final mass) = e^(delta V/exhaust velocity)
Where e is Euler's number, about 2.72.
The dramatic power of exponential growth is illustrated by The Legend of Paal Pasam. An east Indian king enjoyed challenging his guests to a game of chess along with a friendly wager. Unknown to the
king, one of his guests was Krishna. Krishna offered this wager: 1 grain of rice on the first square, 2 on the second, 4 on the third, doubling the grains each square of the chess board. The king
agreed. Only after losing to Krishna did the king realize the enormity of his bet. Krishna revealed his true identity and told the king he could pay his debt over time. To this day the king’s estate
gives rice to Krishna’s followers during their pilgrimages through that land.
Exhaust velocity of hydrogen and oxygen is about 4.4 km/s. 3 / 4.4 = ~ln(2). Each 3 km/s added to the delta V budget is a square on the above chess board. That is, each 3 km/s doubles the starting
Murphy's 6 km/s error quadruples the starting mass.
Refuel In Space?
If you can get propellant along the way, it changes the picture:
At each square with a propellant depot, you get to start over at 1 grain of rice.
Murphy takes a look at refueling in space. A good propellant source would be close to earth in terms of delta V. So what does Murphy suggest? Jupiter or Titan! If he is looking for the most absurd
propellant sources to debunk, he would do better to look at sources from Alpha Centauri. Or better yet, the Andromeda galaxy.
What are potential propellant sources that are close in terms of delta V? Earth’s moon is one.
At the lunar poles are craters floors which never see sunlight. Temperatures in these basins are as low as 40 degrees Kelvin, colder than Pluto. After a comet impact, volatile gases that don’t escape
spread over the lunar surface. Gases reaching the cold traps will freeze and stay there. India’s Chandrayaan-1 lunar orbiter found evidence of thick, relatively pure ice sheets in many of these cold
traps. It is estimated the anomalous north pole craters have at least 600 millions tonnes of ice.
These lunar volatiles are potential propellant only 2.5 km/s from Earth Moon Lagrange 1 (EML1) and Earth Moon Lagrange 2 (EML2). Using 3 body mechanics, there are paths that enjoy delta V savings
over Hohmann orbits. And EML1 and EML2 are hubs for this Interplanetary Transport Network.
Lunar volatiles can also provide water for radiation shielding, water to drink, as well as nitrogen and oxygen to breath. All 2.5 km/s from EML1. This is a huge mass that doesn’t have to be lifted
from the bottom of earth’s gravity well.
Are there other potential propellant sources?
The low density of Mars’ moons Phobos and Deimos could indicate volatile ices. The low density could also be caused by voids within the moons, so the jury’s still out. If these do have ice, they are
potential propellant sources quite close in terms of delta V. It is about 3 km/s from EML1 to Deimos. Possibly a little less if aerobraking is used.
Murphy looks at delta V from one low planet orbit to another. This is common, Atomic Rockets does the same, for example. But there are a multitude of possible parking orbits. Parking in a low
circular orbit takes the maximum delta V. A high apogee capture orbit can take much less. Given the possibility of departing from propellant sources high on the slopes of a gravity well and shedding
velocity using aerobraking, he would do better to look at delta V between elliptical capture orbits.
Grab That Asteroid!
Murphy suggests 5 km/s to capture an asteroid in earth orbit. There are near earth asteroids that could be captured with much less. The comet Oterma suggests a possible capture method using 3 body
mechanics. Oterma will sometimes fall through the Sun-Jupiter L1 (SJL1) neck into Jupiter’s realm. It spends some time in Jupiter’s realm and then exits through the Sun Jupiter L2 (SJL2) neck. Then
later it will fall back into the SJL2 gate, dwell in Jupiter’s realm, then exit trhough the SJL1 gate. This is described in the online textbook Dynamical Systems, The Three-Body Problem and Space
Mission Design, a 17 Mb pdf.
An asteroid slowly drifting by the Sun-Earth L1 (SEL1) or Sun-Earth L2 (SEL2) could be parked in these regions with a minute nudge. From SEL1 or 2, a tiny amount of delta V suffices for delivery to
EML1 or 2. For some asteroids .3 km/s can suffice for capture.
Only a small number of asteroids are amenable to capture this way though. A much larger number of Near Earth Asteroids pass within 1 km/s of EML1.
Murphy’s hypothetical asteroid is a cubic kilometer. The Tunguska object is thought to have been about 50 meters in diameter. Murphy’s asteroid is about 10,000 times larger than a meteorite big
enough to wipe out a major city. So his absurd asteroid is a nonstarter due to safety considerations as well as the difficulty of moving such an enormous mass.
If we find a 20 meter asteroid of value, this could more safely be parked in earth’s orbit. This is small enough to burn up in earth’s upper atmosphere.
If we find a large ore body, it makes no sense to park the entire asteroid in earth orbit. Rather import the resources in small enough loads that it’s safe and doable. This also avoids flooding the
market and thus devaluing the commodity.
Given a 20 meter object and 1 km/s delta V, the energy required differs by a factor of about two and half million from Murphy’s scenario -- somewhat less difficult.
Murphy ignores a number of things: 1) The Oberth Effect. 2) Aerobraking. 3) Moving between capture orbits rather than low circular orbits. 4) Nearby propellant sources. 5) Exploiting 3 body mechanics
for delta V savings. 6) Small asteroids close to EML1 or EML2 in terms of delta V.
Tom Murphy does use weasle words like "simplified, approximate terms" or "crudely speaking". But his errors are truly enormous, too big to be salvaged by these disclaimers.
So I have to give Stranded Resources a grade of F.
Which is a shame. Murphy is correct to urge less consumption. But he doesn't have to resort to wrong arguments to support his view. That only subtracts from his credibility.
6 comments:
Van Kane said...
Hop David - A nice blog. Would be interested in your take on how much mass has to be launched to put the entire system in place.
Van Kane, you are the first person to comment on my blog. Thank you.
An interesting pdf:
One of the authors is Chris Lewicki who is also of Planetary Resources. I'm guessing Planetary Resources hopes to use something like this to park an asteroid in high lunar orbit or at EML1.
On page 14 of they describe the Asteroid Capture and Return (ACR): 5.5 tonnes dry mass, 13 tonnes Xenon propellant for a total of 18.5 tonnes. In the illustration it's launched by an Atlas V, but
perhaps this won't be the only option by the time they launch.
I am encouraged that one of planetary resources first goals is returning a water rich asteroid:
A propellant source high on the slopes of earth's gravity well has the potential to make spaceflight much less difficult. And less expensive spaceflight is a prerequisite for achieving ROI on
asteroidal platinum and other minerals.
What is the mass of infrastructure needed to make propellant? An open question, but I am anxious about this. Cracking water into hydrogen and oxygen takes 237 kJ per mole. I understand the ISS
solar array wings put out about 30 watts/kg. I've heard of solar arrays that have a specific power of 200 watts/kg, but haven't seen yet seen convincing cites. In any case it looks like we'll
need a massive power source to crack water at an appreciable rate.
I've seen proposals to bake the water out of hydrated clays using sunlight: placing asteroid material to an airtight kiln at the focus of a parabolic mirror. I believe high specific power is more
doable for thermal watts than electric watts, but this is still an ambitious undertaking.
To summarize: ACR mass about 19 tonnes. I don't know what the mining infrastructure mass will be. I know mining the asteroid will be difficult but I haven't yet seen persuasive arguments that ROI
is impossible.
Funny how he never expanded upon that original post.
Mass drivers ? Solar electric propulsion ( effectively a different type of mass driver ? )
Nuclear propulsion ? ( i.e. something as simple as steam rocket ) etc etc.
Lunar materials as propellants ? ( Aluminium, oxygen )
The SEP you mention is a big one. The Keck study suggests using SEP with xenon as a propellant with an exhaust velocity of 30 km/s. The study looks at retrieving some objects that can be diverted
from the heliocentric orbits to a high lunar orbit for around .2 km/s.
In Murphy's silly asteroid retrieval scenario he looks at fetching a kilometer sized asteroid that would take 5 km/s. He wants to use a propellant with 3 km/s exhaust velocity.
I don't know if Murphy is capable of plugging .2 km/s delta V and 30 km/s exhaust velocity into the rocket equation. If he were, he'd discover his "rough approximation" is off by several orders
of magnitude.
This may be some time down the track, but another issue that is ignored is the distant potential for self-replicating droid ships of some sort. Fire a few self replicating ships (or colonies,
depending on your scenario for AI technologies) at the asteroid belt with all the fuel you need to brake it there, wait a decade or 2, and eventually it will manufacturer all the ships and fuel
necessary to fire resources back to almost any point in the solar system.
Eclipse, perhaps Von Neumann machines will come to pass one day. But now they're science fiction. Brin's existence is a nice yarn on such devices.
But I do believe telerobots and robots will be game changers. I talk about that on several of my blog entries, the most recent being Who Needs Humans?
Perhaps as robots become more able they will eventually become able to extract resources from their environments and use them to replicate themselves. That's well beyond present state of the art, | {"url":"https://hopsblog-hop.blogspot.com/2012/02/in-his-blog-stranded-resources-tom.html","timestamp":"2024-11-11T18:06:42Z","content_type":"text/html","content_length":"88099","record_id":"<urn:uuid:882dab44-8629-454f-bfd9-d2c9e2aef371>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00576.warc.gz"} |
07-23-2018 01:16 PM
Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-17-2018 02:17 PM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-19-2018 12:41 AM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-19-2018 01:00 AM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-19-2018 10:30 PM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-19-2018 10:39 PM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-20-2018 12:11 AM
Re: Minimizing an unconstrained NLP problem on Mathematical Optimization, Discrete-Event Simulation, and OR. 07-22-2018 04:47 PM
Indeed the solution that you recommended was brilliant and I thank you for that. I never had that idea before 🙂 In line with this, I tried this approach in evaluating the optimal values for another
unconstrained minimization, however, it says that the solution status is failed. I guess I've had some problem with formulation? or in dealing with this type should require other type of solver? proc
optmodel; var x{1..4} >=0; impvar y {i in 1..8} = if i = 1 then -((-0.6*4.5)/(1.6*5.5))*x[1]-(4.5/5.5)*x[3] else if i = 2 then -((-0.7*5.2)/(1.7*6.2))*x[2]-(5.2/6.2)*x[4] else if i = 3 then (-1/0.72)
+((1/0.72)+((-0.6*15.5)/(0.72*1.6*5.5)))*x[1]+((15.5/(0.72*5.5))-((1/0.72)+1))*x[3] else if i = 4 then (-1/0.78)+((1/0.78)+((-0.7*17.6)/(0.78*1.7*6.2)))*x[2]+((17.6/(0.78*6.2))-((1/0.78)+1))*x[4]
else if i = 5 then (0.6/(1.6*5.5))*x[1]-(1/5.5)*x[3] else if i = 6 then (0.7/(1.7*6.2))*x[2]-(1/6.2)*x[4] else if i = 7 then (-1/0.72)+((1/0.72)+(((4.5*5.5)/0.72)+(2/0.72)+1)*(-0.6/1.6))*x[1]-
(((5.5*4.5)+1)/0.72)*x[3] else if i = 8 then (-1/0.78)+((1/0.78)+(((5.2*6.2)/0.78)+(2/0.78)+1)*(-0.7/1.7))*x[2]-(((5.2*6.2)+1)/0.78)*x[4]; min log_h = -( - x[1]*log(3500/(300*x[1])) - x[2]*log(4000/
(300*x[2])) + x[3]*log(175/x[3]) + x[4]*log(64000/(300*x[4])) + y[1]*log(3500/(300*y[1])) + y[2]*log(4000/(300*y[2])) + y[3]*log(0.15/(600*y[3])) + y[4]*log(0.2/(600*y[4])) + y[5]*log(5250/y[5]) + y
[6]*log(2000000/(300*y[6])) - y[7]*log(3.9/(45*y[7])) - y[8]*log(4.3/(45*y[8])) ); solve with nlp / ms; print x; quit;
... View more | {"url":"https://communities.sas.com/t5/user/viewprofilepage/user-id/221546","timestamp":"2024-11-11T10:54:53Z","content_type":"text/html","content_length":"178998","record_id":"<urn:uuid:6d587eb0-2682-471f-92e2-dc0c3a271704>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00864.warc.gz"} |
© 2009,2012-2013,2016-2018,2021,2022 John Abbott, Anna M. Bigatti
GNU Free Documentation License, Version 1.2
CoCoALib Documentation Index
User documentation
The functions in the NumTheory file are predominantly basic operations from number theory. Most of the functions may be applied to machine integers or big integers (i.e. values of type BigInt).
Please recall that computational number theory is not the primary remit of CoCoALib, so do not expect to find a complete collection of operations here -- you would do better to look at Victor Shoup's
NTL (Number Theory Library), or PARI/GP, or some other specialized library/system.
See also BigIntOps for very basic arithmetic operations on integers, and BigRat for very basic arithmetic operations on rational numbers.
The Functions Available For Use
Several of these functions give errors if they are handed unsuitable values: unless otherwise indicated below the error is of type ERR::BadArg. All functions expecting a modulus will throw an error
if the modulus is less than 2 (or an unsigned long value too large to fit into a long).
GCD, LCM, etc.
The main functions available are:
• gcd(m,n) computes the non-negative gcd of m and n. If both args are machine integers, the result is of type long (or error if it does not fit); otherwise result is of type BigInt.
• IsCoprime(m,n) returns true iff abs(gcd(m,n)) == 1
• ExtGcd(a,b,m,n) computes the non-negative gcd of m and n; also sets a and b so that gcd = a*m+b*n. If m and n are machine integers then a and b must be of type (signed) long. If m and n are of
type BigInt then a and b must also be of type BigInt. The cofactors a and b satisfy abs(a) <= abs(n)/2g and abs(b) <= abs(m)/2g where g is the gcd (inequalities are strict if possible). Error if
• InvMod(r,m) computes the least positive inverse of r modulo m; throws error (ERR::DivByZero) if the inverse does not exist. Throws error (ERR::BadModulus) if m < 2 (or too big for long). Result
is of type long if m is a machine integer; otherwise result is of type BigInt.
• InvMod(r,m, RtnZeroOnError) same as InvMod(r,m) except that it returns 0 if the inverse does not exist; RtnZeroOnError comes from an enum.
• InvModNoArgCheck(r,m) computes the least positive inverse of r modulo m; ASSUMES 0 <= r < m and 2 <= m <= MaxLong; result is a long. Throws error ERR::DivByZero if gcd(r,m) is not 1.
• lcm(m,n) computes the non-negative lcm of m and n. If both args are machine integers, the result is of type long; otherwise result is of type BigInt. Gives error ERR::ArgTooBig if the lcm of two
machine integers is too large to fit into a long.
There is a class called CoprimeFactorBasis_BigInt for computing a coprime factor basis of a set of integers:
• CoprimeFactorBasis_BigInt() default ctor; base is initially empty.
• CFB.myAddInfo(n) use also the integer n when computing the factor base.
• CFB.myAddInfo(v) use also the elements of std::vector<long> v or std::vector<BigInt> v when computing the factor base.
• FactorBase(CFB) returns the factor base obtained so far (as vector<BigInt>).
Prime generation and tests
These functions are in NumTheory-prime. The functions NextPrime, PrevPrime, RandomNBitPrime and RandomSmallPrime each produce a result of type SmallPrime (essentially a long which is known to be
• eratosthenes(n) build vector<bool> sieve of Eratosthenes up to n; entry k corresponds to integer 2*k+1; max valid index is n/2
• EratosthenesRange(lo, hi) build vector<bool> sieve of Eratosthenes from lo up to hi; if lo is odd, it is replaced by lo+1; similarly for hi. In returned vector entry k corresponds to integer
lo+2*k; max valid index is (hi-lo)/2
• IsPrime(n) tests the positive number n for primality (may be very slow for larger numbers). Gives error if n <= 0.
• IsProbPrime(n) tests the positive number n for primality (fairly fast for large numbers, but in very rare cases may falsely declare a number to be prime). Gives error if n <= 0.
• IsProbPrime(n,iters) tests the positive number n for primality; performs iters iterations of the Miller-Rabin test (default value is 25). Gives error if n <= 0.
• NextPrime(n) and PrevPrime(n) compute next or previous positive prime (fitting into a machine long). NextPrime returns 0 if no next "small" prime exists; PrevPrime throws OutOfRange if arg is
less than 3. Both throw BadArg if n < 0.
• RandomSmallPrime(N) -- generate a (uniform) random prime from 5 up to N; error if N < 5 or N >= 2^31. Result is of type SmallPrime (essentially a long).
• RandomNBitPrime(N) -- generate a (uniform) random prime in range 2^(N-1) to 2^N; error if N < 10 or N >= 31. Result is of type SmallPrime (essentially a long).
• NextProbPrime(N) and PrevProbPrime(N) compute next or previous positive probable prime (uses IsProbPrime). PrevProbPrime throws OutOfRange error if 0 <= N < 3. Both throw BadArg error if N < 0.
• NextProbPrime(N,iters) and PrevProbPrime(N,iters) compute next or previous positive probable prime (uses IsProbPrime with second arg iters). PrevProbPrime throws OutOfRange error if 0 <= N < 3.
Both throw BadArg error if N < 0.
There are also iterators for generating primes (or almost primes) in increasing order:
• PrimeSeq() the sequence of primes starting with 2.
• PrimeSeq1ModN(n) the sequence of primes 1 modulo n
• PrimeSeqForCRT() a sequence of primes starting with some "large" value, and going upwards.
• FastFinitePrimeSeq() a sequence containing all primes up to some limit (much faster than PrimeSeq); limit is given by mem fn myLastPrime.
• FastMostlyPrimeSeq() a sequence containing all primes and a few composites (much faster than PrimeSeq).
• NoSmallFactorSeq() a sequence of positive integers which have no small factors.
If pseq is one of these iterator objects then
• *pseq gives the current prime in the sequence (as a value of type SmallPrime, or of type long for FastMostlyPrimeSeq and NoSmallFactorSeq)
• ++pseq advances 1 step along the sequence
• IsEnded(pseq) returns true if the end of the sequence has been reached
• CurrPrime(pseq) same as *pseq (only for PrimeSeq and PrimeSeqForCRT)
• NextPrime(pseq) advances 1 step along the sequence, and returns the new "current prime" (only for PrimeSeq and PrimeSeqForCRT)
• pseq.JumpTo(n) advance to n or beyond (will rewind if n is smaller than current value)
• factor(n) finds the complete factorization of n (WARNING may be very slow for large numbers); NB implementation incomplete
• factor_TrialDiv(n,limit) finds small prime factors of n (up to & including the specified limit); result is a factorization object. Gives error if limit is not positive or too large to fit into a
long. See also MultiplicityOf2 in BigIntOps.
• factor_PollardRho(n,niters) attempt to find a (single) factor of n (not nec. prime) using at most niters iterations; returns "empty" factorization if no factor was found; factor found may not be
• AllFactors(n) a vector<long> containing all positive factors of n in increasing; error if n <= 0
• SumOfFactors(n,k) compute sum of k-th powers of positive factors of n
• SmallestNonDivisor(n) finds smallest (positive prime) non-divisor of n; if n=0 throws ERR::NotNonZero.
• IsSqFree(n) returns true if n is square-free, otherwise false; for BigInt result is a bool3
• FactorMultiplicity(b,n) find largest k such that power(b,k) divides n (error if b < 2 or n is zero)
• CoprimeFactor(N,b) effectively N/gcd(N,power(b,INFINITY)); for the "odd part" just compute CoprimeFactor(N,2)
Pollard Rho Sequence
• PollardRhoSeq(N, start, incr) create a sequence starting from start and with increment incr
• PRS.myAdvance(k) advance the sequence by k steps
• GetFactor(PRS) returns a factor of N (may be 1 or N)
• GetNumIters(PRS) returns number of steps/iters performed
Other Functions on Integers
• EulerTotient(n) computes Euler's totient function of the positive number n (i.e. the number of integers up to n which are coprime to n, or the degree of the n-th cyclotomic polynomial). Gives
error if n <= 0.
• InvTotientBound(n) -- returns a BigInt representing an upper bound for the inverse EulerTotient of n -- using OEIS sequence A355667.
• InvTotientBound_ulong(n) -- returns an unsigned long representing an upper bound for the inverse EulerTotient of n
• InvTotientBoundUpto(n) -- returns a BigInt representing an upper bound for the inverse EulerTotient of all k <= n -- related to OEIS sequence A355667.
• InvTotient(n) -- returns a vector<long> containing all possible preimages of n.
• InvTotient(n, InvTotientMode::SqFreePreimages) -- returns a vector<long> containing all possible square-free preimages of n.
• MoebiusFn(n) computes the Moebius function value of the positive number n (i.e. the sum of the primitive n-th roots of unity). Gives error if n <= 0.
• PrimitiveRoot(p) computes the least positive primitive root for the positive prime p. Gives error if p is not a positive prime. WARNING May be very slow for large p (because it must factorize
• KroneckerSymbol(res,mod) (test if res is a quadratic residue) computes the Kronecker symbol, generalization of Jacobi symbol, generalization of Legendre symbol
• MultiplicativeOrderMod(res,mod) computes multiplicative order of res modulo mod. Throws error ERR::BadArg if mod < 2 or gcd(res,mod) is not 1.
• PowerMod(base,exp,modulus) computes base to the power exp modulo modulus; result is least non-negative residue. If modulus is a machine integer then the result is of type long (or error if it
does not fit), otherwise the result is of type BigInt. Gives error if modulus <= 1. Gives ERR::DivByZero if exp is negative and base cannot be inverted. If base and exp are both zero, it produces
• BinomialRepr(N,r) produces the repr of N as a sum of binomial coeffs with "denoms" r, r-1, r-2, ...
Functions on Rationals
• SimplestBigRatBetween(A,B) computes the simplest rational between A and B
• SimplestBinaryRatBetween(A,B) computes the simplest binary rational between A and B; result is a rational of form N*2^k where the integer N is minimal.
Continued Fractions
Several of these functions give errors if they are handed unsuitable values: unless otherwise indicated below the error is of type ERR::BadArg.
Recall that any real number has an expansion as a continued fraction (e.g. see Hardy & Wright for definition and many properties). This expansion is finite for any rational number. We adopt the
following conventions which guarantee that the expansion is unique:
• the last partial quotient is greater than 1 (except for the expansion of integers <= 1)
• only the very first partial quotient may be non-positive.
For example, with these conventions the expansion of -7/3 is (-3, 1, 2).
The main functions available are:
• ContFracIter(q) constructs a new continued fraction iterator object
• IsEnded(CFIter) true iff the iterator has moved past the last partial quotient
• IsFinal(CFIter) true iff the iterator is at the last partial quotient
• quot(CFIter) gives the current partial quotient as a BigInt (or throws ERR::IterEnded)
• *CFIter gives the current partial quotient as a BigInt (or throws ERR::IterEnded)
• ++CFIter moves to next partial quotient (or throws ERR::IterEnded)
• ContFracApproximant() for constructing a rational from its continued fraction quotients
• CFA.myAppendQuot(q) appends the quotient q to the continued fraction
• CFA.myRational() returns the rational associated to the continued fraction
• CFApproximantsIter(q) constructs a new continued fraction approximant iterator
• IsEnded(CFAIter) true iff the iterator has moved past the last "partial quotient"
• *CFAIter gives the current continued fraction approximant as a BigRat (or throws ERR::IterEnded)
• ++CFAIter moves to next approximant (or throws ERR::IterEnded)
• CFApprox(q,eps) gives the simplest cont. frac. approximant to q with relative error at most eps
Chinese Remaindering -- Integer Reconstruction
CoCoALib offers the class CRTMill for reconstructing an integer from several residue-modulus pairs via Chinese Remaindering. At the moment the moduli from distinct pairs must be coprime.
The operations available are:
• CRTMill() ctor; initially the residue is 0 and the modulus is 1
• CRT.myAddInfo(res,mod) give a new residue-modulus pair to the CRTMill (error if mod is not coprime to all previous moduli)
• CRT.myAddInfo(res,mod,CRTMill::CoprimeModulus) give a new residue-modulus pair to the CRTMill asserting that mod is coprime to all previous moduli -- CRTMill::CoprimeModulus is a constant
• CombinedResidue(CRT) the combined residue with absolute value less than or equal to CombinedModulus(CRT)/2
• CombinedModulus(CRT) the product of the moduli of all pairs given to the mill
Rational Reconstruction
CoCoALib offers two heuristic methods for reconstructing rationals from residue-modulus pairs; they have the same user interface but internally one algorithm is based on continued fractions while the
other uses lattice reduction. The methods are heuristic, so may (rarely) produce an incorrect result.
NOTE the heuristic requires the (combined) modulus to be a certain amount larger than strictly necessary to reconstruct the correct answer (assuming perfect bounds are known). In practice, this means
that the methods always fail if the combined modulus is too small.
The constructors available are:
• RatReconstructByContFrac() ctor for continued fraction method mill log-epsilon equal to 20
• RatReconstructByContFrac(LogEps) ctor for continued fraction method mill with given log-epsilon (must be at least 3)
• RatReconstructByLattice(SafetyFactor) ctor for lattice method mill with given SafetyFactor (0 --> use default)
The operations available are:
• reconstructor.myAddInfo(res,mod) give a new residue-modulus pair to the reconstructor
• IsConvincing(reconstructor) gives true iff the mill can produce a convincing result
• ReconstructedRat(reconstructor) gives the reconstructed rational (or an error if IsConvincing is not true).
• BadMFactor(reconstructor) gives the "bad factor" of the combined modulus.
There is also a function for deterministic rational reconstruction which requires certain bounds to be given in input. It uses the continued fraction method.
• RatReconstructWithBounds(e,P,Q,res,mod) where e is upper bound for number of "bad" moduli, P and Q are upper bounds for numerator and denominator of the rational to be reconstructed, and (res
[i],mod[i]) is a residue-modulus pair with distinct moduli being coprime.
Maintainer Documentation
• Correctness of ExtendedEuclideanAlg is not immediately clear, because the cofactor variables could conceivably overflow -- in fact this cannot happen (at least on a binary computer): for a proof
see Shoup's book A Computational Introduction to Number Theory and Algebra, in particular Theorem 4.3 and the comment immediately following it. There is just one line where a harmless "overflow"
could occur -- it is commented in the code.
• I have decided to make ExtGcd give an error if the inputs are both zero because this is an exceptional case, and so should be handled specially. I note that mpz_exgcd accepts this case, and
returns two zero cofactors; so if we decide to accept this case, we should do the same -- this all fits in well with the (reasonable/good) principle that "zero inputs have zero cofactors". -
Several functions are more complicated than you might expect because I wanted them to be correct for all possible machine integer inputs (e.g. including the most negative long value).
• In some cases the function which does all the work is implemented as a file local function operating on unsigned long values: the function should normally be used only via the "dispatch"
functions whose args are of type MachineInt or BigInt.
• The fns for primes (testing and generating) are in the file NumTheory-prime.
• The impl of eratosthenes is fairly straightforward given that I chose to represent only odd numbers in the table: the k-th entry corresponds to the integer 2*k+1. Overflow cannot occur: recall
that the table size is at most half the biggest long. I'm hoping that C++11 will avoid the cost of copying the result upon returning. Anyway, I think the code is a decent compromise between
readability, speed and space efficiency. The impl of EratosthenesRange is similar but the table covers just the given range (only odd numbers are represented; index 0 corresponds to smallest odd
integer greater than or equal to the start of the range).
• The "prime sequence" classes are a bit messier than I'd like. It was delicate getting correct the switch-over from one technique to the other (in those classes where 2 techniques were used). The
limit of 23 for NoSmallFactorSeq is somewhat arbitrary. Not sure the code is 32-bit safe.
• The continued fraction functions are all pretty simple. The only tricky part is that the "end" of the ContFracIter is represented by both myFrac and myQuot being zero. This means that a newly
created iterator for zero is already ended.
• CFApproximantsIter delegates most of the work to ContFracIter.
Bugs, Shortcomings, etc.
• Several functions return long values when perhaps unsigned long would possibly be better choice (since it offers a greater range, and in the case of gcd it would permit the fn to return a result
always, rather than report "overflow"). The choice of return type was dictated by the coding conventions, which were in turn dictated by the risks of nasty surprises to unwary users unfamiliar
with the foibles of unsigned values in C++.
• NextPrime has dodgy semantics: what happens when the end of the iterator is reached? In fact, all the non-finite "prime seq" iterators do not handle end-of-iterator properly!
• Should there also be procedural forms of functions which return BigInt values? (e.g. gcd, lcm, InvMod, PowerMod, and so on). (2016-06-27 this will probably become irrelevant when using "move"
semantics in C++11).
• Certain implementations of PowerMod should be improved (e.g. to use PowerModSmallModulus whenever possible). Is behaviour for 0^0 correct?
• KroneckerSymbol I have chosen to make available just KroneckerSymbol rather than the more widely-known LegendreSymbol because GMP makes Kronecker available, and it is always defined; whereas
LegendreSymbol would have to check that its 2nd arg is a prime (which would dominate the cost of the call)
• LucasTest should produce a certificate, and be made publicly accessible.
• How should the cont frac iterators be printed out???
• ContFracIter could be rather more efficient for rationals having very large numerator and denominator. One way would be to compute with num and den divided by the same large factor (probably a
power of 2), and taking care to monitor how accurate these "scaled" num and den are. I'll wait until there is a real need before implementing (as I expect it will turn out a bit messy).
• CFApproximantsIter::operator++() should be made more efficient.
Main changes
• Feb (v0.99720):
□ SmoothFactor has been renamed factor_TrialDiv
□ added factor_PollardRho - | {"url":"http://cocoa.altervista.org/cocoalib/doc/html/NumTheory.html","timestamp":"2024-11-12T01:51:13Z","content_type":"text/html","content_length":"27407","record_id":"<urn:uuid:db030f4c-29bb-4498-8539-781c6c632b4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00624.warc.gz"} |
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Polyhedron -- from Wolfram MathWorld
The word polyhedron has slightly different meanings in geometry and algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons,
usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the
geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").
The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and
their higher dimensional analogs by "gluing them together" along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space of a simplicial complex (with the
additional constraint sometimes imposed that the complex be finite; Munkres 1993, p. 9). In the usual definition, a polyhedron can be viewed as an intersection of half-spaces, while a polytope is a
bounded polyhedron.
In the Wolfram Language, Polyhedron[] objects represent filled regions founded by closed surfaces with polygonal faces.
A convex polyhedron can be formally defined as the set of solutions to a system of linear inequalities
where matrix and vector. Although usage varies, most authors additionally require that a solution be bounded for it to define a convex polyhedron. An example of a convex polyhedron is illustrated
The following table lists the name given to a polyhedron having
4 tetrahedron
5 pentahedron
6 hexahedron
7 heptahedron
8 octahedron
9 enneahedron
10 decahedron
11 undecahedron
12 dodecahedron
14 tetradecahedron
20 icosahedron
24 icositetrahedron
30 triacontahedron
32 icosidodecahedron
60 hexecontahedron
90 enneacontahedron
A polyhedron is said to be regular if its faces and vertex figures are regular (not necessarily convex) polygons (Coxeter 1973, p. 16). Using this definition, there are a total of nine regular
polyhedra, five being the convex Platonic solids and four being the concave (stellated) Kepler-Poinsot polyhedra. However, the term "regular polyhedra" is sometimes used to refer exclusively to the
Platonic solids (Cromwell 1997, p. 53). The dual polyhedra of the Platonic solids are not new polyhedra, but are themselves Platonic solids.
A convex polyhedron is called semiregular if its faces have a similar arrangement of nonintersecting regular planar convex polygons of two or more different types about each polyhedron vertex (Holden
1991, p. 41). These solids are more commonly called the Archimedean solids, and there are 13 of them. The dual polyhedra of the Archimedean solids are 13 new (and beautiful) solids, sometimes called
the Catalan solids.
A quasiregular polyhedron is the solid region interior to two dual regular polyhedra (Coxeter 1973, pp. 17-20). There are only two convex quasiregular polyhedra: the cuboctahedron and
icosidodecahedron. There are also infinite families of prisms and antiprisms.
There exist exactly 92 convex polyhedra with regular polygonal faces (and not necessarily equivalent vertices). They are known as the Johnson solids. Polyhedra with identical polyhedron vertices
related by a symmetry operation are known as uniform polyhedra. There are 75 such polyhedra in which only two faces may meet at an polyhedron edge, and 76 in which any even number of faces may meet.
Of these, 37 were discovered by Badoureau in 1881 and 12 by Coxeter and Miller ca. 1930.
Polyhedra can be superposed on each other (with the sides allowed to pass through each other) to yield additional polyhedron compounds. Those made from regular polyhedra have symmetries which are
especially aesthetically pleasing. The graphs corresponding to polyhedra skeletons are called Schlegel graphs.
Behnke et al. (1974) have determined the symmetry groups of all polyhedra symmetric with respect to their polyhedron vertices. | {"url":"https://mathworld.wolfram.com/Polyhedron.html","timestamp":"2024-11-04T04:10:00Z","content_type":"text/html","content_length":"72309","record_id":"<urn:uuid:3c779431-0080-4e17-ad69-5a239a7401f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00075.warc.gz"} |
Is this real maths or somebody winding me up?
If u = <3,-2> and v = <4,5> then u · v = (3)(4) + (-2)(5) = 12 - 10 = 2. (b) If u = 2i + j and v = 5i - 6j then u · v = (2)(5) + (1)(-6) = 10 - 6=4. Proof: We prove only the last property. Let u =
<a, b> . Then u · u = <a, b>·<a, b> = a · a + b · b = a2 + b2 = (/a2 + b2)2 except when quantified by any 7point artimace exemplified by stasus elements found in field mortification parameters.
Let u, v and w be three vectors in R3 and let λ be a scalar. (1) v × w = − w × v. (2) u × ( v + w) = u × v + u × w. (3) ( u + v) × w = u × w + v × w. (4) λ( v × w)=(λ v) × w = v × (λ w). We then end
up in obvious paradoxity instigated by v x y intigers elevated beyond tartus secondary aspects.
Can anyone ''read'' that?
... 7point artimace exemplified by stasus elements found in field mortification parameters ...
... obvious paradoxity instigated by v x y intigers elevated beyond tartus secondary aspects ...
These are not things.
Someone's having someone on.
BTW, simply Goggling a few of those terms would clear up the mystery.
"Artimace" is a righteous name, gotta say. Artimace Godby just had to be worth knowing.
Although any kid saddled with it is likely to find the parameters of their field mortification stretched a bit, in school.
call me arf
Valued Senior Member
It looks like badly translated Russian, or perhaps Greek. Maybe Latvian.
That said, some of the equations look ok except for some of the notation and the lack of superscripts, i.e. exponents.
That said, some of the equations look ok except for some of the notation and the lack of superscripts, i.e. exponents.
Yea, the ideas behind convincing gibberish is to make it look plausibly real.
It looks like badly translated Russian, or perhaps Greek. Maybe Latvian.
Hadn't thought of that.
But I assume any auto-translator would leave words alone it didn't understand, so we still shouldn't be seeing nonsense words.
And if it were a human translator, we're back to the same problem.
Actually, I guess that's the same thing. I see no way for gibberish words to get into a translated text - automatically
manually - unless it's deliberate.
New year. PRESENT is 72 years oldl
Valued Senior Member
If u = <3,-2> and v = <4,5> then u · v = (3)(4) + (-2)(5) = 12 - 10 = 2. (b) If u = 2i + j and v = 5i - 6j then u · v = (2)(5) + (1)(-6) = 10 - 6=4. Proof: We prove only the last property. Let u
= <a, b> . Then u · u = <a, b>·<a, b> = a · a + b · b = a2 + b2 = (/a2 + b2)2 except when quantified by any 7point artimace exemplified by stasus elements found in field mortification parameters.
Let u, v and w be three vectors in R3 and let λ be a scalar. (1) v × w = − w × v. (2) u × ( v + w) = u × v + u × w. (3) ( u + v) × w = u × w + v × w. (4) λ( v × w)=(λ v) × w = v × (λ w). We then
end up in obvious paradoxity instigated by v x y intigers elevated beyond tartus secondary aspects.
Can anyone ''read'' that?
It's the upgrade Time Travel expodent oscillation modified frequency generator calculations formula giving new extra power to the
(TARDIS Mk 2) while being more eco friendly
Release date to be announced
Time And Relative Dimension Under Space
which is the next generation of the
Time And Relative Dimension In Space
call me arf
Valued Senior Member
Let's untranslate some of it:
If u = <3,-2> and v = <4,5> then u · v = (3)(4) + (-2)(5) = 12 - 10 = 2.
Looks like the dot product of two vectors, u and v. But, <u,v> (the inner product) is another way to write the dot product (usually restricted to 2 or 3 dimensional vectors). Hence it
be: If
= (2, -2) and
= (4,5) . . ., otherwise it looks ok.
If u = 2i + j and v = 5i - 6j then u · v = (2)(5) + (1)(-6) = 10 - 6=4.
This uses the i,j,k unit vector notation, looks pretty standard for 2 dimensions.
Let u = <a, b> . Then u · u = <a, b>·<a, b> = a · a + b · b
ok so far, but the rest goes off the rails more than a little.
Yep. The cross product is antisymmetric. There seems to be no problem with the rest of it, including the scalar multiplication. I have no idea what the "paradoxicity" is. Perhaps it means you
shouldn't take any without food or a parachute (or something).
Last edited:
Let's untranslate some of it:
Looks like the dot product of two vectors, u and v. But, <u,v> (the inner product) is another way to write the dot product (usually restricted to 2 or 3 dimensional vectors). Hence it should be:
If u = (2, -2) and v = (4,5) . . ., otherwise it looks ok.
This uses the i,j,k unit vector notation, looks pretty standard for 2 dimensions.
ok so far, but the rest goes off the rails more than a little.
Yep. The cross product is antisymmetric. There seems to be no problem with the rest of it, including the scalar multiplication. I have no idea what the "paradoxicity" is. Perhaps it means you
shouldn't take any without food or a parachute (or something).
HUh, all the other people said is gibberish?
I am not surprised it is difficult to learn on the net when some people are teaching false information.
Any mathematicians on who fancy a challenge?
Can any mathematician explain a 0*0 matrice that is in continuous expansion from 0 to infinitely?
matrice Au []
matrice Bu []
I want to explain that both these matrices expand on manifestation and vanish . (gone in a puff a smoke )
A and B are just tags, u is internal energy
Come on guys somebody must know how to explain
Δ0=Δ1u/t=1u/k where k is space
I have just learnt it is called an empty matrice, I simply want to expand this matrice at the speed of c proportional to the inverse then it is back to an empty matrice.
So far I have my abstraction as
Δ0=(+1u/t)/k how do I put at the speed of c to the end of this?
The visual looks like this
or simply 010
or simply 0→1u→0/t
Last edited:
so can I express?
HUh, all the other people said is gibberish?
The rest of us are examining the
Arfa brane is taking a stab at the
to see if it is as ... gibberish
Let's untranslate some of it:
Looks like the dot product of two vectors, u and v. But, <u,v> (the inner product) is another way to write the dot product (usually restricted to 2 or 3 dimensional vectors). Hence it should be:
If u = (2, -2) and v = (4,5) . . ., otherwise it looks ok.
This uses the i,j,k unit vector notation, looks pretty standard for 2 dimensions.
ok so far, but the rest goes off the rails more than a little.
Yep. The cross product is antisymmetric. There seems to be no problem with the rest of it, including the scalar multiplication. I have no idea what the "paradoxicity" is. Perhaps it means you
shouldn't take any without food or a parachute (or something).
Let u = <a> and v = <b>. Then u · v = <a>·<b> = a · b?
Would the above be meaningful in anyway? I am trying to learn this .
Let u = <a> and v = <b>. Then u · v = <a>·<b> = a · b?
Would the above be meaningful in anyway? I am trying to learn this .
I'm not sure what the < and > are for (matrix?), but assuming it doesn't destroy commutativity, then yes:
If u=a, and v=b then
u·v = a·b
I'm not sure what the < and > are for (matrix?), but assuming it doesn't destroy commutativity, then yes:
If u=a, and v=b then
u·v = a·b
Thank you, much appreciated, I am trying to learn and practice this subject. I am not sure what the <> meant myself , I presumed it was to represent the force direction and showed a and b was in a
state of expansion. Now I am at a loss for what it meant if you do not know yourself.
How would I explain that (a) manifests then inversely proportionally disperses at the speed of light c?
Coordinates a=0,0,0
I want to try and explain 0 point energy.
How would I explain that (a) manifests then inversely proportionally disperses at the speed of light c?
I am not sure what that means. Particularly, use of the word 'disperse'.
If c is inversely proportional to a, then its simply c ~1/a.
I am not sure what that means. Particularly, use of the word 'disperse'.
If c is inversely proportional to a, then its simply c ~1/a.
How would I describe
example : Δu=(+1)-(+1) at c ~1/a
Thank you, that is the link I am already using, I must of missed linear span. I think it means what I meant it to mean. | {"url":"https://www.sciforums.com/threads/is-this-real-maths-or-somebody-winding-me-up.160524/","timestamp":"2024-11-04T01:48:31Z","content_type":"text/html","content_length":"148761","record_id":"<urn:uuid:3ac2b9d9-9b04-4cbe-8d79-436935a27e51>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00654.warc.gz"} |
Handling Non-Normally Distributed Data by Removing Outliers - KANDA DATA
Handling Non-Normally Distributed Data by Removing Outliers
The topic I’m writing about today is prompted by questions on how to handle data that is not normally distributed. We know that in quantitative analysis, several statistical tests require that the
data be normally distributed. This is an interesting topic that we will delve deeper into in this article.
In quantitative data analysis, one of the commonly used assumptions is that the data is normally distributed. A normal distribution, when understood more deeply, can be observed in data forming a
symmetrical bell curve. On this bell curve, most values will be centered around the mean.
However, in reality, the results of normality tests do not always match our expectations. In some cases, we may encounter issues where the data is not normally distributed.
Based on experience in conducting a series of analyses and data processing, one of the causes is the presence of outliers in our data. The presence of outliers or data points that significantly
differ from the majority of the data can impact the results of normality tests.
These outliers can cause biased data interpretation, even in descriptive statistical analysis. Therefore, identifying and addressing outliers is an essential step in data analysis.
This article will discuss how to handle non-normal data by removing outliers. I will provide a case study example to help readers better understand the concept.
Assumption of Normally Distributed Data
As mentioned earlier, several quantitative analyses require normally distributed data. For example, researchers using t-tests, ANOVA, and linear regression require normally distributed data to ensure
consistent and unbiased estimation results.
In normally distributed data, the data spread forms a symmetrical bell curve around the mean. Most normally distributed data lie within a standard deviation close to the mean.
If the data is not normally distributed, the analysis results can become invalid and biased. Therefore, it is important for us to evaluate whether the data meets the normal distribution assumption
according to the statistical test we choose.
Solutions for Non-Normally Distributed Data
At the beginning, I mentioned that outliers are data points that significantly differ from most data. Thus, removing outliers is an effective method to make data closer to a normal distribution.
The first step is to identify and detect outliers. The easiest way to identify outliers is through descriptive statistical tests, including mean, median, and standard deviation. We can mark data
points with large standard deviations or those significantly different from the mean as potential outliers.
We can also visualize data using boxplots and observe data points outside the whiskers. Additionally, we can calculate the Z-score to identify outliers in our dataset.
Outliers can be removed or replaced with new data following scientific principles. Outliers may arise due to incorrect sampling techniques or errors during questionnaire completion. Therefore,
validation and verification by researchers are crucial steps.
In cross-sectional data, we can remove outliers or replace them with new respondent data. If outliers are due to errors or are irrelevant to the analysis, they can be removed from the dataset.
However, it is important to justify or provide reasons for the removal of these outliers.
Case Study and Interpretation
To provide a clearer picture, we will conduct a case study with cross-sectional data consisting of 30 observations. Suppose we have a dataset containing weekly sales values from 30 stores. The data
can be seen in the table below:
Based on the above data, we know that in the last observation, there is a value significantly higher than most weekly sales data from other stores. This data point is likely an outlier.
Normality tests will be conducted using the Kolmogorov-Smirnov and Shapiro-Wilk tests with the results shown in the image below:
Based on the analysis results, it is known that according to both tests, Kolmogorov-Smirnov and Shapiro-Wilk, the p-value < 0.05. This indicates that the null hypothesis is rejected (accepting the
alternative hypothesis) which means the data is not normally distributed.
In the output below, we also know that the value of 1000 is an extreme outlier. Therefore, this outlier is suspected to cause the data to be not normally distributed.
Now let’s try removing this outlier. We will see how removing the outlier impacts data normality.
After removing the outlier, it is expected that the mean and median values will be closer and show a more normal data distribution. The results of the normality test using Kolmogorov-Smirnov and
Shapiro-Wilk tests using SPSS are shown in the image below:
Based on the output above, after removing the outlier, the Kolmogorov-Smirnov test p-value is 0.200 and the Shapiro-Wilk test p-value is 0.374. Based on these results, both p-values are > 0.05,
indicating that the null hypothesis is accepted, meaning the data is normally distributed.
Additionally, the outlier test shows no extreme values. By removing the outlier, we achieve a data distribution closer to normal. This enables more valid statistical analysis and more reliable
results. It is important to note that removing outliers should be done carefully and based on proper evaluation.
Handling non-normally distributed data is a crucial step in statistical analysis. Outliers often cause non-normal data distribution. By identifying and removing outliers, we can make the data closer
to a normal distribution.
That concludes the article I can write for this occasion. I hope it is useful and adds knowledge value for readers in need. See you in the next educational article from Kanda Data!
Leave a Comment
You must be logged in to post a comment. | {"url":"https://kandadata.com/handling-non-normally-distributed-data-by-removing-outliers/","timestamp":"2024-11-11T22:53:31Z","content_type":"text/html","content_length":"192408","record_id":"<urn:uuid:2ad8c183-e5e5-4ac2-a5ac-20edc494dd65>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00119.warc.gz"} |
\], AD = 2 cm, DB = 3cm and AE = 1.6 cm. Then EC is equal to
Hint: In this question we will simply use the basic proportionality theorem of triangles to find out EC.
As there is a triangle given to us and \[DE\parallel BC\]
By using the theorem we will find out ratios to sides and thus we will find EC.
Complete step-by-step answer:
In triangle ABC,
D is point on AB and E is point on AC and \[DE\parallel BC\]
Basic Proportionality theorem states that if a line is drawn parallel to the one side of the to intersect the other two sides in distinct points, the other two sides are divided in the same ratio,
which implies \[\dfrac{{AD}}{{DB}} = \dfrac{{AE}}{{EC}}\]……………………… (1)
putting the values of AD, DB, AE, taking EC as EC
we have
$\Rightarrow$$\dfrac{2}{3}\, = \,\dfrac{{1.6}}{{EC}}$ …………………………….(2)
Solving this above ratio found in (2)
$\Rightarrow$$EC$ = $\dfrac{{1.6\, \times \,3}}{2}$ cm
$\Rightarrow$$EC$ = $(0.8 \times 3 )$ cm
$\Rightarrow$$EC = 2.4 cm$
Therefore length of EC = 2.4 cm
Hence option (c) is the correct option.
Note: In case you did not remember basic proportionality theorem you can go for solving the question just by making ratios with respect to sides. Another method is using algebraic operations and
using algebraic methods lots of complication will be there and it can also cause lots of error so it is always better either keep theorem in your mind or use the trick .
Trick whatsoever points are given on sides, go for creating ratio for them and then solve for unknown values. As D was pointed to AB , therefore the ratio came out to be AD/DB. | {"url":"https://www.vedantu.com/question-answer/in-the-given-figure-deparallel-bc-ad-2-cm-db-3cm-class-9-maths-cbse-5f894184177aeb6799ee1211","timestamp":"2024-11-05T10:13:57Z","content_type":"text/html","content_length":"161555","record_id":"<urn:uuid:5cd7956c-a6e2-4c4b-82da-83c837be5e18>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00380.warc.gz"} |
1. Página Inicial
2. Inscrições Científicas
3. Trabalhos
Dados do Trabalho
Evaluation of dosimetric parameters in an ophthalmic brachytherapy source, obtained in water and perspex, using Monte Carlo simulation
Dosimetry in brachytherapy is important to ensure equality between the planned dose and that administered to the patient. However, experimental determination is complex due to the high dose gradient
in regions close to the source. In this sense, the American Association of Physicists in Medicine (AAPM) proposed a formalism for calculating the dose using dosimetric parameters, recommending the
use of Monte Carlo simulation to calculate these parameters.
In this work, dosimetric parameters were determined: anisotropy function and radial dose function, including relative dose profiles, for an ophthalmic brachytherapy source, in two materials, water
and perspex, using Monte Carlo simulation with PENELOPE package.
The brachytherapy source, model IR06-103Pd, commonly used in ophthalmic treatments, was modeled in the center of a cubic phantom with 30 cm of side, filled with water and perspex, in different
simulations. The cut-off energy of photons, electrons and positrons was 5 keV, using the 8 main photons emission lines; the number of primary particles remained constant at 5x10⁸, and the history
condensation parameters were 0.3 and voxel size was 0.3 cm³ in all simulations.
Comparing the relative doses, the greatest uncertainty was found to be less than 4% for regions with a high dose gradient, and the greatest difference found was approximately 5% greater for perspex,
at 6 cm from the source. The anisotropy function was calculated at the reference distance in brachytherapy (1 cm) and compared between the analyzed materials.. The highest relative uncertainty found
in this dosimetric parameter was approximately 2.2%, when the phantom was filled with water, below the maximum relative uncertainty value recommended by the AAPM. When the results were compared,
differences up to 0.03 (absolute value) in the anisotropy function were found from 0º to 40º and from 140º to 180º. And for the radial dose function, the maximum uncertainty found was 0.97%, for the
reference distance in brachytherapy, noting that perspex presented a difference 10% greater than water, 1.5 cm from the source.
The results of this work show that the PENELOPE package as a promising tool in the dosimetric calculation of brachytherapy and the importance of choosing the material on which the dosimetric
parameter is calculated, in order to minimize errors in the dosimetric calculations of this type of source.
Dosimetry, brachytherapy, PENELOPE | {"url":"https://ccm.iweventos.com.br/evento/sbrt2021/trabalhosaprovados/naintegra/101574","timestamp":"2024-11-06T12:38:43Z","content_type":"text/html","content_length":"34572","record_id":"<urn:uuid:a168c02d-5116-4e09-836d-def2823c9525>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00476.warc.gz"} |
Printable linear equation test
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Author Message
flootect Posted: Thursday 18th of Sep 10:02
Hey People out there I really hope some math expert reads this. I am stuck on this homework that I have to take in the next week and I can’t seem to find a way to complete it. You see, my
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to some private tutor to help me solve it. If one of you friends can show me how to do it, I will highly grateful.
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AllejHat Posted: Friday 19th of Sep 09:08
You don’t need to ask anyone to solve any sample questions for you; as a matter of fact all you need is Algebrator. I’ve tried many such algebra simulation software but Algebrator is a
lot better than most of them. It’ll solve any question that you have and it’ll even explain each and every step involved in reaching that answer. You can work out as many examples as you
would like to, and unlike us human beings, it would never say, Oh! I’ve had enough for the day! ;) I used to have some problems in solving questions on perpendicular lines and exponential
equations, but this software really helped me get over those.
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3Di Posted: Saturday 20th of Sep 07:11
I am a student turned tutor; I give classes to high school children. Along with the traditional mode of explanation, I use Algebrator to solve examples practically in front of the
45°26' N,
09°10' E
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ToanVusion Posted: Sunday 21st of Sep 16:46
Hello again. Thanks a lot for the useful advice. I normally never trust math tools ; however, this piece of software seems worth trying. Can I get a link to it?
From: A
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Jot Posted: Monday 22nd of Sep 10:14
Hello Guys , Based on your reviews , I bought the Algebrator to get myself educated with the fundamental theory of Pre Algebra. The explanations on gcf and difference of cubes were not
only graspable but made the whole topic pretty exciting . Thanks a bunch for all of you who directed me to check Algebrator!
From: Ubik
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Mov Posted: Tuesday 23rd of Sep 09:25
Algebrator is a very remarkable product and is surely worth a try. You will also find many exciting stuff there. I use it as reference software for my math problems and can say that it
has made learning math more fun .
Back to top | {"url":"https://softmath.com/algebra-software-5/printable-linear-equation-test.html","timestamp":"2024-11-10T12:18:42Z","content_type":"text/html","content_length":"43081","record_id":"<urn:uuid:57cfebd9-54c7-423d-b94b-0f9142115e88>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00279.warc.gz"} |
forp – formula prover
forp [ -m ] [ file ]
Forp is a tool for proving formulae involving finite-precision arithmetic. Given a formula it will attempt to find a counterexample; if it can’t find one the formula has been proven correct.
Forp is invoked on an input file with the syntax as defined below. If no input file is provided, standard input is used instead. The -m flag instructs forp to produce a table of all counterexamples
rather than report just one. Note that counterexamples may report bits as ?, meaning that either value will lead to a counterexample.
The input file consists of statements terminated by semicolons and comments using C syntax (using // or /* */ syntax). Valid statements are
Variable definitions, roughly: type var ;
Expressions (including assignments): expr ;
Assertions: obviously expr ;
Assumptions: assume expr ;
Assertions are formulae to be proved. If multiple assertions are given, they are effectively "and"-ed together. Each input file must have at least one assertion to be valid. Assumptions are formulae
that are assumed, i.e. counterexamples that would violate assumptions are never considered. Exercise care with them, as contradictory assumptions will lead to any formula being true (the logician’s
principle of explosion).
Variables can be defined with C notation, but the only types supported are bit and 1D arrays of bit (corresponding to machine integers of the specified size). Signed integers are indicated with the
keyword signed. Like int in C, the bit keyword can be omitted in the presence of signed. For example,
bit a, b[4], c[8];
signed bit d[3];
signed e[16];
is a set of valid declarations.
Unlike a programming language, it is perfectly legitimate to use a variable before it is assigned value; this means the variable is an "input" variable. Forp tries to find assignments for all input
variables that render the assertions invalid.
Expressions can be formed just as in C, however when used in an expression, all variables are automatically promoted to an infinite size signed type. The valid operators are listed below, in
decreasing precedence. Note that logical operations treat all non-zero values as 1, whereas bitwise operators operate on all bits independently.
Array indexing. The syntax is var[a:b], with a denoting the MSB and b denoting the LSB. Omiting :b addresses a single bit. The result is always treated as unsigned.
!, ~, +, -
(Unary operators) Logical and bitwise "not", unary plus (no-op), arithmetic negation. Because of promotion, ~ and - operate beyond the width of variables.
*, /, %
Multiplication, division, modulo. Division and modulo add an assumption that the divisor is non-zero.
+, -
Addition, subtraction.
<<, >>
Left shift, arithmetic right shift. Because of promotion, this is effectively a logical right shift on unsigned variables.
<, <=, >, >=
Less than, less than or equal to, greater than, greather than or equal to.
==, !=
Equal to, not equal to.
Bitwise "and".
Bitwise "xor".
Bitwise "or".
Logical "and"
Logical "or".
<=>, =>
Logical equivalence and logical implication (equivalent to (a != 0) == (b != 0) and !a || b , respectively).
Ternary operator (a?b:c equals b if a is true and c otherwise).
One subtle point concerning assignments is that they forcibly override any previous values, i.e. expressions use the value of the latest assignments preceding them. Note that the values reported as
the counterexample are always the values given by the last assignment.
We know that, mathematically, a + b ≥ a if b ≥ 0 (which is always true for an unsigned number). We can ask forp to prove this using
bit a[32], b[32];
obviously a + b >= a;
Forp will report "Proved", since it cannot find a counterexample for which this is not true. In C, on the other hand, we know that this is not necessarily true. The reason is that, depending on the
types involved, results are truncated. We can emulate this by writing
bit a[32], b[32], c[32];
c = a + b;
obviously c >= a;
Given this, forp will now report it as incorrect by providing a counterexample, for example
a = 10000000000000000000000000000000
b = 10000000000000000000000000000000
c = 00000000000000000000000000000000
Can we use c < a to check for overflow? We can ask forp to confirm this using
bit a[32], b[32], c[32];
c = a + b;
obviously c < a <=> c != a+b;
Here the statement to be proved is "c is less than a if and only if c does not equal the mathematical sum a + b (i.e. overflow has occured)".
Any proof is only as good as the assumptions made, in particular care has to be taken with respect to truncation of intermediate results.
Array indices must be constants.
Left shifting can produce a huge number of intermediate bits. Forp will try to identify the minimum needed number but it may be a good idea to help it by assigning the result of a left shift to a
Forp first appeared in 9front in March, 2018. | {"url":"http://man2.aiju.de/1/forp","timestamp":"2024-11-08T21:16:00Z","content_type":"text/html","content_length":"14479","record_id":"<urn:uuid:4f642977-3ed5-45b4-9087-6f302046712e>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00013.warc.gz"} |
Is it possible to pay someone to complete my instrumentation assignment? | Pay Someone To Do Electrical Engineering Assignment
Is it possible to pay someone to complete my instrumentation assignment? If not, I’d appreciate it if someone really could remind me to do this. I still lack the ability to write any new way to view
and input/execute a function. With most read what he said having this ability, I’d already probably have enough. This isn’t all that hard to happen. Please let me know how I would approach it I’d
probably need an expert on how to work with the language. Assuming it is “mainstream” language we would definitely need additional manual coding assistance. I am capable of writing a bunch of code,
so there should be a lot of input and output needed there. If they are asked to provide some extra help, I would give it a try so that I don’t have to waste my efforts getting it shipped out – or
even losing my company since I have to do this already. Thanks, If this is not a written solution, then the only way to work 100% is by taking out your input, and figuring out how to feed it. If
somebody has the same ability, but I can get close to the core language support, then that would probably be about right but easier to work on if everyone were involved. I would then ideally be able
to finish all the work I’d need, I have that much experience. Make sure you don’t focus on math and keep my ability grounded. So people saying you don’t have the best skills to deal with is not worth
treating any other people as better – I’d better focus on math and make it more clear how you have you are – both of which are helpful when making choices – if you don’t have as deep a knowledge of
the language as you want to, then you don’t have as many things to think about. Lastly, if possible, put me a nice PayPal monthly payment to donate. The fact you have that means you wouldn’t need to
pay for me to do the instrumentation. Doing the same thing before comes out results in total waste.Is it possible to pay someone to complete my instrumentation assignment? I’m doing it as part of the
instrumentation project on a piece of equipment for a computer based project. Now this piece of equipment only has 3300 seconds for the tasks I’m trying to complete. Here is a file that I am using
for a script (make). EDIT: Following the instructions from the online textbook.
Math Homework Done For You
The number of seconds I am using a dollar-and-doll program is 99999 instead of the 32000-99999 program found on the Internet. Here is the relevant page on the math2spark wiki over at my own site. I
have been posting links for a while now about the problem of the 64000-99999 is the only available line, but the first two things I will describe today is how to actually use “comprised
intallability”. Clicking on the text box should be able to generate what I want (in that order). To reproduce the above code, click “Next” then right-click there and type an answer. I have this
question as well, so let me know if you understand. EDIT: I came across the tutorial a while back on this. The numbers are getting shorter and shorter each time I’m using the manual instruction, and
I definitely want the number of seconds, when this is possible, to be as short as possible. A: I found the closest answer. This is how I came up with in a short while discussion. The results of the
hours I went through are to be reported in an appropriate order. I figured out an easier way is to answer this question using the following variables: time = int((“N” + time + “:”), True) `time` Now
we know what time the times pass. time += 1.50*100 time += 3.00*10000 Which tells Time units to be equal to 1.50*1001…. (as opposed to the equivalent number ofIs it possible to pay someone to
complete my instrumentation assignment? I do not want to pay them to complete the project.
My Classroom
Please help. Hi V., The below is how I have to do the instrumentation. The number is given as follows: $E(F(P))=\frac{1}{18}\left(1-\frac{5}{18}\right)\times\left(|P|-|Q|\right)$ You should change
that for this statement, $E(F)$ should calculate $E(P)$ for both elements and assign the number of P’s. $E(P)$ is the intersection of functions given in order of precedence. I would like to know how
to do the following for my simple language: $\equals=\left( 1-2\sqrt[3]{4p}\right)$ is equal to $\equals=\left(\sqrt[3]{4p}-2\sqrt{2}$$p\right)+(4-2\sqrt{3}p)$$$^*$ is equal to $+4-3\sqrt{3}p$ also
here’s the function square bracket. $-6+p+9+21+41\sqrt{3}p+32p^2$ = -36 = 168 If this function can handle all elements and reduce anything to that, thats right. It means that the above would be
doable if you use $6^*$ or something else. Also, give me the way to do it up to the level you want. I know that some people I know who did the language programming but didn’t get the idea. Thanks! A:
One of the reasons I use the square brackets works for “language.” I need to mention one problem I discussed at the Wikipedia page with help should be that it does not have a good idea when the | {"url":"https://electricalassignments.com/is-it-possible-to-pay-someone-to-complete-my-instrumentation-assignment","timestamp":"2024-11-13T22:54:25Z","content_type":"text/html","content_length":"144521","record_id":"<urn:uuid:b460bb33-9f96-4465-ba05-17c051910af3>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00776.warc.gz"} |
Logic in computer science - SILO.PUB
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LOGIC IN COMPUTER SCIENCE Modelling and Reasoning about Systems
LOGIC IN COMPUTER SCIENCE Modelling and Reasoning about Systems MICHAEL HUTH Department of Computing Imperial College London, United Kingdom
MARK RYAN School of Computer Science University of Birmingham, United Kingdom
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge
University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521543101 © Cambridge University Press 2004 This publication is in copyright. Subject to statutory
exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in
print format 2004 ISBN-13 ISBN-10
978-0-511-26401-6 eBook (EBL) 0-511-26401-1 eBook (EBL)
ISBN-13 ISBN-10
978-0-521-54310-1 paperback 0-521-54310-X paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any
content on such websites is, or will remain, accurate or appropriate.
Foreword to the first edition Preface to the second edition Acknowledgements 1 Propositional logic 1.1 Declarative sentences 1.2 Natural deduction 1.2.1 Rules for natural deduction 1.2.2 Derived rules
1.2.3 Natural deduction in summary 1.2.4 Provable equivalence 1.2.5 An aside: proof by contradiction 1.3 Propositional logic as a formal language 1.4 Semantics of propositional logic 1.4.1 The
meaning of logical connectives 1.4.2 Mathematical induction 1.4.3 Soundness of propositional logic 1.4.4 Completeness of propositional logic 1.5 Normal forms 1.5.1 Semantic equivalence, satisfiability
and validity 1.5.2 Conjunctive normal forms and validity 1.5.3 Horn clauses and satisfiability 1.6 SAT solvers 1.6.1 A linear solver 1.6.2 A cubic solver 1.7 Exercises 1.8 Bibliographic notes 2
Predicate logic 2.1 The need for a richer language v
page ix xi xiii 1 2 5 6 23 26 29 29 31 36 36 40 45 49 53 54 58 65 68 69 72 78 91 93 93
2.2 Predicate logic as a formal language 2.2.1 Terms 2.2.2 Formulas 2.2.3 Free and bound variables 2.2.4 Substitution 2.3 Proof theory of predicate logic 2.3.1 Natural deduction rules 2.3.2 Quantifier
equivalences 2.4 Semantics of predicate logic 2.4.1 Models 2.4.2 Semantic entailment 2.4.3 The semantics of equality 2.5 Undecidability of predicate logic 2.6 Expressiveness of predicate logic 2.6.1
Existential second-order logic 2.6.2 Universal second-order logic 2.7 Micromodels of software 2.7.1 State machines 2.7.2 Alma – re-visited 2.7.3 A software micromodel 2.8 Exercises 2.9 Bibliographic
notes 3 Verification by model checking 3.1 Motivation for verification 3.2 Linear-time temporal logic 3.2.1 Syntax of LTL 3.2.2 Semantics of LTL 3.2.3 Practical patterns of specifications 3.2.4
Important equivalences between LTL formulas 3.2.5 Adequate sets of connectives for LTL 3.3 Model checking: systems, tools, properties 3.3.1 Example: mutual exclusion 3.3.2 The NuSMV model checker
3.3.3 Running NuSMV 3.3.4 Mutual exclusion revisited 3.3.5 The ferryman 3.3.6 The alternating bit protocol 3.4 Branching-time logic 3.4.1 Syntax of CTL
3.4.2 Semantics of CTL 3.4.3 Practical patterns of specifications 3.4.4 Important equivalences between CTL formulas 3.4.5 Adequate sets of CTL connectives 3.5 CTL* and the expressive powers of LTL and
CTL 3.5.1 Boolean combinations of temporal formulas in CTL 3.5.2 Past operators in LTL 3.6 Model-checking algorithms 3.6.1 The CTL model-checking algorithm 3.6.2 CTL model checking with fairness
3.6.3 The LTL model-checking algorithm 3.7 The fixed-point characterisation of CTL 3.7.1 Monotone functions 3.7.2 The correctness of SATEG 3.7.3 The correctness of SATEU 3.8 Exercises 3.9
Bibliographic notes 4 Program verification 4.1 Why should we specify and verify code? 4.2 A framework for software verification 4.2.1 A core programming language 4.2.2 Hoare triples 4.2.3 Partial and
total correctness 4.2.4 Program variables and logical variables 4.3 Proof calculus for partial correctness 4.3.1 Proof rules 4.3.2 Proof tableaux 4.3.3 A case study: minimal-sum section 4.4 Proof
calculus for total correctness 4.5 Programming by contract 4.6 Exercises 4.7 Bibliographic notes 5 Modal logics and agents 5.1 Modes of truth 5.2 Basic modal logic 5.2.1 Syntax 5.2.2 Semantics 5.3
Logic engineering 5.3.1 The stock of valid formulas
5.3.2 Important properties of the accessibility relation 5.3.3 Correspondence theory 5.3.4 Some modal logics 5.4 Natural deduction 5.5 Reasoning about knowledge in a multi-agent system 5.5.1 Some
examples 5.5.2 The modal logic KT45n 5.5.3 Natural deduction for KT45n 5.5.4 Formalising the examples 5.6 Exercises 5.7 Bibliographic notes 6 Binary decision diagrams 6.1 Representing boolean
functions 6.1.1 Propositional formulas and truth tables 6.1.2 Binary decision diagrams 6.1.3 Ordered BDDs 6.2 Algorithms for reduced OBDDs 6.2.1 The algorithm reduce 6.2.2 The algorithm apply 6.2.3
The algorithm restrict 6.2.4 The algorithm exists 6.2.5 Assessment of OBDDs 6.3 Symbolic model checking 6.3.1 Representing subsets of the set of states 6.3.2 Representing the transition relation
6.3.3 Implementing the functions pre∃ and pre∀ 6.3.4 Synthesising OBDDs 6.4 A relational mu-calculus 6.4.1 Syntax and semantics 6.4.2 Coding CTL models and specifications 6.5 Exercises 6.6
Bibliographic notes Bibliography Index
Foreword to the first edition
by Edmund M. Clarke FORE Systems Professor of Computer Science Carnegie Mellon University Pittsburgh, PA Formal methods have finally come of age! Specification languages, theorem provers, and model
checkers are beginning to be used routinely in industry. Mathematical logic is basic to all of these techniques. Until now textbooks on logic for computer scientists have not kept pace with the
development of tools for hardware and software specification and verification. For example, in spite of the success of model checking in verifying sequential circuit designs and communication
protocols, until now I did not know of a single text, suitable for undergraduate and beginning graduate students, that attempts to explain how this technique works. As a result, this material is
rarely taught to computer scientists and electrical engineers who will need to use it as part of their jobs in the near future. Instead, engineers avoid using formal methods in situations where the
methods would be of genuine benefit or complain that the concepts and notation used by the tools are complicated and unnatural. This is unfortunate since the underlying mathematics is generally quite
simple, certainly no more difficult than the concepts from mathematical analysis that every calculus student is expected to learn. Logic in Computer Science by Huth and Ryan is an exceptional book. I
was amazed when I looked through it for the first time. In addition to propositional and predicate logic, it has a particularly thorough treatment of temporal logic and model checking. In fact, the
book is quite remarkable in how much of this material it is able to cover: linear and branching time temporal logic, explicit state model checking, fairness, the basic fixpoint ix
Foreword to the first edition
theorems for computation tree logic (CTL), even binary decision diagrams and symbolic model checking. Moreover, this material is presented at a level that is accessible to undergraduate and beginning
graduate students. Numerous problems and examples are provided to help students master the material in the book. Since both Huth and Ryan are active researchers in logics of programs and program
verification, they write with considerable authority. In summary, the material in this book is up-to-date, practical, and elegantly presented. The book is a wonderful example of what a modern text on
logic for computer science should be like. I recommend it to the reader with greatest enthusiasm and predict that the book will be an enormous success. (This foreword is re-printed in the second
edition with its author’s permission.)
Preface to the second edition
Our motivation for (re)writing this book One of the leitmotifs of writing the first edition of our book was the observation that most logics used in the design, specification and verification of
computer systems fundamentally deal with a satisfaction relation Mφ where M is some sort of situation or model of a system, and φ is a specification, a formula of that logic, expressing what should be
true in situation M. At the heart of this set-up is that one can often specify and implement algorithms for computing . We developed this theme for propositional, first-order, temporal, modal, and
program logics. Based on the encouraging feedback received from five continents we are pleased to hereby present the second edition of this text which means to preserve and improve on the original
intent of the first edition.
What’s new and what’s gone Chapter 1 now discusses the design, correctness, and complexity of a SAT solver (a marking algorithm similar to St˚ almarck’s method [SS90]) for full propositional logic.
Chapter 2 now contains basic results from model theory (Compactness Theorem and L¨owenheim–Skolem Theorem); a section on the transitive closure and the expressiveness of existential and universal
second-order logic; and a section on the use of the object modelling language Alloy and its analyser for specifying and exploring under-specified first-order logic models with respect to properties
written in first-order logic with transitive closure. The Alloy language is executable which makes such exploration interactive and formal. xi
Preface to the second edition
Chapter 3 has been completely restructured. It now begins with a discussion of linear-time temporal logic; features the open-source NuSMV modelchecking tool throughout; and includes a discussion on
planning problems, more material on the expressiveness of temporal logics, and new modelling examples. Chapter 4 contains more material on total correctness proofs and a new section on the
programming-by-contract paradigm of verifying program correctness. Chapters 5 and 6 have also been revised, with many small alterations and corrections.
The interdependence of chapters and prerequisites The book requires that students know the basics of elementary arithmetic and naive set theoretic concepts and notation. The core material of Chapter
1 (everything except Sections 1.4.3 to 1.6.2) is essential for all of the chapters that follow. Other than that, only Chapter 6 depends on Chapter 3 and a basic understanding of the static scoping
rules covered in Chapter 2 – although one may easily cover Sections 6.1 and 6.2 without having done Chapter 3 at all. Roughly, the interdependence diagram of chapters is
WWW page This book is supported by a Web page, which contains a list of errata; text files for all the program code; ancillary technical material and links; all the figures; an interactive tutor based
on multiple-choice questions; and details of how instructors can obtain the solutions to exercises in this book which are marked with a ∗. The URL for the book’s page is www.cs.bham.ac.uk/research/
lics/. See also www.cambridge.org/ 052154310x
Many people have, directly or indirectly, assisted us in writing this book. David Schmidt kindly provided serveral exercises for Chapter 4. Krysia Broda has pointed out some typographical errors and
she and the other authors of [BEKV94] have allowed us to use some exercises from that book. We have also borrowed exercises or examples from [Hod77] and [FHMV95]. Susan Eisenbach provided a first
description of the Package Dependency System that we model in Alloy in Chapter 2. Daniel Jackson make very helpful comments on versions of that section. Zena Matilde Ariola, Josh Hodas, Jan
Komorowski, Sergey Kotov, Scott A. Smolka and Steve Vickers have corresponded with us about this text; their comments are appreciated. Matt Dwyer and John Hatcliff made useful comments on drafts of
Chapter 3. Kevin Lucas provided insightful comments on the content of Chapter 6, and notified us of numerous typographical errors in several drafts of the book. Achim Jung read several chapters and
gave useful feedback. Additionally, a number of people read and provided useful comments on several chapters, including Moti Ben-Ari, Graham Clark, Christian Haack, Anthony Hook, Roberto Segala, Alan
Sexton and Allen Stoughton. Numerous students at Kansas State University and the University of Birmingham have given us feedback of various kinds, which has influenced our choice and presentation of
the topics. We acknowledge Paul Taylor’s LATEX package for proof boxes. About half a dozen anonymous referees made critical, but constructive, comments which helped to improve this text in various
ways. In spite of these contributions, there may still be errors in the book, and we alone must take responsibility for those. Added for second edition Many people have helped improve this text by
pointing out typos and making other useful comments after the publication date. Among them,
we mention Wolfgang Ahrendt, Yasuhiro Ajiro, Torben Amtoft, Stephan Andrei, Bernhard Beckert, Jonathan Brown, James Caldwell, Ruchira Datta, Amy Felty, Dimitar Guelev, Hirotsugu Kakugawa, Kamran
Kashef, Markus Kr¨ otzsch, Jagun Kwon, Ranko Lazic, David Makinson, Alexander Miczo, Aart Middeldorp, Robert Morelli, Prakash Panangaden, Aileen Paraguya, Frank Pfenning, Shekhar Pradhan, Koichi
Takahashi, Kazunori Ueda, Hiroshi Watanabe, Fuzhi Wang and Reinhard Wilhelm.
1 Propositional logic
The aim of logic in computer science is to develop languages to model the situations we encounter as computer science professionals, in such a way that we can reason about them formally. Reasoning
about situations means constructing arguments about them; we want to do this formally, so that the arguments are valid and can be defended rigorously, or executed on a machine. Consider the following
argument: Example 1.1 If the train arrives late and there are no taxis at the station, then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there
were taxis at the station. Intuitively, the argument is valid, since if we put the first sentence and the third sentence together, they tell us that if there are no taxis, then John will be late. The
second sentence tells us that he was not late, so it must be the case that there were taxis. Much of this book will be concerned with arguments that have this structure, namely, that consist of a
number of sentences followed by the word ‘therefore’ and then another sentence. The argument is valid if the sentence after the ‘therefore’ logically follows from the sentences before it. Exactly
what we mean by ‘follows from’ is the subject of this chapter and the next one. Consider another example: Example 1.2 If it is raining and Jane does not have her umbrella with her, then she will get
wet. Jane is not wet. It is raining. Therefore, Jane has her umbrella with her. This is also a valid argument. Closer examination reveals that it actually has the same structure as the argument of
the previous example! All we have 1
1 Propositional logic
done is substituted some sentence fragments for others: Example 1.1 the train is late there are taxis at the station John is late for his meeting
Example 1.2 it is raining Jane has her umbrella with her Jane gets wet.
The argument in each example could be stated without talking about trains and rain, as follows: If p and not q, then r. Not r. p. Therefore, q.
In developing logics, we are not concerned with what the sentences really mean, but only in their logical structure. Of course, when we apply such reasoning, as done above, such meaning will be of
great interest.
1.1 Declarative sentences In order to make arguments rigorous, we need to develop a language in which we can express sentences in such a way that brings out their logical structure. The language we
begin with is the language of propositional logic. It is based on propositions, or declarative sentences which one can, in principle, argue as being true or false. Examples of declarative sentences
are: (1) (2) (3) (4) (5) (6)
The sum of the numbers 3 and 5 equals 8. Jane reacted violently to Jack’s accusations. Every even natural number >2 is the sum of two prime numbers. All Martians like pepperoni on their pizza. Albert
Camus ´etait un ´ecrivain fran¸cais. Die W¨ urde des Menschen ist unantastbar.
These sentences are all declarative, because they are in principle capable of being declared ‘true’, or ‘false’. Sentence (1) can be tested by appealing to basic facts about arithmetic (and by
tacitly assuming an Arabic, decimal representation of natural numbers). Sentence (2) is a bit more problematic. In order to give it a truth value, we need to know who Jane and Jack are and perhaps to
have a reliable account from someone who witnessed the situation described. In principle, e.g., if we had been at the scene, we feel that we would have been able to detect Jane’s violent reaction,
provided that it indeed occurred in that way. Sentence (3), known as Goldbach’s conjecture, seems straightforward on the face of it. Clearly, a fact about all even numbers >2 is either true or false.
But to this day nobody knows whether sentence (3) expresses a truth or not. It is even not clear whether this could be shown by some finite means, even if it were true. However, in
1.1 Declarative sentences
this text we will be content with sentences as soon as they can, in principle, attain some truth value regardless of whether this truth value reflects the actual state of affairs suggested by the
sentence in question. Sentence (4) seems a bit silly, although we could say that if Martians exist and eat pizza, then all of them will either like pepperoni on it or not. (We have to introduce
predicate logic in Chapter 2 to see that this sentence is also declarative if no Martians exist; it is then true.) Again, for the purposes of this text sentence (4) will do. Et alors, qu’est-ce qu’on
pense des phrases (5) et (6)? Sentences (5) and (6) are fine if you happen to read French and German a bit. Thus, declarative statements can be made in any natural, or artificial, language. The kind of
sentences we won’t consider here are non-declarative ones, like r Could you please pass me the salt? r Ready, steady, go! r May fortune come your way.
Primarily, we are interested in precise declarative sentences, or statements about the behaviour of computer systems, or programs. Not only do we want to specify such statements but we also want to
check whether a given program, or system, fulfils a specification at hand. Thus, we need to develop a calculus of reasoning which allows us to draw conclusions from given assumptions, like initialised
variables, which are reliable in the sense that they preserve truth: if all our assumptions are true, then our conclusion ought to be true as well. A much more difficult question is whether, given any
true property of a computer program, we can find an argument in our calculus that has this property as its conclusion. The declarative sentence (3) above might illuminate the problematic aspect of
such questions in the context of number theory. The logics we intend to design are symbolic in nature. We translate a certain sufficiently large subset of all English declarative sentences into strings
of symbols. This gives us a compressed but still complete encoding of declarative sentences and allows us to concentrate on the mere mechanics of our argumentation. This is important since
specifications of systems or software are sequences of such declarative sentences. It further opens up the possibility of automatic manipulation of such specifications, a job that computers just love
to do1 . Our strategy is to consider certain declarative sentences as 1
There is a certain, slightly bitter, circularity in such endeavours: in proving that a certain computer program P satisfies a given property, we might let some other computer program Q try to find a
proof that P satisfies the property; but who guarantees us that Q satisfies the property of producing only correct proofs? We seem to run into an infinite regress.
1 Propositional logic
being atomic, or indecomposable, like the sentence ‘The number 5 is even.’ We assign certain distinct symbols p, q, r, . . ., or sometimes p1 , p2 , p3 , . . . to each of these atomic sentences and
we can then code up more complex sentences in a compositional way. For example, given the atomic sentences p: ‘I won the lottery last week.’ q: ‘I purchased a lottery ticket.’ r: ‘I won last week’s
we can form more complex sentences according to the rules below: ¬: The negation of p is denoted by ¬p and expresses ‘I did not win the lottery last week,’ or equivalently ‘It is not true that I won
the lottery last week.’ ∨: Given p and r we may wish to state that at least one of them is true: ‘I won the lottery last week, or I won last week’s sweepstakes;’ we denote this declarative sentence
by p ∨ r and call it the disjunction of p and r2 . ∧: Dually, the formula p ∧ r denotes the rather fortunate conjunction of p and r: ‘Last week I won the lottery and the sweepstakes.’ →: Last, but
definitely not least, the sentence ‘If I won the lottery last week, then I purchased a lottery ticket.’ expresses an implication between p and q, suggesting that q is a logical consequence of p. We
write p → q for that3 . We call p the assumption of p → q and q its conclusion.
Of course, we are entitled to use these rules of constructing propositions repeatedly. For example, we are now in a position to form the proposition p ∧ q → ¬r ∨ q which means that ‘if p and q then
not r or q’. You might have noticed a potential ambiguity in this reading. One could have argued that this sentence has the structure ‘p is the case and if q then . . . ’ A computer would require the
insertion of brackets, as in (p ∧ q) → ((¬r) ∨ q) 2
Its meaning should not be confused with the often implicit meaning of or in natural language discourse as either . . . or. In this text or always means at least one of them and should not be
confounded with exclusive or which states that exactly one of the two statements holds. The natural language meaning of ‘if . . . then . . . ’ often implicitly assumes a causal role of the assumption
somehow enabling its conclusion. The logical meaning of implication is a bit different, though, in the sense that it states the preservation of truth which might happen without any causal
relationship. For example, ‘If all birds can fly, then Bob Dole was never president of the United States of America.’ is a true statement, but there is no known causal connection between the flying
skills of penguins and effective campaigning.
1.2 Natural deduction
to disambiguate this assertion. However, we humans get annoyed by a proliferation of such brackets which is why we adopt certain conventions about the binding priorities of these symbols. Convention
1.3 ¬ binds more tightly than ∨ and ∧, and the latter two bind more tightly than →. Implication → is right-associative: expressions of the form p → q → r denote p → (q → r).
1.2 Natural deduction How do we go about constructing a calculus for reasoning about propositions, so that we can establish the validity of Examples 1.1 and 1.2? Clearly, we would like to have a set
of rules each of which allows us to draw a conclusion given a certain arrangement of premises. In natural deduction, we have such a collection of proof rules. They allow us to infer formulas from
other formulas. By applying these rules in succession, we may infer a conclusion from a set of premises. Let’s see how this works. Suppose we have a set of formulas4 φ1 , φ2 , φ3 , . . . , φn , which
we will call premises, and another formula, ψ, which we will call a conclusion. By applying proof rules to the premises, we hope to get some more formulas, and by applying more proof rules to those,
to eventually obtain the conclusion. This intention we denote by φ1 , φ2 , . . . , φn ψ. This expression is called a sequent; it is valid if a proof for it can be found. The sequent for Examples 1.1
and 1.2 is p ∧ ¬q → r, ¬r, p q. Constructing such a proof is a creative exercise, a bit like programming. It is not necessarily obvious which rules to apply, and in what order, to obtain the desired
conclusion. Additionally, our proof rules should be carefully chosen; otherwise, we might be able to ‘prove’ invalid patterns of argumentation. For 4
It is traditional in logic to use Greek letters. Lower-case letters are used to stand for formulas and upper-case letters are used for sets of formulas. Here are some of the more commonly used Greek
letters, together with their pronunciation: Lower-case φ phi ψ psi χ chi η eta α alpha β beta γ gamma
Upper-case Φ Phi Ψ Psi Γ Gamma ∆ Delta
1 Propositional logic
example, we expect that we won’t be able to show the sequent p, q p ∧ ¬q. For example, if p stands for ‘Gold is a metal.’ and q for ‘Silver is a metal,’ then knowing these two facts should not allow
us to infer that ‘Gold is a metal whereas silver isn’t.’ Let’s now look at our proof rules. We present about fifteen of them in total; we will go through them in turn and then summarise at the end of
this section.
1.2.1 Rules for natural deduction The rules for conjunction Our first rule is called the rule for conjunction (∧): and-introduction. It allows us to conclude φ ∧ ψ, given that we have already
concluded φ and ψ separately. We write this rule as φ ψ φ∧ψ
Above the line are the two premises of the rule. Below the line goes the conclusion. (It might not yet be the final conclusion of our argument; we might have to apply more rules to get there.) To the
right of the line, we write the name of the rule; ∧i is read ‘and-introduction’. Notice that we have introduced a ∧ (in the conclusion) where there was none before (in the premises). For each of the
connectives, there is one or more rules to introduce it and one or more rules to eliminate it. The rules for and-elimination are these two: φ∧ψ φ
φ∧ψ ψ
∧e2 .
The rule ∧e1 says: if you have a proof of φ ∧ ψ, then by applying this rule you can get a proof of φ. The rule ∧e2 says the same thing, but allows you to conclude ψ instead. Observe the dependences
of these rules: in the first rule of (1.1), the conclusion φ has to match the first conjunct of the premise, whereas the exact nature of the second conjunct ψ is irrelevant. In the second rule it is
just the other way around: the conclusion ψ has to match the second conjunct ψ and φ can be any formula. It is important to engage in this kind of pattern matching before the application of proof
rules. Example 1.4 Let’s use these rules to prove that p ∧ q, r |− q ∧ r is valid. We start by writing down the premises; then we leave a gap and write the
1.2 Natural deduction
conclusion: p∧q r q∧r The task of constructing the proof is to fill the gap between the premises and the conclusion by applying a suitable sequence of proof rules. In this case, we apply ∧e2 to the
first premise, giving us q. Then we apply ∧i to this q and to the second premise, r, giving us q ∧ r. That’s it! We also usually number all the lines, and write in the justification for each line,
producing this: 1
∧e2 1
∧i 3, 2
Demonstrate to yourself that you’ve understood this by trying to show on your own that (p ∧ q) ∧ r, s ∧ t |− q ∧ s is valid. Notice that the φ and ψ can be instantiated not just to atomic sentences,
like p and q in the example we just gave, but also to compound sentences. Thus, from (p ∧ q) ∧ r we can deduce p ∧ q by applying ∧e1 , instantiating φ to p ∧ q and ψ to r. If we applied these proof
rules literally, then the proof above would actually be a tree with root q ∧ r and leaves p ∧ q and r, like this: p∧q q
∧e2 q∧r
However, we flattened this tree into a linear presentation which necessitates the use of pointers as seen in lines 3 and 4 above. These pointers allow us to recreate the actual proof tree. Throughout
this text, we will use the flattened version of presenting proofs. That way you have to concentrate only on finding a proof, not on how to fit a growing tree onto a sheet of paper. If a sequent is
valid, there may be many different ways of proving it. So if you compare your solution to these exercises with those of others, they need not coincide. The important thing to realise, though, is that
any putative proof can be checked for correctness by checking each individual line, starting at the top, for the valid application of its proof rule.
1 Propositional logic
The rules of double negation Intuitively, there is no difference between a formula φ and its double negation ¬¬φ, which expresses no more and nothing less than φ itself. The sentence ‘It is not true
that it does not rain.’ is just a more contrived way of saying ‘It rains.’ Conversely, knowing ‘It rains,’ we are free to state this fact in this more complicated manner if we wish. Thus, we obtain
rules of elimination and introduction for double negation: ¬¬φ φ
φ ¬¬φ
(There are rules for single negation on its own, too, which we will see later.) Example 1.5 The proof of the sequent p, ¬¬(q ∧ r) ¬¬p ∧ r below uses most of the proof rules discussed so far: 1
¬¬(q ∧ r) premise
¬¬i 1
¬¬e 2
∧e2 4
¬¬p ∧ r
∧i 3, 5
Example 1.6 We now prove the sequent (p ∧ q) ∧ r, s ∧ t |− q ∧ s which you were invited to prove by yourself in the last section. Please compare the proof below with your solution: 1
(p ∧ q) ∧ r
∧e1 1
∧e2 3
∧e1 2
∧i 4, 5
1.2 Natural deduction
The rule for eliminating implication There is one rule to introduce → and one to eliminate it. The latter is one of the best known rules of propositional logic and is often referred to by its Latin
name modus ponens. We will usually call it by its modern name, implies-elimination (sometimes also referred to as arrow-elimination). This rule states that, given φ and knowing that φ implies ψ, we
may rightfully conclude ψ. In our calculus, we write this as φ
φ→ψ ψ
Let us justify this rule by spelling out instances of some declarative sentences p and q. Suppose that p : It rained. p → q : If it rained, then the street is wet. so q is just ‘The street is wet.’
Now, if we know that it rained and if we know that the street is wet in the case that it rained, then we may combine these two pieces of information to conclude that the street is indeed wet. Thus,
the justification of the →e rule is a mere application of common sense. Another example from programming is: p : The value of the program’s input is an integer. p → q : If the program’s input is an
integer, then the program outputs a boolean. Again, we may put all this together to conclude that our program outputs a boolean value if supplied with an integer input. However, it is important to
realise that the presence of p is absolutely essential for the inference to happen. For example, our program might well satisfy p → q, but if it doesn’t satisfy p – e.g. if its input is a surname –
then we will not be able to derive q. As we saw before, the formal parameters φ and the ψ for →e can be instantiated to any sentence, including compound ones: 1
¬p ∧ q
¬p ∧ q → r ∨ ¬p
r ∨ ¬p
→e 2, 1
1 Propositional logic
Of course, we may use any of these rules as often as we wish. For example, given p, p → q and p → (q → r), we may infer r: 1
p → (q → r) premise
→e 1, 3
→e 2, 3
→e 4, 5
Before turning to implies-introduction, let’s look at a hybrid rule which has the Latin name modus tollens. It is like the →e rule in that it eliminates an implication. Suppose that p → q and ¬q are
the case. Then, if p holds we can use →e to conclude that q holds. Thus, we then have that q and ¬q hold, which is impossible. Therefore, we may infer that p must be false. But this can only mean
that ¬p is true. We summarise this reasoning into the rule modus tollens, or MT for short:5 φ → ψ ¬ψ ¬φ
Again, let us see an example of this rule in the natural language setting: ‘If Abraham Lincoln was Ethiopian, then he was African. Abraham Lincoln was not African; therefore he was not Ethiopian.’
Example 1.7 In the following proof of p → (q → r), p, ¬r ¬q we use several of the rules introduced so far:
p → (q → r) premise
→e 1, 2
MT 4, 3
We will be able to derive this rule from other ones later on, but we introduce it here because it allows us already to do some pretty slick proofs. You may think of this rule as one on a higher level
insofar as it does not mention the lower-level rules upon which it depends.
1.2 Natural deduction
Examples 1.8 Here are two example proofs which combine the rule MT with either ¬¬e or ¬¬i: 1
¬p → q
MT 1, 2
¬¬e 3
proves that the sequent ¬p → q, ¬q p is valid; and 1
p → ¬q
¬¬i 2
MT 1, 3
shows the validity of the sequent p → ¬q, q ¬p. Note that the order of applying double negation rules and MT is different in these examples; this order is driven by the structure of the particular
sequent whose validity one is trying to show. The rule implies introduction The rule MT made it possible for us to show that p → q, ¬q ¬p is valid. But the validity of the sequent p → q ¬q → ¬p seems
just as plausible. That sequent is, in a certain sense, saying the same thing. Yet, so far we have no rule which builds implications that do not already occur as premises in our proofs. The mechanics
of such a rule are more involved than what we have seen so far. So let us proceed with care. Let us suppose that p → q is the case. If we temporarily assume that ¬q holds, we can use MT to infer ¬p.
Thus, assuming p → q we can show that ¬q implies ¬p; but the latter we express symbolically as ¬q → ¬p. To summarise, we have found an argumentation for p → q ¬q → ¬p: 1
MT 1, 2
¬q → ¬p
→i 2−3
The box in this proof serves to demarcate the scope of the temporary assumption ¬q. What we are saying is: let’s make the assumption of ¬q. To
1 Propositional logic
do this, we open a box and put ¬q at the top. Then we continue applying other rules as normal, for example to obtain ¬p. But this still depends on the assumption of ¬q, so it goes inside the box.
Finally, we are ready to apply →i. It allows us to conclude ¬q → ¬p, but that conclusion no longer depends on the assumption ¬q. Compare this with saying that ‘If you are French, then you are
European.’ The truth of this sentence does not depend on whether anybody is French or not. Therefore, we write the conclusion ¬q → ¬p outside the box. This works also as one would expect if we think
of p → q as a type of a procedure. For example, p could say that the procedure expects an integer value x as input and q might say that the procedure returns a boolean value y as output. The validity
of p → q amounts now to an assume-guarantee assertion: if the input is an integer, then the output is a boolean. This assertion can be true about a procedure while that same procedure could compute
strange things or crash in the case that the input is not an integer. Showing p → q using the rule →i is now called type checking, an important topic in the construction of compilers for typed
programming languages. We thus formulate the rule →i as follows: φ .. . ψ φ→ψ
It says: in order to prove φ → ψ, make a temporary assumption of φ and then prove ψ. In your proof of ψ, you can use φ and any of the other formulas such as premises and provisional conclusions that
you have made so far. Proofs may nest boxes or open new boxes after old ones have been closed. There are rules about which formulas can be used at which points in the proof. Generally, we can only
use a formula φ in a proof at a given point if that formula occurs prior to that point and if no box which encloses that occurrence of φ has been closed already. The line immediately following a
closed box has to match the pattern of the conclusion of the rule that uses the box. For implies-introduction, this means that we have to continue after the box with φ → ψ, where φ was the first and ψ
the last formula of that box. We will encounter two more proof rules involving proof boxes and they will require similar pattern matching.
1.2 Natural deduction
Example 1.9 Here is another example of a proof using →i: 1
¬q → ¬p
¬¬i 2
MT 1, 3
p → ¬¬q
→i 2−4
which verifies the validity of the sequent ¬q → ¬p p → ¬¬q. Notice that we could apply the rule MT to formulas occurring in or above the box: at line 4, no box has been closed that would enclose line
1 or 3. At this point it is instructive to consider the one-line argument p
which demonstrates p p. The rule →i (with conclusion φ → ψ) does not prohibit the possibility that φ and ψ coincide. They could both be instantiated to p. Therefore we may extend the proof above to 1
→i 1 − 1
We write p → p to express that the argumentation for p → p does not depend on any premises at all. Definition 1.10 Logical formulas φ with valid sequent φ are theorems. Example 1.11 Here is an
example of a theorem whose proof utilises most of the rules introduced so far: 1
¬q → ¬p
¬¬i 3
MT 2, 4
¬¬e 5
→e 1, 6
→i 3−7
(¬q → ¬p) → (p → r)
→i 2−8
(q → r) → ((¬q → ¬p) → (p → r)) →i 1−9
1 Propositional logic
→ q→r
¬q → ¬p r p
Figure 1.1. Part of the structure of the formula (q → r) → ((¬q → ¬p) → (p → r)) to show how it determines the proof structure.
Therefore the sequent (q → r) → ((¬q → ¬p) → (p → r)) is valid, showing that (q → r) → ((¬q → ¬p) → (p → r)) is another theorem. Remark 1.12 Indeed, this example indicates that we may transform any
proof of φ1 , φ2 , . . . , φn ψ in such a way into a proof of the theorem φ1 → (φ2 → (φ3 → (· · · → (φn → ψ) . . . ))) by ‘augmenting’ the previous proof with n lines of the rule →i applied to φn ,
φn−1 ,. . . , φ1 in that order. The nested boxes in the proof of Example 1.11 reveal a pattern of using elimination rules first, to deconstruct assumptions we have made, and then introduction rules to
construct our final conclusion. More difficult proofs may involve several such phases. Let us dwell on this important topic for a while. How did we come up with the proof above? Parts of it are
determined by the structure of the formulas we have, while other parts require us to be creative. Consider the logical structure of (q → r) → ((¬q → ¬p) → (p → r)) schematically depicted in Figure
1.1. The formula is overall an implication since → is the root of the tree in Figure 1.1. But the only way to build an implication is by means
1.2 Natural deduction
of the rule →i. Thus, we need to state the assumption of that implication as such (line 1) and have to show its conclusion (line 9). If we managed to do that, then we know how to end the proof in
line 10. In fact, as we already remarked, this is the only way we could have ended it. So essentially lines 1, 9 and 10 are completely determined by the structure of the formula; further, we have
reduced the problem to filling the gaps in between lines 1 and 9. But again, the formula in line 9 is an implication, so we have only one way of showing it: assuming its premise in line 2 and trying
to show its conclusion in line 8; as before, line 9 is obtained by →i. The formula p → r in line 8 is yet another implication. Therefore, we have to assume p in line 3 and hope to show r in line 7,
then →i produces the desired result in line 8. The remaining question now is this: how can we show r, using the three assumptions in lines 1–3? This, and only this, is the creative part of this
proof. We see the implication q → r in line 1 and know how to get r (using →e) if only we had q. So how could we get q? Well, lines 2 and 3 almost look like a pattern for the MT rule, which would
give us ¬¬q in line 5; the latter is quickly changed to q in line 6 via ¬¬e. However, the pattern for MT does not match right away, since it requires ¬¬p instead of p. But this is easily accomplished
via ¬¬i in line 4. The moral of this discussion is that the logical structure of the formula to be shown tells you a lot about the structure of a possible proof and it is definitely worth your while
to exploit that information in trying to prove sequents. Before ending this section on the rules for implication, let’s look at some more examples (this time also involving the rules for
conjunction). Example 1.13 Using the rule ∧i, we can prove the validity of the sequent p ∧ q → r p → (q → r) : 1
p∧q →r
∧i 2, 3
→e 1, 4
→i 3−5
p → (q → r) →i 2−6
1 Propositional logic
Example 1.14 Using the two elimination rules ∧e1 and ∧e2 , we can show that the ‘converse’ of the sequent above is valid, too: 1
p → (q → r)
∧e1 2
∧e2 2
→e 1, 3
→e 5, 4
p∧q →r
→i 2−6
The validity of p → (q → r) p ∧ q → r and p ∧ q → r p → (q → r) means that these two formulas are equivalent in the sense that we can prove one from the other. We denote this by p ∧ q → r p → (q →
r). Since there can be only one formula to the right of , we observe that each instance of can only relate two formulas to each other. Example 1.15 Here is an example of a proof that uses
introduction and elimination rules for conjunction; it shows the validity of the sequent p → q p ∧ r → q ∧ r: 1
∧e1 2
∧e2 2
→e 1, 3
∧i 5, 4
p∧r →q∧r
→i 2−6
The rules for disjunction The rules for disjunction are different in spirit from those for conjunction. The case for conjunction was concise and clear: proofs of φ ∧ ψ are essentially nothing but a
concatenation of a proof of φ and a proof of ψ, plus an additional line invoking ∧i. In the case of disjunctions, however, it turns out that the introduction of disjunctions is by far easier to grasp
than their elimination. So we begin with the rules ∨i1 and ∨i2 . From the premise φ we can infer that ‘φ or ψ’ holds, for we already know
1.2 Natural deduction
that φ holds. Note that this inference is valid for any choice of ψ. By the same token, we may conclude ‘φ or ψ’ if we already have ψ. Similarly, that inference works for any choice of φ. Thus, we
arrive at the proof rules φ φ∨ψ
ψ φ∨ψ
∨i2 .
So if p stands for ‘Agassi won a gold medal in 1996.’ and q denotes the sentence ‘Agassi won Wimbledon in 1996.’ then p ∨ q is the case because p is true, regardless of the fact that q is false.
Naturally, the constructed disjunction depends upon the assumptions needed in establishing its respective disjunct p or q. Now let’s consider or-elimination. How can we use a formula of the form φ ∨
ψ in a proof? Again, our guiding principle is to disassemble assumptions into their basic constituents so that the latter may be used in our argumentation such that they render our desired
conclusion. Let us imagine that we want to show some proposition χ by assuming φ ∨ ψ. Since we don’t know which of φ and ψ is true, we have to give two separate proofs which we need to combine into
one argument: 1. First, we assume φ is true and have to come up with a proof of χ. 2. Next, we assume ψ is true and need to give a proof of χ as well. 3. Given these two proofs, we can infer χ from
the truth of φ ∨ ψ, since our case analysis above is exhaustive.
Therefore, we write the rule ∨e as follows:
φ .. .
ψ .. .
It is saying that: if φ ∨ ψ is true and – no matter whether we assume φ or we assume ψ – we can get a proof of χ, then we are entitled to deduce χ anyway. Let’s look at a proof that p ∨ q q ∨ p is
valid: 1
∨i2 2
∨i1 4
∨e 1, 2−3, 4−5
1 Propositional logic
Here are some points you need to remember about applying the ∨e rule. r For it to be a sound argument we have to make sure that the conclusions in each of the two cases (the χ in the rule) are
actually the same formula. r The work done by the rule ∨e is the combining of the arguments of the two cases into one. r In each case you may not use the temporary assumption of the other case,
unless it is something that has already been shown before those case boxes began. r The invocation of rule ∨e in line 6 lists three things: the line in which the disjunction appears (1), and the
location of the two boxes for the two cases (2–3 and 4–5).
If we use φ ∨ ψ in an argument where it occurs only as an assumption or a premise, then we are missing a certain amount of information: we know φ, or ψ, but we don’t know which one of the two it is.
Thus, we have to make a solid case for each of the two possibilities φ or ψ; this resembles the behaviour of a CASE or IF statement found in most programming languages. Example 1.16 Here is a more
complex example illustrating these points. We prove that the sequent q → r p ∨ q → p ∨ r is valid: 1
∨i1 3
→e 1, 5
∨i2 6
∨e 2, 3−4, 5−7
p∨q →p∨r
→i 2−8
Note that the propositions in lines 4, 7 and 8 coincide, so the application of ∨e is legitimate. We give some more example proofs which use the rules ∨e, ∨i1 and ∨i2 . Example 1.17 Proving the
validity of the sequent (p ∨ q) ∨ r p ∨ (q ∨ r) is surprisingly long and seemingly complex. But this is to be expected, since
1.2 Natural deduction
the elimination rules break (p ∨ q) ∨ r up into its atomic constituents p, q and r, whereas the introduction rules then built up the formula p ∨ (q ∨ r). 1
(p ∨ q) ∨ r
(p ∨ q)
p ∨ (q ∨ r) ∨i1 3
∨i1 5
p ∨ (q ∨ r) ∨i2 6
p ∨ (q ∨ r) ∨e 2, 3−4, 5−7
∨i2 9
p ∨ (q ∨ r) ∨i2 10
p ∨ (q ∨ r) ∨e 1, 2−8, 9−11
Example 1.18 From boolean algebra, or circuit theory, you may know that disjunctions distribute over conjunctions. We are now able to prove this in natural deduction. The following proof: 1
p ∧ (q ∨ r)
∧e1 1
∧e2 1
∧i 2, 4
(p ∧ q) ∨ (p ∧ r) ∨i1 5
∧i 2, 7
(p ∧ q) ∨ (p ∧ r) ∨i2 8
(p ∧ q) ∨ (p ∧ r) ∨e 3, 4−6, 7−9
verifies the validity of the sequent p ∧ (q ∨ r) (p ∧ q) ∨ (p ∧ r) and you are encouraged to show the validity of the ‘converse’ (p ∧ q) ∨ (p ∧ r) p ∧ (q ∨ r) yourself.
1 Propositional logic
A final rule is required in order to allow us to conclude a box with a formula which has already appeared earlier in the proof. Consider the sequent p → (q → p), whose validity may be proved as
follows: 1
copy 1
→i 2−3
p → (q → p)
→i 1−4
The rule ‘copy’ allows us to repeat something that we know already. We need to do this in this example, because the rule →i requires that we end the inner box with p. The copy rule entitles us to
copy formulas that appeared before, unless they depend on temporary assumptions whose box has already been closed. Though a little inelegant, this additional rule is a small price to pay for the
freedom of being able to use premises, or any other ‘visible’ formulas, more than once. The rules for negation We have seen the rules ¬¬i and ¬¬e, but we haven’t seen any rules that introduce or
eliminate single negations. These rules involve the notion of contradiction. This detour is to be expected since our reasoning is concerned about the inference, and therefore the preservation, of
truth. Hence, there cannot be a direct way of inferring ¬φ, given φ. Definition 1.19 Contradictions are expressions of the form φ ∧ ¬φ or ¬φ ∧ φ, where φ is any formula. Examples of such
contradictions are r ∧ ¬r, (p → q) ∧ ¬(p → q) and ¬(r ∨ s → q) ∧ (r ∨ s → q). Contradictions are a very important notion in logic. As far as truth is concerned, they are all equivalent; that means we
should be able to prove the validity of ¬(r ∨ s → q) ∧ (r ∨ s → q) (p → q) ∧ ¬(p → q)
since both sides are contradictions. We’ll be able to prove this later, when we have introduced the rules for negation. Indeed, it’s not just that contradictions can be derived from contradictions;
actually, any formula can be derived from a contradiction. This can be
1.2 Natural deduction
confusing when you first encounter it; why should we endorse the argument p ∧ ¬p q, where p : The moon is made of green cheese. q : I like pepperoni on my pizza. considering that our taste in pizza
doesn’t have anything to do with the constitution of the moon? On the face of it, such an endorsement may seem absurd. Nevertheless, natural deduction does have this feature that any formula can be
derived from a contradiction and therefore it makes this argument valid. The reason it takes this stance is that tells us all the things we may infer, provided that we can assume the formulas to the
left of it. This process does not care whether such premises make any sense. This has at least the advantage that we can match to checks based on semantic intuitions which we formalise later by using
truth tables: if all the premises compute to ‘true’, then the conclusion must compute ‘true’ as well. In particular, this is not a constraint in the case that one of the premises is (always) false.
The fact that ⊥ can prove anything is encoded in our calculus by the proof rule bottom-elimination: ⊥ φ
The fact that ⊥ itself represents a contradiction is encoded by the proof rule not-elimination: φ
¬φ ⊥
Example 1.20 We apply these rules to show that ¬p ∨ q |− p → q is valid: 1
¬p ∨ q
¬e 3, 2
copy 2
⊥e 4
→i 3−4
→i 3−5
∨e 1, 2−6
1 Propositional logic
Notice how, in this example, the proof boxes for ∨e are drawn side by side instead of on top of each other. It doesn’t matter which way you do it. What about introducing negations? Well, suppose we
make an assumption which gets us into a contradictory state of affairs, i.e. gets us ⊥. Then our assumption cannot be true; so it must be false. This intuition is the basis for the proof rule ¬i: φ ..
. ⊥ ¬φ
Example 1.21 We put these rules in action, demonstrating that the sequent p → q, p → ¬q ¬p is valid: 1
p → ¬q
→e 1, 3
→e 2, 3
¬e 4, 5
¬i 3−6
Lines 3–6 contain all the work of the ¬i rule. Here is a second example, showing the validity of a sequent, p → ¬p ¬p, with a contradictory formula as sole premise: 1
p → ¬p
→e 1, 2
¬e 2, 3
¬i 2−4
Example 1.22 We prove that the sequent p → (q → r), p, ¬r |− ¬q is valid,
1.2 Natural deduction
without using the MT rule: 1
p → (q → r) premise
→e 1, 2
→e 5, 4
¬e 6, 3
¬i 4−7
Example 1.23 Finally, we return to the argument of Examples 1.1 and 1.2, which can be coded up by the sequent p ∧ ¬q → r, ¬r, p |− q whose validity we now prove: 1
p ∧ ¬q → r
p ∧ ¬q
∧i 3, 4
→e 1, 5
¬e 6, 2
¬i 4−7
¬¬e 8
1.2.2 Derived rules When describing the proof rule modus tollens (MT), we mentioned that it is not a primitive rule of natural deduction, but can be derived from some of the other rules. Here is the
derivation of φ → ψ ¬ψ ¬φ
1 Propositional logic
from →e, ¬e and ¬i: 1
→e 1, 3
¬e 4, 2
¬i 3−5
We could now go back through the proofs in this chapter and replace applications of MT by this combination of →e, ¬e and ¬i. However, it is convenient to think of MT as a shorthand (or a macro). The
same holds for the rule φ ¬¬i. ¬¬φ It can be derived from the rules ¬i and ¬e, as follows: 1
¬e 1, 2
¬i 2−3
There are (unboundedly) many such derived rules which we could write down. However, there is no point in making our calculus fat and unwieldy; and some purists would say that we should stick to a
minimum set of rules, all of which are independent of each other. We don’t take such a purist view. Indeed, the two derived rules we now introduce are extremely useful. You will find that they crop up
frequently when doing exercises in natural deduction, so it is worth giving them names as derived rules. In the case of the second one, its derivation from the primitive proof rules is not very
obvious. The first one has the Latin name reductio ad absurdum. It means ‘reduction to absurdity’ and we will simply call it proof by contradiction (PBC for short). The rule says: if from ¬φ we obtain
a contradiction, then we are entitled to deduce φ: ¬φ .. . ⊥ φ
1.2 Natural deduction
This rule looks rather similar to ¬i, except that the negation is in a different place. This is the clue to how to derive PBC from our basic proof rules. Suppose we have a proof of ⊥ from ¬φ. By →i,
we can transform this into a proof of ¬φ → ⊥ and proceed as follows: 1
¬φ → ⊥ given
→e 1, 2
¬i 2−3
¬¬e 4
This shows that PBC can be derived from →i, ¬i, →e and ¬¬e. The final derived rule we consider in this section is arguably the most useful to use in proofs, because its derivation is rather long and
complicated, so its usage often saves time and effort. It also has a Latin name, tertium non datur ; the English name is the law of the excluded middle, or LEM for short. It simply says that φ ∨ ¬φ is
true: whatever φ is, it must be either true or false; in the latter case, ¬φ is true. There is no third possibility (hence excluded middle): the sequent φ ∨ ¬φ is valid. Its validity is implicit, for
example, whenever you write an if-statement in a programming language: ‘if B {C1 } else {C2 }’ relies on the fact that B ∨ ¬B is always true (and that B and ¬B can never be true at the same time).
Here is a proof in natural deduction that derives the law of the excluded middle from basic proof rules: 1
¬(φ ∨ ¬φ)
φ ∨ ¬φ
∨i1 2
¬e 3, 1
¬i 2−4
φ ∨ ¬φ
∨i2 5
¬e 6, 1
¬¬(φ ∨ ¬φ) ¬i 1−7
φ ∨ ¬φ
¬¬e 8
1 Propositional logic
Example 1.24 Using LEM, we show that p → q ¬p ∨ q is valid: 1
¬p ∨ p
¬p ∨ q
∨i1 3
→e 1, 5
¬p ∨ q
∨i2 6
¬p ∨ q
∨e 2, 3−4, 5−7
It can be difficult to decide which instance of LEM would benefit the progress of a proof. Can you re-do the example above with q ∨ ¬q as LEM?
1.2.3 Natural deduction in summary The proof rules for natural deduction are summarised in Figure 1.2. The explanation of the rules we have given so far in this chapter is declarative; we have
presented each rule and justified it in terms of our intuition about the logical connectives. However, when you try to use the rules yourself, you’ll find yourself looking for a more procedural
interpretation; what does a rule do and how do you use it? For example, r ∧i says: to prove φ ∧ ψ, you must first prove φ and ψ separately and then use the rule ∧i. r ∧e says: to prove φ, try proving
φ ∧ ψ and then use the rule ∧e . Actually, 1 1 this doesn’t sound like very good advice because probably proving φ ∧ ψ will be harder than proving φ alone. However, you might find that you already
have φ ∧ ψ lying around, so that’s when this rule is useful. Compare this with the example sequent in Example 1.15. r ∨i1 says: to prove φ ∨ ψ, try proving φ. Again, in general it is harder to prove
φ than it is to prove φ ∨ ψ, so this will usually be useful only if you’ve already managed to prove φ. For example, if you want to prove q |− p ∨ q, you certainly won’t be able simply to use the rule
∨i1 , but ∨i2 will work. r ∨e has an excellent procedural interpretation. It says: if you have φ ∨ ψ, and you want to prove some χ, then try to prove χ from φ and from ψ in turn. (In those subproofs,
of course you can use the other prevailing premises as well.) r Similarly, →i says, if you want to prove φ → ψ, try proving ψ from φ (and the other prevailing premises). r ¬i says: to prove ¬φ, prove
⊥ from φ (and the other prevailing premises).
1.2 Natural deduction
The basic rules of natural deduction: introduction φ ψ φ∧ψ
φ φ∨ψ
elimination φ∧ψ φ
ψ φ∨ψ
φ∧ψ ψ
φ .. .
ψ .. .
χ ∨e
φ .. . ψ
φ φ→ψ ψ
φ .. . ⊥
¬ ⊥
φ ¬φ ⊥ ⊥ φ
(no introduction rule for ⊥)
Some useful derived rules: φ → ψ ¬ψ ¬φ
φ ¬¬φ
¬¬φ φ
¬φ .. . ⊥ φ
φ ∨ ¬φ
Figure 1.2. Natural deduction rules for propositional logic.
1 Propositional logic
At any stage of a proof, it is permitted to introduce any formula as assumption, by choosing a proof rule that opens a box. As we saw, natural deduction employs boxes to control the scope of
assumptions. When an assumption is introduced, a box is opened. Discharging assumptions is achieved by closing a box according to the pattern of its particular proof rule. It’s useful to make
assumptions by opening boxes. But don’t forget you have to close them in the manner prescribed by their proof rule. OK, but how do we actually go about constructing a proof? Given a sequent, you
write its premises at the top of your page and its conclusion at the bottom. Now, you’re trying to fill in the gap, which involves working simultaneously on the premises (to bring them towards the
conclusion) and on the conclusion (to massage it towards the premises). Look first at the conclusion. If it is of the form φ → ψ, then apply6 the rule →i. This means drawing a box with φ at the top
and ψ at the bottom. So your proof, which started out like this: .. . premises .. . φ→ψ now looks like this: .. . premises .. . φ
ψ φ→ψ
You still have to find a way of filling in the gap between the φ and the ψ. But you now have an extra formula to work with and you have simplified the conclusion you are trying to reach. 6
Except in situations such as p → (q → ¬r), p q → ¬r where →e produces a simpler proof.
1.2 Natural deduction
The proof rule ¬i is very similar to →i and has the same beneficial effect on your proof attempt. It gives you an extra premise to work with and simplifies your conclusion. At any stage of a proof,
several rules are likely to be applicable. Before applying any of them, list the applicable ones and think about which one is likely to improve the situation for your proof. You’ll find that →i and ¬i
most often improve it, so always use them whenever you can. There is no easy recipe for when to use the other rules; often you have to make judicious choices.
1.2.4 Provable equivalence Definition 1.25 Let φ and ψ be formulas of propositional logic. We say that φ and ψ are provably equivalent iff (we write ‘iff’ for ‘if, and only if’ in the sequel) the
sequents φ ψ and ψ φ are valid; that is, there is a proof of ψ from φ and another one going the other way around. As seen earlier, we denote that φ and ψ are provably equivalent by φ ψ. Note that, by
Remark 1.12, we could just as well have defined φ ψ to mean that the sequent (φ → ψ) ∧ (ψ → φ) is valid; it defines the same concept. Examples of provably equivalent formulas are ¬(p ∧ q) ¬q ∨ ¬p p → q
¬q → ¬p p ∧ q → p r ∨ ¬r
¬(p ∨ q) ¬q ∧ ¬p p → q ¬p ∨ q p ∧ q → r p → (q → r).
The reader should prove all of these six equivalences in natural deduction.
1.2.5 An aside: proof by contradiction Sometimes we can’t prove something directly in the sense of taking apart given assumptions and reasoning with their constituents in a constructive way. Indeed,
the proof system of natural deduction, summarised in Figure 1.2, specifically allows for indirect proofs that lack a constructive quality: for example, the rule ¬φ .. . ⊥ φ
1 Propositional logic
allows us to prove φ by showing that ¬φ leads to a contradiction. Although ‘classical logicians’ argue that this is valid, logicians of another kind, called ‘intuitionistic logicians,’ argue that to
prove φ you should do it directly, rather than by arguing merely that ¬φ is impossible. The two other rules on which classical and intuitionistic logicians disagree are φ ∨ ¬φ
¬¬φ φ
Intuitionistic logicians argue that, to show φ ∨ ¬φ, you have to show φ, or ¬φ. If neither of these can be shown, then the putative truth of the disjunction has no justification. Intuitionists reject
¬¬e since we have already used this rule to prove LEM and PBC from rules which the intuitionists do accept. In the exercises, you are asked to show why the intuitionists also reject PBC. Let us look
at a proof that shows up this difference, involving real numbers. Real numbers are floating point numbers like 23.54721, only some of them might actually be infinitely long such as
23.138592748500123950734 . . ., with no periodic behaviour after the decimal point. Given a positive real number a and a natural (whole) number b, we can calculate ab : it is just a times itself, b
times, so 22 = 2 · 2 = 4, 23 = 2 · 2 · 2 = 8 and so on. When b is a real number, we can also define ab , as follows. def We say that a0√= 1 and, for a non-zero rational number k/n, where n = 0, √ def
n we let ak/n = ak where n x is the real number y such that y n = x. From real analysis one knows that any real number b can be approximated by a sequence of rational numbers k0 /n0 , k1 /n1 , . . .
Then we define ab to be the real number approximated by the sequence ak0 /n0 , ak1 /n1 , . . . (In calculus, one can show that this ‘limit’ ab is unique and independent of the choice of approximating
sequence.) Also, one calls a real number irrational if it can’t be written in the form k/n for some integers k and n = 0. √ In the exercises you will be asked to find a semi-formal proof showing that
2 is irrational. We now present a proof of a fact about real numbers in the informal style used by mathematicians (this proof can be formalised as a natural deduction proof in the logic presented in
Chapter 2). The fact we prove is: Theorem 1.26 There exist irrational numbers a and b such that ab is rational. √ Proof: We choose b to be 2 and proceed by a case analysis. Either bb is irrational,
or it is not. (Thus, our proof uses ∨e on an instance of LEM.)
1.3 Propositional logic as a formal language
(i) Assume that bb is rational. Then √ this proof is easy since we can choose irrational numbers a and b to be 2 and see that ab is just bb which was assumed to be rational. (ii) Assume that bb is ir
rational. Then we change our strategy slightly and choose √ √2 a to be 2 . Clearly, a is irrational by the assumption of case (ii). But we know that b is irrational (this was known by the ancient
Greeks; see the proof outline in the exercises). So a and b are both irrational numbers and √ √2 b a =( 2 )
√ √ ( 2· 2)
√ 2 = ( 2) = 2
is rational, where we used the law (xy )z = x(y·z) .
Since the two cases above are exhaustive (either bb is irrational, or it isn’t) we have proven the theorem. 2 This proof is perfectly legitimate and mathematicians use arguments like that all the
time. The exhaustive nature of the case analysis above rests on the use of the rule LEM, which we use to prove that either b is rational or it is not. Yet, there is something puzzling about it.
Surely, we have secured the fact that there are irrational numbers a and b such that ab is rational, but are we in a position to specify an actual pair of such numbers satisfying this theorem? More
precisely, which of the pairs (a, b) above fulfils the assertion √ √ √ √2 √ of the theorem, the pair ( 2, 2), or the pair ( 2 , 2)? Our proof tells us nothing about which of them is the right choice;
it just says that at least one of them works. Thus, the intuitionists favour a calculus containing the introduction and elimination rules shown in Figure 1.2 and excluding the rule ¬¬e and the
derived rules. Intuitionistic logic turns out to have some specialised applications in computer science, such as modelling type-inference systems used in compilers or the staged execution of program
code; but in this text we stick to the full so-called classical logic which includes all the rules.
1.3 Propositional logic as a formal language In the previous section we learned about propositional atoms and how they can be used to build more complex logical formulas. We were deliberately
informal about that, for our main focus was on trying to understand the precise mechanics of the natural deduction rules. However, it should have been clear that the rules we stated are valid for any
formulas we can form, as long as they match the pattern required by the respective rule. For example,
1 Propositional logic
the application of the proof rule →e in 1
→e 1, 2
is equally valid if we substitute p with p ∨ ¬r and q with r → p: 1
p ∨ ¬r → (r → p) premise
p ∨ ¬r
→e 1, 2
This is why we expressed such rules as schemes with Greek symbols standing for generic formulas. Yet, it is time that we make precise the notion of ‘any formula we may form.’ Because this text
concerns various logics, we will introduce in (1.3) an easy formalism for specifying well-formed formulas. In general, we need an unbounded supply of propositional atoms p, q, r, . . ., or p1 , p2 ,
p3 , . . . You should not be too worried about the need for infinitely many such symbols. Although we may only need finitely many of these propositions to describe a property of a computer program
successfully, we cannot specify how many such atomic propositions we will need in any concrete situation, so having infinitely many symbols at our disposal is a cheap way out. This can be compared
with the potentially infinite nature of English: the number of grammatically correct English sentences is infinite, but finitely many such sentences will do in whatever situation you might be in
(writing a book, attending a lecture, listening to the radio, having a dinner date, . . . ). Formulas in our propositional logic should certainly be strings over the alphabet {p, q, r, . . . } ∪ {p1
, p2 , p3 , . . . } ∪ {¬, ∧, ∨, →, (, )}. This is a trivial observation and as such is not good enough for what we are trying to capture. For example, the string (¬)() ∨ pq → is a word over that
alphabet, yet, it does not seem to make a lot of sense as far as propositional logic is concerned. So what we have to define are those strings which we want to call formulas. We call such formulas
well-formed. Definition 1.27 The well-formed formulas of propositional logic are those which we obtain by using the construction rules below, and only those, finitely many times:
1.3 Propositional logic as a formal language
atom: Every propositional atom p, q, r, . . . and p1 , p2 , p3 , . . . is a wellformed formula. ¬: If φ is a well-formed formula, then so is (¬φ). ∧: If φ and ψ are well-formed formulas, then so is
(φ ∧ ψ). ∨: If φ and ψ are well-formed formulas, then so is (φ ∨ ψ). →: If φ and ψ are well-formed formulas, then so is (φ → ψ). It is most crucial to realize that this definition is the one a
computer would expect and that we did not make use of the binding priorities agreed upon in the previous section. Convention. In this section we act as if we are a rigorous computer and we call
formulas well-formed iff they can be deduced to be so using the definition above. Further, note that the condition ‘and only those’ in the definition above rules out the possibility of any other means
of establishing that formulas are well-formed. Inductive definitions, like the one of well-formed propositional logic formulas above, are so frequent that they are often given by a defining grammar in
Backus Naur form (BNF). In that form, the above definition reads more compactly as φ ::= p | (¬φ) | (φ ∧ φ) | (φ ∨ φ) | (φ → φ)
where p stands for any atomic proposition and each occurrence of φ to the right of ::= stands for any already constructed formula. So how can we show that a string is a well-formed formula? For
example, how do we answer this for φ being (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))) ?
Such reasoning is greatly facilitated by the fact that the grammar in (1.3) satisfies the inversion principle, which means that we can invert the process of building formulas: although the grammar
rules allow for five different ways of constructing more complex formulas – the five clauses in (1.3) – there is always a unique clause which was used last. For the formula above, this last operation
was an application of the fifth clause, for φ is an implication with the assumption ((¬p) ∧ q) and conclusion (p ∧ (q ∨ (¬r))). By applying the inversion principle to the assumption, we see that it is
a conjunction of (¬p) and q. The former has been constructed using the second clause and is well-formed since p is well-formed by the first clause in (1.3). The latter is well-formed for the same
reason. Similarly, we can apply the inversion
1 Propositional logic
Figure 1.3. A parse tree representing a well-formed formula.
principle to the conclusion (p ∧ (q ∨ (¬r))), inferring that it is indeed wellformed. In summary, the formula in (1.4) is well-formed. For us humans, dealing with brackets is a tedious task. The
reason we need them is that formulas really have a tree-like structure, although we prefer to represent them in a linear way. In Figure 1.3 you can see the parse tree7 of the well-formed formula φ in
(1.4). Note how brackets become unnecessary in this parse tree since the paths and the branching structure of this tree remove any possible ambiguity in interpreting φ. In representing φ as a linear
string, the branching structure of the tree is retained by the insertion of brackets as done in the definition of well-formed formulas. So how would you go about showing that a string of symbols ψ is
not wellformed? At first sight, this is a bit trickier since we somehow have to make sure that ψ could not have been obtained by any sequence of construction rules. Let us look at the formula (¬)() ∨
pq → from above. We can decide this matter by being very observant. The string (¬)() ∨ pq → contains ¬) and ¬ cannot be the rightmost symbol of a well-formed formula (check all the rules to verify
this claim!); but the only time we can put a ‘)’ to the right of something is if that something is a well-formed formula (again, check all the rules to see that this is so). Thus, (¬)() ∨ pq → is not
well-formed. Probably the easiest way to verify whether some formula φ is well-formed is by trying to draw its parse tree. In this way, you can verify that the 7
We will use this name without explaining it any further and are confident that you will understand its meaning through the examples.
1.3 Propositional logic as a formal language
formula in (1.4) is well-formed. In Figure 1.3 we see that its parse tree has → as its root, expressing that the formula is, at its top level, an implication. Using the grammar clause for
implication, it suffices to show that the left and right subtrees of this root node are well-formed. That is, we proceed in a top-down fashion and, in this case, successfully. Note that the parse trees
of well-formed formulas have either an atom as root (and then this is all there is in the tree), or the root contains ¬, ∨, ∧ or →. In the case of ¬ there is only one subtree coming out of the root.
In the cases ∧, ∨ or → we must have two subtrees, each of which must behave as just described; this is another example of an inductive definition. Thinking in terms of trees will help you understand
standard notions in logic, for example, the concept of a subformula. Given the well-formed formula φ above, its subformulas are just the ones that correspond to the subtrees of its parse tree in
Figure 1.3. So we can list all its leaves p, q (occurring twice), and r, then (¬p) and ((¬p) ∧ q) on the left subtree of → and (¬r), (q ∨ (¬r)) and ((p ∧ (q ∨ (¬p)))) on the right subtree of →. The
whole tree is a subtree of itself as well. So we can list all nine subformulas of φ as p q r (¬p) ((¬p) ∧ q) (¬r) (q ∨ (¬r)) ((p ∧ (q ∨ (¬r)))) (((¬p) ∧ q) → (p ∧ (q ∨ (¬r)))). Let us consider the
tree in Figure 1.4. Why does it represent a well-formed formula? All its leaves are propositional atoms (p twice, q and r), all branching nodes are logical connectives (¬ twice, ∧, ∨ and →) and the
numbers of subtrees are correct in all those cases (one subtree for a ¬ node and two subtrees for all other non-leaf nodes). How do we obtain the linear representation of this formula? If we ignore
brackets, then we are seeking nothing but the in-order representation of this tree as a list8 . The resulting well-formed formula is ((¬(p ∨ (q → (¬p)))) ∧ r). 8
The other common ways of flattening trees to lists are preordering and postordering. See any text on binary trees as data structures for further details.
1 Propositional logic
Figure 1.4. Given: a tree; wanted: its linear representation as a logical formula.
The tree in Figure 1.21 on page 82, however, does not represent a wellformed formula for two reasons. First, the leaf ∧ (and a similar argument applies to the leaf ¬), the left subtree of the node →,
is not a propositional atom. This could be fixed by saying that we decided to leave the left and right subtree of that node unspecified and that we are willing to provide those now. However, the second
reason is fatal. The p node is not a leaf since it has a subtree, the node ¬. This cannot make sense if we think of the entire tree as some logical formula. So this tree does not represent a
well-formed logical formula.
1.4 Semantics of propositional logic 1.4.1 The meaning of logical connectives In the second section of this chapter, we developed a calculus of reasoning which could verify that sequents of the form
φ1 , φ2 , . . . , φn ψ are valid, which means: from the premises φ1 , φ2 , . . . , φn , we may conclude ψ. In this section we give another account of this relationship between the premises φ1 , φ2 ,
. . . , φn and the conclusion ψ. To contrast with the sequent
1.4 Semantics of propositional logic
above, we define a new relationship, written φ1 , φ2 , . . . , φn ψ. This account is based on looking at the ‘truth values’ of the atomic formulas in the premises and the conclusion; and at how the
logical connectives manipulate these truth values. What is the truth value of a declarative sentence, like sentence (3) ‘Every even natural number > 2 is the sum of two prime numbers’ ? Well,
declarative sentences express a fact about the real world, the physical world we live in, or more abstract ones such as computer models, or our thoughts and feelings. Such factual statements either
match reality (they are true), or they don’t (they are false). If we combine declarative sentences p and q with a logical connective, say ∧, then the truth value of p ∧ q is determined by three
things: the truth value of p, the truth value of q and the meaning of ∧. The meaning of ∧ is captured by the observation that p ∧ q is true iff p and q are both true; otherwise p ∧ q is false. Thus,
as far as ∧ is concerned, it needs only to know whether p and q are true, it does not need to know what p and q are actually saying about the world out there. This is also the case for all the other
logical connectives and is the reason why we can compute the truth value of a formula just by knowing the truth values of the atomic propositions occurring in it. Definition 1.28 1. The set of truth
values contains two elements T and F, where T represents ‘true’ and F represents ‘false’. 2. A valuation or model of a formula φ is an assignment of each propositional atom in φ to a truth value.
Example 1.29 The map which assigns T to q and F to p is a valuation for p ∨ ¬q. Please list the remaining three valuations for this formula. We can think of the meaning of ∧ as a function of two
arguments; each argument is a truth value and the result is again such a truth value. We specify this function in a table, called the truth table for conjunction, which you can see in Figure 1.5. In
the first column, labelled φ, we list all possible φ T T F F
ψ φ∧ψ T T F F T F F F
Figure 1.5. The truth table for conjunction, the logical connective ∧.
1 Propositional logic
φ T T F F φ T T F F
ψ φ∧ψ T T F F T F F F
ψ φ→ψ T T F F T T F T
φ T T F F φ ¬φ T F F T
ψ φ∨ψ T T F T T T F F T
⊥ F
Figure 1.6. The truth tables for all the logical connectives discussed so far.
truth values of φ. Actually we list them twice since we also have to deal with another formula ψ, so the possible number of combinations of truth values for φ and ψ equals 2 · 2 = 4. Notice that the
four pairs of φ and ψ values in the first two columns really exhaust all those possibilities (TT, TF, FT and FF). In the third column, we list the result of φ ∧ ψ according to the truth values of φ
and ψ. So in the first line, where φ and ψ have value T, the result is T again. In all other lines, the result is F since at least one of the propositions φ or ψ has value F. In Figure 1.6 you find the
truth tables for all logical connectives of propositional logic. Note that ¬ turns T into F and vice versa. Disjunction is the mirror image of conjunction if we swap T and F, namely, a disjunction
returns F iff both arguments are equal to F, otherwise (= at least one of the arguments equals T) it returns T. The behaviour of implication is not quite as intuitive. Think of the meaning of → as
checking whether truth is being preserved. Clearly, this is not the case when we have T → F, since we infer something that is false from something that is true. So the second entry in the column φ →
ψ equals F. On the other hand, T → T obviously preserves truth, but so do the cases F → T and F → F, because there is no truth to be preserved in the first place as the assumption of the implication
is false. If you feel slightly uncomfortable with the semantics (= the meaning) of →, then it might be good to think of φ → ψ as an abbreviation of the formula ¬φ ∨ ψ as far as meaning is concerned;
these two formulas are very different syntactically and natural deduction treats them differently as well. But using the truth tables for ¬ and ∨ you can check that φ → ψ evaluates
1.4 Semantics of propositional logic
to T iff ¬φ ∨ ψ does so. This means that φ → ψ and ¬φ ∨ ψ are semantically equivalent; more on that in Section 1.5. Given a formula φ which contains the propositional atoms p1 , p2 , . . . , pn , we
can construct a truth table for φ, at least in principle. The caveat is that this truth table has 2n many lines, each line listing a possible combination of truth values for p1 , p2 , . . . , pn ;
and for large n this task is impossible to complete. Our aim is thus to compute the value of φ for each of these 2n cases for moderately small values of n. Let us consider the example φ in Figure
1.3. It involves three propositional atoms (n = 3) so we have 23 = 8 cases to consider. We illustrate how things go for one particular case, namely for the valuation in which q evaluates to F; and p
and r evaluate to T. What does ¬p ∧ q → p ∧ (q ∨ ¬r) evaluate to? Well, the beauty of our semantics is that it is compositional. If we know the meaning of the subformulas ¬p ∧ q and p ∧ (q ∨ ¬r),
then we just have to look up the appropriate line of the → truth table to find the value of φ, for φ is an implication of these two subformulas. Therefore, we can do the calculation by traversing the
parse tree of φ in a bottom-up fashion. We know what its leaves evaluate to since we stated what the atoms p, q and r evaluated to. Because the meaning of p is T, we see that ¬p computes to F. Now q
is assumed to represent F and the conjunction of F and F is F. Thus, the left subtree of the node → evaluates to F. As for the right subtree of →, r stands for T so ¬r computes to F and q means F, so
the disjunction of F and F is still F. We have to take that result, F, and compute its conjunction with the meaning of p which is T. Since the conjunction of T and F is F, we get F as the meaning of
the right subtree of →. Finally, to evaluate the meaning of φ, we compute F → F which is T. Figure 1.7 shows how the truth values propagate upwards to reach the root whose associated truth value is
the truth value of φ given the meanings of p, q and r above. It should now be quite clear how to build a truth table for more complex formulas. Figure 1.8 contains a truth table for the formula (p →
¬q) → (q ∨ ¬p). To be more precise, the first two columns list all possible combinations of values for p and q. The next two columns compute the corresponding values for ¬p and ¬q. Using these four
columns, we may compute the column for p → ¬q and q ∨ ¬p. To do so we think of the first and fourth columns as the data for the → truth table and compute the column of p → ¬q accordingly. For example,
in the first line p is T and ¬q is F so the entry for p → ¬q is T → F = F by definition of the meaning of →. In this fashion, we can fill out the rest of the fifth column. Column 6 works similarly, only
we now need to look up the truth table for ∨ with columns 2 and 3 as input.
1 Propositional logic
Figure 1.7. The evaluation of a logical formula under a given valuation.
p T T F F
q ¬p ¬q p → ¬q q ∨ ¬p (p → ¬q) → (q ∨ ¬p) T F F F T T F F T T F F T T F T T T F T T T T T
Figure 1.8. An example of a truth table for a more complex logical formula.
Finally, column 7 results from applying the truth table of → to columns 5 and 6.
1.4.2 Mathematical induction Here is a little anecdote about the German mathematician Gauss who, as a pupil at age 8, did not pay attention in class (can you imagine?), with the result that his
teacher made him sum up all natural numbers from 1 to 100. The story has it that Gauss came up with the correct answer 5050 within seconds, which infuriated his teacher. How did Gauss do it? Well,
possibly he knew that 1 + 2 + 3 + 4 + ··· + n =
n · (n + 1) 2
1.4 Semantics of propositional logic
for all natural numbers n.9 Thus, taking n = 100, Gauss could easily calculate: 100 · 101 = 5050. 1 + 2 + 3 + 4 + · · · + 100 = 2 Mathematical induction allows us to prove equations, such as the one
in (1.5), for arbitrary n. More generally, it allows us to show that every natural number satisfies a certain property. Suppose we have a property M which we think is true of all natural numbers. We
write M (5) to say that the property is true of 5, etc. Suppose that we know the following two things about the property M : 1. Base case: The natural number 1 has property M , i.e. we have a proof
of M (1). 2. Inductive step: If n is a natural number which we assume to have property M (n), then we can show that n + 1 has property M (n + 1); i.e. we have a proof of M (n) → M (n + 1).
Definition 1.30 The principle of mathematical induction says that, on the grounds of these two pieces of information above, every natural number n has property M (n). The assumption of M (n) in the
inductive step is called the induction hypothesis. Why does this principle make sense? Well, take any natural number k. If k equals 1, then k has property M (1) using the base case and so we are
done. Otherwise, we can use the inductive step, applied to n = 1, to infer that 2 = 1 + 1 has property M (2). We can do that using →e, for we know that 1 has the property in question. Now we use that
same inductive step on n = 2 to infer that 3 has property M (3) and we repeat this until we reach n = k (see Figure 1.9). Therefore, we should have no objections about using the principle of
mathematical induction for natural numbers. Returning to Gauss’ example we claim that the sum 1 + 2 + 3 + 4 + · · · + n equals n · (n + 1)/2 for all natural numbers n. Theorem 1.31 The sum 1 + 2 + 3
+ 4 + · · · + n equals n · (n + 1)/2 for all natural numbers n. 9
There is another way of finding the sum 1 + 2 + · · · + 100, which works like this: write the sum backwards, as 100 + 99 + · · · + 1. Now add the forwards and backwards versions, obtaining 101 + 101 +
· · · + 101 (100 times), which is 10100. Since we added the sum to itself, we now divide by two to get the answer 5050. Gauss probably used this method; but the method of mathematical induction that
we explore in this section is much more powerful and can be applied in a wide variety of situations.
1 Propositional logic
M →
(n )
(n an d
M (n ) M in g us 1) + (n M ov e pr W e
pr W e
pr W e
an d 1) − (n M us in g (n )
ov e
M ov e
ov e pr W e
... 1
(3 )
(2 ) M
(1 ) M ov e pr W e
us in g
us in g
an d
an d
(2 )
(1 )
(3 )
(2 )
(n )
... n
Figure 1.9. How the principle of mathematical induction works. By proving just two facts, M (1) and M (n) → M (n + 1) for a formal (and unconstrained) parameter n, we are able to deduce M (k) for
each natural number k.
Proof: We use mathematical induction. In order to reveal the fine structure of our proof we write LHSn for the expression 1 + 2 + 3 + 4 + · · · + n and RHSn for n · (n + 1)/2. Thus, we need to show
LHSn = RHSn for all n ≥ 1. Base case: If n equals 1, then LHS1 is just 1 (there is only one summand), which happens to equal RHS1 = 1 · (1 + 1)/2. Inductive step: Let us assume that LHSn = RHSn .
Recall that this assumption is called the induction hypothesis; it is the driving force of our argument. We need to show LHSn+1 = RHSn+1 , i.e. that the longer sum 1 + 2 + 3 + 4 + · · · + (n + 1)
equals (n + 1) · ((n + 1) + 1)/2. The key observation is that the sum 1 + 2 + 3 + 4 + · · · + (n + 1) is nothing but the sum (1 + 2 + 3 + 4 + · · · + n) + (n + 1) of two summands, where the first one
is the sum of our induction hypothesis. The latter says that 1 + 2 + 3 + 4 + · · · + n equals n · (n + 1)/2, and we are certainly entitled to substitute equals for equals in our reasoning. Thus, we
compute LHSn+1 = 1 + 2 + 3 + 4 + · · · + (n + 1) = LHSn + (n + 1) regrouping the sum
1.4 Semantics of propositional logic
= RHSn + (n + 1) by our induction hypothesis = = = =
n·(n+1) + (n + 1) 2 n·(n+1) + 2·(n+1) arithmetic 2 2 (n+2)·(n+1) arithmetic 2 ((n+1)+1)·(n+1) arithmetic 2
= RHSn+1 . Since we successfully showed the base case and the inductive step, we can use mathematical induction to infer that all natural numbers n have the property stated in the theorem above. 2
Actually, there are numerous variations of this principle. For example, we can think of a version in which the base case is n = 0, which would then cover all natural numbers including 0. Some
statements hold only for all natural numbers, say, greater than 3. So you would have to deal with a base case 4, but keep the version of the inductive step (see the exercises for such an example).
The use of mathematical induction typically suceeds on properties M (n) that involve inductive definitions (e.g. the definition of k l with l ≥ 0). Sentence (3) on page 2 suggests there may be true
properties M (n) for which mathematical induction won’t work. Course-of-values induction. There is a variant of mathematical induction in which the induction hypothesis for proving M (n + 1) is not
just M (n), but the conjunction M (1) ∧ M (2) ∧ · · · ∧ M (n). In that variant, called courseof-values induction, there doesn’t have to be an explicit base case at all – everything can be done in the
inductive step. How can this work without a base case? The answer is that the base case is implicitly included in the inductive step. Consider the case n = 3: the inductive-step instance is M (1) ∧ M
(2) ∧ M (3) → M (4). Now consider n = 1: the inductive-step instance is M (1) → M (2). What about the case when n equals 0? In this case, there are zero formulas on the left of the →, so we have to
prove M (1) from nothing at all. The inductive-step instance is simply the obligation to show M (1). You might find it useful to modify Figure 1.9 for course-of-values induction. Having said that the
base case is implicit in course-of-values induction, it frequently turns out that it still demands special attention when you get inside trying to prove the inductive case. We will see precisely this
in the two applications of course-of-values induction in the following pages.
1 Propositional logic
Figure 1.10. A parse tree with height 5.
In computer science, we often deal with finite structures of some kind, data structures, programs, files etc. Often we need to show that every instance of such a structure has a certain property. For
example, the well-formed formulas of Definition 1.27 have the property that the number of ‘(’ brackets in a particular formula equals its number of ‘)’ brackets. We can use mathematical induction on
the domain of natural numbers to prove this. In order to succeed, we somehow need to connect well-formed formulas to natural numbers. Definition 1.32 Given a well-formed formula φ, we define its
height to be 1 plus the length of the longest path of its parse tree. For example, consider the well-formed formulas in Figures 1.3, 1.4 and 1.10. Their heights are 5, 6 and 5, respectively. In
Figure 1.3, the longest path goes from → to ∧ to ∨ to ¬ to r, a path of length 4, so the height is 4 + 1 = 5. Note that the height of atoms is 1 + 0 = 1. Since every well-formed formula has finite
height, we can show statements about all well-formed formulas by mathematical induction on their height. This trick is most often called structural induction, an important reasoning technique in
computer science. Using the notion of the height of a parse tree, we realise that structural induction is just a special case of course-of-values induction.
1.4 Semantics of propositional logic
Theorem 1.33 For every well-formed propositional logic formula, the number of left brackets is equal to the number of right brackets. Proof: We proceed by course-of-values induction on the height of
wellformed formulas φ. Let M (n) mean ‘All formulas of height n have the same number of left and right brackets.’ We assume M (k) for each k < n and try to prove M (n). Take a formula φ of height n.
r Base case: Then n = 1. This means that φ is just a propositional atom. So there are no left or right brackets, 0 equals 0. r Course-of-values inductive step: Then n > 1 and so the root of the parse
tree of φ must be ¬, →, ∨ or ∧, for φ is well-formed. We assume that it is →, the other three cases are argued in a similar way. Then φ equals (φ1 → φ2 ) for some wellformed formulas φ1 and φ2 (of
course, they are just the left, respectively right, linear representations of the root’s two subtrees). It is clear that the heights of φ1 and φ2 are strictly smaller than n. Using the induction
hypothesis, we therefore conclude that φ1 has the same number of left and right brackets and that the same is true for φ2 . But in (φ1 → φ2 ) we added just two more brackets, one ‘(’ and one ‘)’.
Thus, the number of occurrences of ‘(’ and ‘)’ in φ is the same. 2
The formula (p → (q ∧ ¬r)) illustrates why we could not prove the above directly with mathematical induction on the height of formulas. While this formula has height 4, its two subtrees have heights
1 and 3, respectively. Thus, an induction hypothesis for height 3 would have worked for the right subtree but failed for the left subtree.
1.4.3 Soundness of propositional logic The natural deduction rules make it possible for us to develop rigorous threads of argumentation, in the course of which we arrive at a conclusion ψ assuming
certain other propositions φ1 , φ2 , . . . , φn . In that case, we said that the sequent φ1 , φ2 , . . . , φn ψ is valid. Do we have any evidence that these rules are all correct in the sense that
valid sequents all ‘preserve truth’ computed by our truth-table semantics? Given a proof of φ1 , φ2 , . . . , φn ψ, is it conceivable that there is a valuation in which ψ above is false although all
propositions φ1 , φ2 , . . . , φn are true? Fortunately, this is not the case and in this subsection we demonstrate why this is so. Let us suppose that some proof in our natural deduction calculus
has established that the sequent φ1 , φ2 , . . . , φn ψ is valid. We need to show: for all valuations in which all propositions φ1 , φ2 , . . . , φn evaluate to T, ψ evaluates to T as well.
1 Propositional logic
Definition 1.34 If, for all valuations in which all φ1 , φ2 , . . . , φn evaluate to T, ψ evaluates to T as well, we say that φ1 , φ2 , . . . , φn ψ holds and call the semantic entailment relation.
Let us look at some examples of this notion. 1. Does p ∧ q p hold? Well, we have to inspect all assignments of truth values to p and q; there are four of these. Whenever such an assignment computes T
for p ∧ q we need to make sure that p is true as well. But p ∧ q computes T only if p and q are true, so p ∧ q p is indeed the case. 2. What about the relationship p ∨ q p? There are three
assignments for which p ∨ q computes T, so p would have to be true for all of these. However, if we assign T to q and F to p, then p ∨ q computes T, but p is false. Thus, p ∨ q p does not hold. 3.
What if we modify the above to ¬q, p ∨ q p? Notice that we have to be concerned only about valuations in which ¬q and p ∨ q evaluate to T. This forces q to be false, which in turn forces p to be
true. Hence ¬q, p ∨ q p is the case. 4. Note that p q ∨ ¬q holds, despite the fact that no atomic proposition on the right of occurs on the left of .
From the discussion above we realize that a soundness argument has to show: if φ1 , φ2 , . . . , φn ψ is valid, then φ1 , φ2 , . . . , φn ψ holds. Theorem 1.35 (Soundness) Let φ1 , φ2 , . . . , φn
and ψ be propositional logic formulas. If φ1 , φ2 , . . . , φn ψ is valid, then φ1 , φ2 , . . . , φn ψ holds. Proof: Since φ1 , φ2 , . . . , φn ψ is valid we know there is a proof of ψ from the
premises φ1 , φ2 , . . . , φn . We now do a pretty slick thing, namely, we reason by mathematical induction on the length of this proof ! The length of a proof is just the number of lines it
involves. So let us be perfectly clear about what it is we mean to show. We intend to show the assertion M (k): ‘For all sequents φ1 , φ2 , . . . , φn ψ (n ≥ 0) which have a proof of length k, it is
the case that φ1 , φ2 , . . . , φn ψ holds.’
by course-of-values induction on the natural number k. This idea requires
1.4 Semantics of propositional logic
some work, though. The sequent p ∧ q → r p → (q → r) has a proof 1
p∧q →r
∧i 2, 3
→e 1, 4
→i 3−5
p → (q → r) →i 2−6
but if we remove the last line or several of the last lines, we no longer have a proof as the outermost box does not get closed. We get a complete proof, though, by removing the last line and
re-writing the assumption of the outermost box as a premise: 1
p∧q →r
∧i 2, 3
→e 1, 4
→i 3−5
This is a proof of the sequent p ∧ q → r, p p → r. The induction hypothesis then ensures that p ∧ q → r, p p → r holds. But then we can also reason that p ∧ q → r p → (q → r) holds as well – why?
Let’s proceed with our proof by induction. We assume M (k ) for each k < k and we try to prove M (k). Base case: a one-line proof. If the proof has length 1 (k = 1), then it must be of the form 1
φ premise
since all other rules involve more than one line. This is the case when n = 1 and φ1 and ψ equal φ, i.e. we are dealing with the sequent φ φ. Of course, since φ evaluates to T so does φ. Thus, φ φ
holds as claimed.
1 Propositional logic
Course-of-values inductive step: Let us assume that the proof of the sequent φ1 , φ2 , . . . , φn ψ has length k and that the statement we want to prove is true for all numbers less than k. Our proof
has the following structure: 1 2
φ1 premise φ2 premise .. .
φn premise .. .
ψ justification
There are two things we don’t know at this point. First, what is happening in between those dots? Second, what was the last rule applied, i.e. what is the justification of the last line? The first
uncertainty is of no concern; this is where mathematical induction demonstrates its power. The second lack of knowledge is where all the work sits. In this generality, there is simply no way of
knowing which rule was applied last, so we need to consider all such rules in turn. 1. Let us suppose that this last rule is ∧i. Then we know that ψ is of the form ψ1 ∧ ψ2 and the justification in
line k refers to two lines further up which have ψ1 , respectively ψ2 , as their conclusions. Suppose that these lines are k1 and k2 . Since k1 and k2 are smaller than k, we see that there exist
proofs of the sequents φ1 , φ2 , . . . , φn ψ1 and φ1 , φ2 , . . . , φn ψ2 with length less than k – just take the first k1 , respectively k2 , lines of our original proof. Using the induction
hypothesis, we conclude that φ1 , φ2 , . . . , φn ψ1 and φ1 , φ2 , . . . , φn ψ2 holds. But these two relations imply that φ1 , φ2 , . . . , φn ψ1 ∧ ψ2 holds as well – why? 2. If ψ has been shown
using the rule ∨e, then we must have proved, assumed or given as a premise some formula η1 ∨ η2 in some line k with k < k, which was referred to via ∨e in the justification of line k. Thus, we have a
shorter proof of the sequent φ1 , φ2 , . . . , φn η1 ∨ η2 within that proof, obtained by turning all assumptions of boxes that are open at line k into premises. In a similar way we obtain proofs of
the sequents φ1 , φ2 , . . . , φn , η1 ψ and φ1 , φ2 , . . . , φn , η2 ψ from the case analysis of ∨e. By our induction hypothesis, we conclude that the relations φ1 , φ2 , . . . , φn η1 ∨ η2 , φ1 ,
φ2 , . . . , φn , η1 ψ and φ1 , φ2 , . . . , φn , η2 ψ hold. But together these three relations then force that φ1 , φ2 , . . . , φn ψ holds as well – why? 3. You can guess by now that the rest of
the argument checks each possible proof rule in turn and ultimately boils down to verifying that our natural deduction
1.4 Semantics of propositional logic
rules behave semantically in the same way as their corresponding truth tables 2 evaluate. We leave the details as an exercise.
The soundness of propositional logic is useful in ensuring the non-existence of a proof for a given sequent. Let’s say you try to prove that φ1 , φ2 , . . . , φ2 ψ is valid, but that your best efforts
won’t succeed. How could you be sure that no such proof can be found? After all, it might just be that you can’t find a proof even though there is one. It suffices to find a valuation in which φi
evaluate to T whereas ψ evaluates to F. Then, by definition of , we don’t have φ1 , φ2 , . . . , φ2 ψ. Using soundness, this means that φ1 , φ2 , . . . , φ2 ψ cannot be valid. Therefore, this sequent
does not have a proof. You will practice this method in the exercises.
1.4.4 Completeness of propositional logic In this subsection, we hope to convince you that the natural deduction rules of propositional logic are complete: whenever φ1 , φ2 , . . . , φn ψ holds, then
there exists a natural deduction proof for the sequent φ1 , φ2 , . . . , φn ψ. Combined with the soundness result of the previous subsection, we then obtain φ1 , φ2 , . . . , φn ψ is valid iff φ1 , φ2
, . . . , φn ψ holds. This gives you a certain freedom regarding which method you prefer to use. Often it is much easier to show one of these two relationships (although neither of the two is
universally better, or easier, to establish). The first method involves a proof search, upon which the logic programming paradigm is based. The second method typically forces you to compute a truth
table which is exponential in the size of occurring propositional atoms. Both methods are intractable in general but particular instances of formulas often respond differently to treatment under these
two methods. The remainder of this section is concerned with an argument saying that if φ1 , φ2 , . . . , φn ψ holds, then φ1 , φ2 , . . . , φn ψ is valid. Assuming that φ1 , φ2 , . . . , φn ψ holds,
the argument proceeds in three steps: Step 1: We show that φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))) holds. Step 2: We show that φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))) is valid. Step 3: Finally,
we show that φ1 , φ2 , . . . , φn ψ is valid. The first and third steps are quite easy; all the real work is done in the second one.
1 Propositional logic
50 F →
F →
T φ1
F →
T φ2
F →
T φn−1
φn ψ
Figure 1.11. The only way this parse tree can evaluate to F. We represent parse trees for φ1 , φ2 , . . . , φn as triangles as their internal structure does not concern us here.
Step 1: Definition 1.36 A formula of propositional logic φ is called a tautology iff it evaluates to T under all its valuations, i.e. iff φ. Supposing that φ1 , φ2 , . . . , φn ψ holds, let us verify
that φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))) is indeed a tautology. Since the latter formula is a nested implication, it can evaluate to F only if all φ1 , φ2 ,. . .,φn evaluate to T and ψ
evaluates to F; see its parse tree in Figure 1.11. But this contradicts the fact that φ1 , φ2 , . . . , φn ψ holds. Thus, φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))) holds. Step 2: Theorem 1.37 If η
holds, then η is valid. In other words, if η is a tautology, then η is a theorem. This step is the hard one. Assume that η holds. Given that η contains n distinct propositional atoms p1 , p2 , . . .
, pn we know that η evaluates to T for all 2n lines in its truth table. (Each line lists a valuation of η.) How can we use this information to construct a proof for η? In some cases this can be done
quite easily by taking a very good look at the concrete structure of η. But here we somehow have to come up with a uniform way of building such a proof. The key insight is to ‘encode’ each line in
the truth table of η
1.4 Semantics of propositional logic
as a sequent. Then we construct proofs for these 2n sequents and assemble them into a proof of η. Proposition 1.38 Let φ be a formula such that p1 , p2 , . . . , pn are its only propositional atoms.
Let l be any line number in φ’s truth table. For all 1 ≤ i ≤ n let pˆi be pi if the entry in line l of pi is T, otherwise pˆi is ¬pi . Then we have 1. pˆ1 , pˆ2 , . . . , pˆn φ is provable if the
entry for φ in line l is T 2. pˆ1 , pˆ2 , . . . , pˆn ¬φ is provable if the entry for φ in line l is F
Proof: This proof is done by structural induction on the formula φ, that is, mathematical induction on the height of the parse tree of φ. 1. If φ is a propositional atom p, we need to show that p p
and ¬p ¬p. These have one-line proofs. 2. If φ is of the form ¬φ1 we again have two cases to consider. First, assume that φ evaluates to T. In this case φ1 evaluates to F. Note that φ1 has the same
atomic propositions as φ. We may use the induction hypothesis on φ1 to conclude that pˆ1 , pˆ2 , . . . , pˆn ¬φ1 ; but ¬φ1 is just φ, so we are done. Second, if φ evaluates to F, then φ1 evaluates to
T and we get pˆ1 , pˆ2 , . . . , pˆn φ1 by induction. Using the rule ¬¬i, we may extend the proof of pˆ1 , pˆ2 , . . . , pˆn φ1 to one for pˆ1 , pˆ2 , . . . , pˆn ¬¬φ1 ; but ¬¬φ1 is just ¬φ, so again
we are done.
The remaining cases all deal with two subformulas: φ equals φ1 ◦ φ2 , where ◦ is →, ∧ or ∨. In all these cases let q1 , . . . , ql be the propositional atoms of φ1 and r1 , . . . , rk be the
propositional atoms of φ2 . Then we certainly have {q1 , . . . , ql } ∪ {r1 , . . . , rk } = {p1 , . . . , pn }. Therefore, whenever qˆ1 , . . . , qˆl ψ1 and rˆ1 , . . . , rˆk ψ2 are valid so is pˆ1
, . . . , pˆn ψ1 ∧ ψ2 using the rule ∧i. In this way, we can use our induction hypothesis and only owe proofs that the conjunctions we conclude allow us to prove the desired conclusion for φ or ¬φ as
the case may be. 3. To wit, let φ be φ1 → φ2 . If φ evaluates to F, then we know that φ1 evaluates to T and φ2 to F. Using our induction hypothesis, we have qˆ1 , . . . , qˆl φ1 and rˆ1 , . . . , rˆk
¬φ2 , so pˆ1 , . . . , pˆn φ1 ∧ ¬φ2 follows. We need to show pˆ1 , . . . , pˆn ¬(φ1 → φ2 ); but using pˆ1 , . . . , pˆn φ1 ∧ ¬φ2 , this amounts to proving the sequent φ1 ∧ ¬φ2 ¬(φ1 → φ2 ), which we
leave as an exercise. If φ evaluates to T, then we have three cases. First, if φ1 evaluates to F and φ2 to F, then we get, by our induction hypothesis, that qˆ1 , . . . , qˆl ¬φ1 and rˆ1 , . . . ,
rˆk ¬φ2 , so pˆ1 , . . . , pˆn ¬φ1 ∧ ¬φ2 follows. Again, we need only to show the sequent ¬φ1 ∧ ¬φ2 φ1 → φ2 , which we leave as an exercise. Second, if φ1 evaluates to F and φ2 to T, we use our
induction hypothesis to arrive at
1 Propositional logic
pˆ1 , . . . , pˆn ¬φ1 ∧ φ2 and have to prove ¬φ1 ∧ φ2 φ1 → φ2 , which we leave as an exercise. Third, if φ1 and φ2 evaluate to T, we arrive at pˆ1 , . . . , pˆn φ1 ∧ φ2 , using our induction
hypothesis, and need to prove φ1 ∧ φ2 φ1 → φ2 , which we leave as an exercise as well. 4. If φ is of the form φ1 ∧ φ2 , we are again dealing with four cases in total. First, if φ1 and φ2 evaluate to
T, we get qˆ1 , . . . , qˆl φ1 and rˆ1 , . . . , rˆk φ2 by our induction hypothesis, so pˆ1 , . . . , pˆn φ1 ∧ φ2 follows. Second, if φ1 evaluates to F and φ2 to T, then we get pˆ1 , . . . , pˆn ¬φ1
∧ φ2 using our induction hypothesis and the rule ∧i as above and we need to prove ¬φ1 ∧ φ2 ¬(φ1 ∧ φ2 ), which we leave as an exercise. Third, if φ1 and φ2 evaluate to F, then our induction hypothesis
and the rule ∧i let us infer that pˆ1 , . . . , pˆn ¬φ1 ∧ ¬φ2 ; so we are left with proving ¬φ1 ∧ ¬φ2 ¬(φ1 ∧ φ2 ), which we leave as an exercise. Fourth, if φ1 evaluates to T and φ2 to F, we obtain
pˆ1 , . . . , pˆn φ1 ∧ ¬φ2 by our induction hypothesis and we have to show φ1 ∧ ¬φ2 ¬(φ1 ∧ φ2 ), which we leave as an exercise. 5. Finally, if φ is a disjunction φ1 ∨ φ2 , we again have four cases.
First, if φ1 and φ2 evaluate to F, then our induction hypothesis and the rule ∧i give us pˆ1 , . . . , pˆn ¬φ1 ∧ ¬φ2 and we have to show ¬φ1 ∧ ¬φ2 ¬(φ1 ∨ φ2 ), which we leave as an exercise. Second,
if φ1 and φ2 evaluate to T, then we obtain pˆ1 , . . . , pˆn φ1 ∧ φ2 , by our induction hypothesis, and we need a proof for φ1 ∧ φ2 φ1 ∨ φ2 , which we leave as an exercise. Third, if φ1 evaluates to
F and φ2 to T, then we arrive at pˆ1 , . . . , pˆn ¬φ1 ∧ φ2 , using our induction hypothesis, and need to establish ¬φ1 ∧ φ2 φ1 ∨ φ2 , which we leave as an exercise. Fourth, if φ1 evaluates to T and
φ2 to F, then pˆ1 , . . . , pˆn φ1 ∧ ¬φ2 results from our induction hypothesis and all we need is a proof for φ1 ∧ ¬φ2 φ1 ∨ φ2 , which we leave as an exercise. 2
We apply this technique to the formula φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))). Since it is a tautology it evaluates to T in all 2n lines of its truth table; thus, the proposition above gives us
2n many proofs of pˆ1 , pˆ2 , . . . , pˆn η, one for each of the cases that pˆi is pi or ¬pi . Our job now is to assemble all these proofs into a single proof for η which does not use any premises.
We illustrate how to do this for an example, the tautology p ∧ q → p. The formula p ∧ q → p has two propositional atoms p and q. By the proposition above, we are guaranteed to have a proof for each
of the four sequents p, q p ∧ q → p ¬p, q p ∧ q → p p, ¬q p ∧ q → p ¬p, ¬q p ∧ q → p. Ultimately, we want to prove p ∧ q → p by appealing to the four proofs of the sequents above. Thus, we somehow
need to get rid of the premises on
1.5 Normal forms
the left-hand sides of these four sequents. This is the place where we rely on the law of the excluded middle which states r ∨ ¬r, for any r. We use LEM for all propositional atoms (here p and q) and
then we separately assume all the four cases, by using ∨e. That way we can invoke all four proofs of the sequents above and use the rule ∨e repeatedly until we have got rid of all our premises. We
spell out the combination of these four phases schematically: 1
p ∨ ¬p
q ∨ ¬q
LEM q ∨ ¬q
q .. .. ..
ass ¬q .. .. ..
¬p q .. .. ..
ass LEM ass ¬q .. .. ..
p∧q →p
p∧q →p
p∧q →p
p∧q →p
p∧q →p ∨e
p∧q →p
p∧q →p ∨e ∨e
As soon as you understand how this particular example works, you will also realise that it will work for an arbitrary tautology with n distinct atoms. Of course, it seems ridiculous to prove p ∧ q →
p using a proof that is this long. But remember that this illustrates a uniform method that constructs a proof for every tautology η, no matter how complicated it is. Step 3: Finally, we need to find
a proof for φ1 , φ2 , . . . , φn ψ. Take the proof for φ1 → (φ2 → (φ3 → (. . . (φn → ψ) . . . ))) given by step 2 and augment its proof by introducing φ1 , φ2 , . . . , φn as premises. Then apply →e
n times on each of these premises (starting with φ1 , continuing with φ2 etc.). Thus, we arrive at the conclusion ψ which gives us a proof for the sequent φ1 , φ2 , . . . , φn ψ. Corollary 1.39
(Soundness and Completeness) Let φ1 , φ2 , . . . , φn , ψ be formulas of propositional logic. Then φ1 , φ2 , . . . , φn ψ is holds iff the sequent φ1 , φ2 , . . . , φn ψ is valid.
1.5 Normal forms In the last section, we showed that our proof system for propositional logic is sound and complete for the truth-table semantics of formulas in Figure 1.6.
1 Propositional logic
Soundness means that whatever we prove is going to be a true fact, based on the truth-table semantics. In the exercises, we apply this to show that a sequent does not have a proof: simply show that
φ1 , φ2 , . . . , φ2 does not semantically entail ψ; then soundness implies that the sequent φ1 , φ2 , . . . , φ2 ψ does not have a proof. Completeness comprised a much more powerful statement: no
matter what (semantically) valid sequents there are, they all have syntactic proofs in the proof system of natural deduction. This tight correspondence allows us to freely switch between working with
the notion of proofs () and that of semantic entailment (). Using natural deduction to decide the validity of instances of is only one of many possibilities. In Exercise 1.2.6 we sketch a non-linear,
tree-like, notion of proofs for sequents. Likewise, checking an instance of by applying Definition 1.34 literally is only one of many ways of deciding whether φ1 , φ2 , . . . , φn ψ holds. We now
investigate various alternatives for deciding φ1 , φ2 , . . . , φn ψ which are based on transforming these formulas syntactically into ‘equivalent’ ones upon which we can then settle the matter by
purely syntactic or algorithmic means. This requires that we first clarify what exactly we mean by equivalent formulas.
1.5.1 Semantic equivalence, satisfiability and validity Two formulas φ and ψ are said to be equivalent if they have the same ‘meaning.’ This suggestion is vague and needs to be refined. For example, p
→ q and ¬p ∨ q have the same truth table; all four combinations of T and F for p and q return the same result. ’Coincidence of truth tables’ is not good enough for what we have in mind, for what
about the formulas p ∧ q → p and r ∨ ¬r? At first glance, they have little in common, having different atomic formulas and different connectives. Moreover, the truth table for p ∧ q → p is four lines
long, whereas the one for r ∨ ¬r consists of only two lines. However, both formulas are always true. This suggests that we define the equivalence of formulas φ and ψ via : if φ semantically entails ψ
and vice versa, then these formulas should be the same as far as our truth-table semantics is concerned. Definition 1.40 Let φ and ψ be formulas of propositional logic. We say that φ and ψ are
semantically equivalent iff φ ψ and ψ φ hold. In that case we write φ ≡ ψ. Further, we call φ valid if φ holds. Note that we could also have defined φ ≡ ψ to mean that (φ → ψ) ∧ (ψ → φ) holds; it
amounts to the same concept. Indeed, because of soundness and completeness, semantic equivalence is identical to provable equivalence
1.5 Normal forms
(Definition 1.25). Examples of equivalent formulas are p → q ≡ ¬q → ¬p p → q ≡ ¬p ∨ q p ∧ q → p ≡ r ∨ ¬r p ∧ q → r ≡ p → (q → r). Recall that a formula η is called a tautology if η holds, so the
tautologies are exactly the valid formulas. The following lemma says that any decision procedure for tautologies is in fact a decision procedure for the validity of sequents as well. Lemma 1.41 Given
formulas φ1 , φ2 , . . . , φn and ψ of propositional logic, φ1 , φ2 , . . . , φn ψ holds iff φ1 → (φ2 → (φ3 → · · · → (φn → ψ))) holds. Proof: First, suppose that φ1 → (φ2 → (φ3 → · · · → (φn → ψ)))
holds. If φ1 , φ2 , . . . , φn are all true under some valuation, then ψ has to be true as well for that same valuation. Otherwise, φ1 → (φ2 → (φ3 → · · · → (φn → ψ))) would not hold (compare this
with Figure 1.11). Second, if φ1 , φ2 , . . . , φn ψ holds, we have already shown that φ1 → (φ2 → (φ3 → 2 · · · → (φn → ψ))) follows in step 1 of our completeness proof. For our current purposes, we
want to transform formulas into ones which don’t contain → at all and the occurrences of ∧ and ∨ are confined to separate layers such that validity checks are easy. This is being done by 1. using the
equivalence φ → ψ ≡ ¬φ ∨ ψ to remove all occurrences of → from a formula and 2. by specifying an algorithm that takes a formula without any → into a normal form (still without →) for which checking
validity is easy.
Naturally, we have to specify which forms of formulas we think of as being ‘normal.’ Again, there are many such notions, but in this text we study only two important ones. Definition 1.42 A literal L
is either an atom p or the negation of an atom ¬p. A formula C is in conjunctive normal form (CNF) if it is a conjunction of clauses, where each clause D is a disjunction of literals: L ::= p | ¬p D
::= L | L ∨ D C ::= D | D ∧ C.
1 Propositional logic
Examples of formulas in conjunctive normal form are (i) (¬q ∨ p ∨ r) ∧ (¬p ∨ r) ∧ q
(ii) (p ∨ r) ∧ (¬p ∨ r) ∧ (p ∨ ¬r).
In the first case, there are three clauses of type D: ¬q ∨ p ∨ r, ¬p ∨ r, and q – which is a literal promoted to a clause by the first rule of clauses in (1.6). Notice how we made implicit use of the
associativity laws for ∧ and ∨, saying that φ ∨ (ψ ∨ η) ≡ (φ ∨ ψ) ∨ η and φ ∧ (ψ ∧ η) ≡ (φ ∧ ψ) ∧ η, since we omitted some parentheses. The formula (¬(q ∨ p) ∨ r) ∧ (q ∨ r) is not in CNF since q ∨ p
is not a literal. Why do we care at all about formulas φ in CNF? One of the reasons for their usefulness is that they allow easy checks of validity which otherwise take times exponential in the
number of atoms. For example, consider the formula in CNF from above: (¬q ∨ p ∨ r) ∧ (¬p ∨ r) ∧ q. The semantic entailment (¬q ∨ p ∨ r) ∧ (¬p ∨ r) ∧ q holds iff all three relations ¬q ∨ p ∨ r
¬p ∨ r
hold, by the semantics of ∧. But since all of these formulas are disjunctions of literals, or literals, we can settle the matter as follows. Lemma 1.43 A disjunction of literals L1 ∨ L2 ∨ · · · ∨ Lm
is valid iff there are 1 ≤ i, j ≤ m such that Li is ¬Lj . Proof: If Li equals ¬Lj , then L1 ∨ L2 ∨ · · · ∨ Lm evaluates to T for all valuations. For example, the disjunct p ∨ q ∨ r ∨ ¬q can never be
made false. To see that the converse holds as well, assume that no literal Lk has a matching negation in L1 ∨ L2 ∨ · · · ∨ Lm . Then, for each k with 1 ≤ k ≤ n, we assign F to Lk , if Lk is an atom;
or T, if Lk is the negation of an atom. For example, the disjunct ¬q ∨ p ∨ r can be made false by assigning F to p and r and T to q. 2 Hence, we have an easy and fast check for the validity of φ,
provided that φ is in CNF; inspect all conjuncts ψk of φ and search for atoms in ψk such that ψk also contains their negation. If such a match is found for all conjuncts, we have φ. Otherwise (= some
conjunct contains no pair Li and ¬Li ), φ is not valid by the lemma above. Thus, the formula (¬q ∨ p ∨ r) ∧ (¬p ∨ r) ∧ q above is not valid. Note that the matching literal has to be found in the same
conjunct ψk . Since there is no free lunch in this universe, we can expect that the computation of a formula φ in CNF, which is equivalent to a given formula φ, is a costly worst-case operation.
Before we study how to compute equivalent conjunctive normal forms, we introduce another semantic concept closely related to that of validity.
1.5 Normal forms
Definition 1.44 Given a formula φ in propositional logic, we say that φ is satisfiable if it has a valuation in which is evaluates to T. For example, the formula p ∨ q → p is satisfiable since it
computes T if we assign T to p. Clearly, p ∨ q → p is not valid. Thus, satisfiability is a weaker concept since every valid formula is by definition also satisfiable but not vice versa. However, these
two notions are just mirror images of each other, the mirror being negation. Proposition 1.45 Let φ be a formula of propositional logic. Then φ is satisfiable iff ¬φ is not valid. Proof: First, assume
that φ is satisfiable. By definition, there exists a valuation of φ in which φ evaluates to T; but that means that ¬φ evaluates to F for that same valuation. Thus, ¬φ cannot be valid. Second, assume
that ¬φ is not valid. Then there must be a valuation of ¬φ in which ¬φ evaluates to F. Thus, φ evaluates to T and is therefore satisfiable. (Note that the valuations of φ are exactly the valuations of
¬φ.) 2 This result is extremely useful since it essentially says that we need provide a decision procedure for only one of these concepts. For example, let’s say that we have a procedure P for
deciding whether any φ is valid. We obtain a decision procedure for satisfiability simply by asking P whether ¬φ is valid. If it is, φ is not satisfiable; otherwise φ is satisfiable. Similarly, we may
transform any decision procedure for satisfiability into one for validity. We will encounter both kinds of procedures in this text. There is one scenario in which computing an equivalent formula in
CNF is really easy; namely, when someone else has already done the work of writing down a full truth table for φ. For example, take the truth table of (p → ¬q) → (q ∨ ¬p) in Figure 1.8 (page 40). For
each line where (p → ¬q) → (q ∨ ¬p) computes F we now construct a disjunction of literals. Since there is only one such line, we have only one conjunct ψ1 . That conjunct is now obtained by a
disjunction of literals, where we include literals ¬p and q. Note that the literals are just the syntactic opposites of the truth values in that line: here p is T and q is F. The resulting formula in
CNF is thus ¬p ∨ q which is readily seen to be in CNF and to be equivalent to (p → ¬q) → (q ∨ ¬p). Why does this always work for any formula φ? Well, the constructed formula will be false iff at least
one of its conjuncts ψi will be false. This means that all the disjuncts in such a ψi must be F. Using the de Morgan
1 Propositional logic
rule ¬φ1 ∨ ¬φ2 ∨ · · · ∨ ¬φn ≡ ¬(φ1 ∧ φ2 ∧ · · · ∧ φn ), we infer that the conjunction of the syntactic opposites of those literals must be true. Thus, φ and the constructed formula have the same
truth table. Consider another example, in which φ is given by the truth table: p T T T T F F F F
q T T F F T T F F
r T F T F T F T F
φ T F T T F F F T
Note that this table is really just a specification of φ; it does not tell us what φ looks like syntactically, but it does tells us how it ought to ‘behave.’ Since this truth table has four entries
which compute F, we construct four conjuncts ψi (1 ≤ i ≤ 4). We read the ψi off that table by listing the disjunction of all atoms, where we negate those atoms which are true in those lines: def
ψ1 = ¬p ∨ ¬q ∨ r (line 2) def
ψ3 = p ∨ ¬q ∨ r
ψ2 = p ∨ ¬q ∨ ¬r (line 5) def
ψ4 = p ∨ q ∨ ¬r.
The resulting φ in CNF is therefore (¬p ∨ ¬q ∨ r) ∧ (p ∨ ¬q ∨ ¬r) ∧ (p ∨ ¬q ∨ r) ∧ (p ∨ q ∨ ¬r). If we don’t have a full truth table at our disposal, but do know the structure of φ, then we would
like to compute a version of φ in CNF. It should be clear by now that a full truth table of φ and an equivalent formula in CNF are pretty much the same thing as far as questions about validity are
concerned – although the formula in CNF may be much more compact.
1.5.2 Conjunctive normal forms and validity We have already seen the benefits of conjunctive normal forms in that they allow for a fast and easy syntactic test of validity. Therefore, one wonders
whether any formula can be transformed into an equivalent formula in CNF. We now develop an algorithm achieving just that. Note that, by Definition 1.40, a formula is valid iff any of its equivalent
formulas is valid. We reduce the problem of determining whether any φ is valid to the problem of computing an equivalent ψ ≡ φ such that ψ is in CNF and checking, via Lemma 1.43, whether ψ is valid.
1.5 Normal forms
Before we sketch such a procedure, we make some general remarks about its possibilities and its realisability constraints. First of all, there could be more or less efficient ways of computing such
normal forms. But even more so, there could be many possible correct outputs, for ψ1 ≡ φ and ψ2 ≡ φ do not generally imply that ψ1 is the same as ψ2 , even if ψ1 and ψ2 are in def def def CNF. For
example, take φ = p, ψ1 = p and ψ2 = p ∧ (p ∨ q); then convince yourself that φ ≡ ψ2 holds. Having this ambiguity of equivalent conjunctive normal forms, the computation of a CNF for φ with minimal
‘cost’ (where ‘cost’ could for example be the number of conjuncts, or the height of φ’s parse tree) becomes a very important practical problem, an issue persued in Chapter 6. Right now, we are
content with stating a deterministic algorithm which always computes the same output CNF for a given input φ. This algorithm, called CNF, should satisfy the following requirements: (1) CNF terminates
for all formulas of propositional logic as input; (2) for each such input, CNF outputs an equivalent formula; and (3) all output computed by CNF is in CNF.
If a call of CNF with a formula φ of propositional logic as input terminates, which is enforced by (1), then (2) ensures that ψ ≡ φ holds for the output ψ. Thus, (3) guarantees that ψ is an
equivalent CNF of φ. So φ is valid iff ψ is valid; and checking the latter is easy relative to the length of ψ. What kind of strategy should CNF employ? It will have to function correctly for all,
i.e. infinitely many, formulas of propositional logic. This strongly suggests to write a procedure that computes a CNF by structural induction on the formula φ. For example, if φ is of the form φ1 ∧
φ2 , we may simply compute conjunctive normal forms ηi for φi (i = 1, 2), whereupon η1 ∧ η2 is a conjunctive normal form which is equivalent to φ provided that ηi ≡ φi (i = 1, 2). This strategy also
suggests to use proof by structural induction on φ to prove that CNF meets the requirements (1–3) stated above. Given a formula φ as input, we first do some preprocessing. Initially, we translate away
all implications in φ by replacing all subformulas of the form ψ → η by ¬ψ ∨ η. This is done by a procedure called IMPL FREE. Note that this procedure has to be recursive, for there might be
implications in ψ or η as well. The application of IMPL FREE might introduce double negations into the output formula. More importantly, negations whose scopes are non-atomic formulas might still be
present. For example, the formula p ∧ ¬(p ∧ q) has such a negation with p ∧ q as its scope. Essentially, the question is whether one can efficiently compute a CNF for ¬φ from a CNF for φ. Since nobody
seems to know the answer, we circumvent the question by translating ¬φ
1 Propositional logic
into an equivalent formula that contains only negations of atoms. Formulas which only negate atoms are said to be in negation normal form (NNF). We spell out such a procedure, NNF, in detail later
on. The key to its specification for implication-free formulas lies in the de Morgan rules. The second phase of the preprocessing, therefore, calls NNF with the implication-free output of IMPL FREE to
obtain an equivalent formula in NNF. After all this preprocessing, we obtain a formula φ which is the result of the call NNF (IMPL FREE (φ)). Note that φ ≡ φ since both algorithms only transform
formulas into equivalent ones. Since φ contains no occurrences of → and since only atoms in φ are negated, we may program CNF by an analysis of only three cases: literals, conjunctions and
disjunctions. r If φ is a literal, it is by definition in CNF and so CNF outputs φ. r If φ equals φ1 ∧ φ2 , we call CNF recursively on each φi to get the respective output ηi and return the CNF η1 ∧
η2 as output for input φ. r If φ equals φ1 ∨ φ2 , we again call CNF recursively on each φi to get the respective output ηi ; but this time we must not simply return η1 ∨ η2 since that formula is
certainly not in CNF, unless η1 and η2 happen to be literals.
So how can we complete the program in the last case? Well, we may resort to the distributivity laws, which entitle us to translate any disjunction of conjunctions into a conjunction of disjunctions.
However, for this to result in a CNF, we need to make certain that those disjunctions generated contain only literals. We apply a strategy for using distributivity based on matching patterns in φ1 ∨
φ2 . This results in an independent algorithm called DISTR which will do all that work for us. Thus, we simply call DISTR with the pair (η1 , η2 ) as input and pass along its result. Assuming that we
already have written code for IMPL FREE, NNF and DISTR, we may now write pseudo code for CNF: function CNF (φ) : /* precondition: φ implication free and in NNF */ /* postcondition: CNF (φ) computes
an equivalent CNF for φ */ begin function case φ is a literal : return φ φ is φ1 ∧ φ2 : return CNF (φ1 ) ∧ CNF (φ2 ) φ is φ1 ∨ φ2 : return DISTR (CNF (φ1 ), CNF (φ2 )) end case end function
1.5 Normal forms
Notice how the calling of DISTR is done with the computed conjunctive normal forms of φ1 and φ2 . The routine DISTR has η1 and η2 as input parameters and does a case analysis on whether these inputs
are conjunctions. What should DISTR do if none of its input formulas is such a conjunction? Well, since we are calling DISTR for inputs η1 and η2 which are in CNF, this can only mean that η1 and η2
are literals, or disjunctions of literals. Thus, η1 ∨ η2 is in CNF. Otherwise, at least one of the formulas η1 and η2 is a conjunction. Since one conjunction suffices for simplifying the problem, we
have to decide which conjunct we want to transform if both formulas are conjunctions. That way we maintain that our algorithm CNF is deterministic. So let us suppose that η1 is of the form η11 ∧ η12
. Then the distributive law says that η1 ∨ η2 ≡ (η11 ∨ η2 ) ∧ (η12 ∨ η2 ). Since all participating formulas η11 , η12 and η2 are in CNF, we may call DISTR again for the pairs (η11 , η2 ) and (η12 ,
η2 ), and then simply form their conjunction. This is the key insight for writing the function DISTR. The case when η2 is a conjunction is symmetric and the structure of the recursive call of DISTR
is then dictated by the equivalence η1 ∨ η2 ≡ (η1 ∨ η21 ) ∧ (η1 ∨ η22 ), where η2 = η21 ∧ η22 : function DISTR (η1 , η2 ) : /* precondition: η1 and η2 are in CNF */ /* postcondition: DISTR (η1 , η2 )
computes a CNF for η1 ∨ η2 */ begin function case η1 is η11 ∧ η12 : return DISTR (η11 , η2 ) ∧ DISTR (η12 , η2 ) η2 is η21 ∧ η22 : return DISTR (η1 , η21 ) ∧ DISTR (η1 , η22 ) otherwise (= no
conjunctions) : return η1 ∨ η2 end case end function Notice how the three clauses are exhausting all possibilities. Furthermore, the first and second cases overlap if η1 and η2 are both conjunctions.
It is then our understanding that this code will inspect the clauses of a case statement from the top to the bottom clause. Thus, the first clause would apply. Having specified the routines CNF and
DISTR, this leaves us with the task of writing the functions IMPL FREE and NNF. We delegate the design
1 Propositional logic
of IMPL FREE to the exercises. The function NNF has to transform any implication-free formula into an equivalent one in negation normal form. Four examples of formulas in NNF are p ¬p ∧ (p ∧ q)
¬p ¬p ∧ (p → q),
although we won’t have to deal with a formula of the last kind since → won’t occur. Examples of formulas which are not in NNF are ¬¬p and ¬(p ∧ q). Again, we program NNF recursively by a case
analysis over the structure of the input formula φ. The last two examples already suggest a solution for two of these clauses. In order to compute a NNF of ¬¬φ, we simply compute a NNF of φ. This is
a sound strategy since φ and ¬¬φ are semantically equivalent. If φ equals ¬(φ1 ∧ φ2 ), we use the de Morgan rule ¬(φ1 ∧ φ2 ) ≡ ¬φ1 ∨ ¬φ2 as a recipe for how NNF should call itself recursively in that
case. Dually, the case of φ being ¬(φ1 ∨ φ2 ) appeals to the other de Morgan rule ¬(φ1 ∨ φ2 ) ≡ ¬φ1 ∧ ¬φ2 and, if φ is a conjunction or disjunction, we simply let NNF pass control to those
subformulas. Clearly, all literals are in NNF. The resulting code for NNF is thus function NNF (φ) : /* precondition: φ is implication free */ /* postcondition: NNF (φ) computes a NNF for φ */ begin
function case φ is a literal : return φ φ is ¬¬φ1 : return NNF (φ1 ) φ is φ1 ∧ φ2 : return NNF (φ1 ) ∧ NNF (φ2 ) φ is φ1 ∨ φ2 : return NNF (φ1 ) ∨ NNF (φ2 ) φ is ¬(φ1 ∧ φ2 ) : return NNF (¬φ1 ) ∨ NNF
(¬φ2 ) φ is ¬(φ1 ∨ φ2 ) : return NNF (¬φ1 ) ∧ NNF (¬φ2 ) end case end function Notice that these cases are exhaustive due to the algorithm’s precondition. Given any formula φ of propositional logic,
we may now convert it into an
1.5 Normal forms
equivalent CNF by calling CNF (NNF (IMPL FREE (φ))). In the exercises, you are asked to show that r all four algorithms terminate on input meeting their preconditions, r the result of CNF (NNF (IMPL
FREE (φ))) is in CNF and r that result is semantically equivalent to φ.
We will return to the important issue of formally proving the correctness of programs in Chapter 4. Let us now illustrate the programs coded above on some concrete examples. We begin by computing CNF
(NNF (IMPL FREE (¬p ∧ q → p ∧ (r → q)))). We show almost all details of this computation and you should compare this with how you would expect the code above to behave. First, we compute IMPL FREE
(φ): IMPL FREE (φ) = ¬IMPL FREE (¬p ∧ q) ∨ IMPL FREE (p ∧ (r → q)) = ¬((IMPL FREE ¬p) ∧ (IMPL FREE q)) ∨ IMPL FREE (p ∧ (r → q)) = ¬((¬p) ∧ IMPL FREE q) ∨ IMPL FREE (p ∧ (r → q)) = ¬(¬p ∧ q) ∨ IMPL
FREE (p ∧ (r → q)) = ¬(¬p ∧ q) ∨ ((IMPL FREE p) ∧ IMPL FREE (r → q)) = ¬(¬p ∧ q) ∨ (p ∧ IMPL FREE (r → q)) = ¬(¬p ∧ q) ∨ (p ∧ (¬(IMPL FREE r) ∨ (IMPL FREE q))) = ¬(¬p ∧ q) ∨ (p ∧ (¬r ∨ (IMPL FREE
q))) = ¬(¬p ∧ q) ∨ (p ∧ (¬r ∨ q)). Second, we compute NNF (IMPL FREE φ): NNF (IMPL FREE φ) = NNF (¬(¬p ∧ q)) ∨ NNF (p ∧ (¬r ∨ q)) = NNF (¬(¬p) ∨ ¬q) ∨ NNF (p ∧ (¬r ∨ q)) = (NNF (¬¬p)) ∨ (NNF (¬q)) ∨
NNF (p ∧ (¬r ∨ q)) = (p ∨ (NNF (¬q))) ∨ NNF (p ∧ (¬r ∨ q)) = (p ∨ ¬q) ∨ NNF (p ∧ (¬r ∨ q)) = (p ∨ ¬q) ∨ ((NNF p) ∧ (NNF (¬r ∨ q))) = (p ∨ ¬q) ∨ (p ∧ (NNF (¬r ∨ q))) = (p ∨ ¬q) ∨ (p ∧ ((NNF (¬r)) ∨
(NNF q))) = (p ∨ ¬q) ∨ (p ∧ (¬r ∨ (NNF q))) = (p ∨ ¬q) ∨ (p ∧ (¬r ∨ q)).
1 Propositional logic
Third, we finish it off with CNF (NNF (IMPL FREE φ)) = CNF ((p ∨ ¬q) ∨ (p ∧ (¬r ∨ q))) = DISTR (CNF (p ∨ ¬q), CNF (p ∧ (¬r ∨ q))) = DISTR (p ∨ ¬q, CNF (p ∧ (¬r ∨ q))) = DISTR (p ∨ ¬q, p ∧ (¬r ∨ q)) =
DISTR (p ∨ ¬q, p) ∧ DISTR (p ∨ ¬q, ¬r ∨ q) = (p ∨ ¬q ∨ p) ∧ DISTR (p ∨ ¬q, ¬r ∨ q) = (p ∨ ¬q ∨ p) ∧ (p ∨ ¬q ∨ ¬r ∨ q) . The formula (p ∨ ¬q ∨ p) ∧ (p ∨ ¬q ∨ ¬r ∨ q) is thus the result of the call CNF
(NNF (IMPL FREE φ)) and is in conjunctive normal form and equivalent to φ. Note that it is satisfiable (choose p to be true) but not valid (choose p to be false and q to be true); it is also
equivalent to the simpler conjunctive normal form p ∨ ¬q. Observe that our algorithm does not do such optimisations so one would need a separate optimiser running on the output. Alternatively, one
might change the code of our functions to allow for such optimisations ‘on the fly,’ a computational overhead which could prove to be counterproductive. You should realise that we omitted several
computation steps in the subcalls CNF (p ∨ ¬q) and CNF (p ∧ (¬r ∨ q)). They return their input as a result since the input is already in conjunctive normal form. def As a second example, consider φ =
r → (s → (t ∧ s → r)). We compute IMPL FREE (φ) = ¬(IMPL FREE r) ∨ IMPL FREE (s → (t ∧ s → r)) = ¬r ∨ IMPL FREE (s → (t ∧ s → r)) = ¬r ∨ (¬(IMPL FREE s) ∨ IMPL FREE (t ∧ s → r)) = ¬r ∨ (¬s ∨ IMPL
FREE (t ∧ s → r)) = ¬r ∨ (¬s ∨ (¬(IMPL FREE (t ∧ s)) ∨ IMPL FREE r)) = ¬r ∨ (¬s ∨ (¬((IMPL FREE t) ∧ (IMPL FREE s)) ∨ IMPL FREE r)) = ¬r ∨ (¬s ∨ (¬(t ∧ (IMPL FREE s)) ∨ (IMPL FREE r))) = ¬r ∨ (¬s ∨
(¬(t ∧ s)) ∨ (IMPL FREE r)) = ¬r ∨ (¬s ∨ (¬(t ∧ s)) ∨ r)
1.5 Normal forms
NNF (IMPL FREE φ) = NNF (¬r ∨ (¬s ∨ ¬(t ∧ s) ∨ r)) = (NNF ¬r) ∨ NNF (¬s ∨ ¬(t ∧ s) ∨ r) = ¬r ∨ NNF (¬s ∨ ¬(t ∧ s) ∨ r) = ¬r ∨ (NNF (¬s) ∨ NNF (¬(t ∧ s) ∨ r)) = ¬r ∨ (¬s ∨ NNF (¬(t ∧ s) ∨ r)) = ¬r ∨
(¬s ∨ (NNF (¬(t ∧ s)) ∨ NNF r)) = ¬r ∨ (¬s ∨ (NNF (¬t ∨ ¬s)) ∨ NNF r) = ¬r ∨ (¬s ∨ ((NNF (¬t) ∨ NNF (¬s)) ∨ NNF r)) = ¬r ∨ (¬s ∨ ((¬t ∨ NNF (¬s)) ∨ NNF r)) = ¬r ∨ (¬s ∨ ((¬t ∨ ¬s) ∨ NNF r)) = ¬r ∨
(¬s ∨ ((¬t ∨ ¬s) ∨ r)) where the latter is already in CNF and valid as r has a matching ¬r.
1.5.3 Horn clauses and satisfiability We have already commented on the computational price we pay for transforming a propositional logic formula into an equivalent CNF. The latter class of formulas
has an easy syntactic check for validity, but its test for satisfiability is very hard in general. Fortunately, there are practically important subclasses of formulas which have much more efficient ways
of deciding their satisfiability. One such example is the class of Horn formulas; the name ‘Horn’ is derived from the logician A. Horn’s last name. We shortly define them and give an algorithm for
checking their satisfiability. Recall that the logical constants ⊥ (‘bottom’) and (‘top’) denote an unsatisfiable formula, respectively, a tautology. Definition 1.46 A Horn formula is a formula φ of
propositional logic if it can be generated as an instance of H in this grammar: P ::= ⊥ | | p A ::= P | P ∧ A C ::= A → P H ::= C | C ∧ H. We call each instance of C a Horn clause.
1 Propositional logic
Horn formulas are conjunctions of Horn clauses. A Horn clause is an implication whose assumption A is a conjunction of propositions of type P and whose conclusion is also of type P . Examples of Horn
formulas are (p ∧ q ∧ s → p) ∧ (q ∧ r → p) ∧ (p ∧ s → s) (p ∧ q ∧ s → ⊥) ∧ (q ∧ r → p) ∧ ( → s) (p2 ∧ p3 ∧ p5 → p13 ) ∧ ( → p5 ) ∧ (p5 ∧ p11 → ⊥). Examples of formulas which are not Horn formulas are
(p ∧ q ∧ s → ¬p) ∧ (q ∧ r → p) ∧ (p ∧ s → s) (p ∧ q ∧ s → ⊥) ∧ (¬q ∧ r → p) ∧ ( → s) (p2 ∧ p3 ∧ p5 → p13 ∧ p27 ) ∧ ( → p5 ) ∧ (p5 ∧ p11 → ⊥) (p2 ∧ p3 ∧ p5 → p13 ∧ p27 ) ∧ ( → p5 ) ∧ (p5 ∧ p11 ∨ ⊥).
The first formula is not a Horn formula since ¬p, the conclusion of the implication of the first conjunct, is not of type P . The second formula does not qualify since the premise of the implication of
the second conjunct, ¬q ∧ r, is not a conjunction of atoms, ⊥, or . The third formula is not a Horn formula since the conclusion of the implication of the first conjunct, p13 ∧ p27 , is not of type P
. The fourth formula clearly is not a Horn formula since it is not a conjunction of implications. The algorithm we propose for deciding the satisfiability of a Horn formula φ maintains a list of all
occurrences of type P in φ and proceeds like this: 1. It marks if it occurs in that list. 2. If there is a conjunct P1 ∧ P2 ∧ · · · ∧ Pki → P of φ such that all Pj with 1 ≤ j ≤ ki are marked, mark P
as well and go to 2. Otherwise (= there is no conjunct P1 ∧ P2 ∧ · · · ∧ Pki → P such that all Pj are marked) go to 3. 3. If ⊥ is marked, print out ‘The Horn formula φ is unsatisfiable.’ and stop.
Otherwise, go to 4. 4. Print out ‘The Horn formula φ is satisfiable.’ and stop.
In these instructions, the markings of formulas are shared by all other occurrences of these formulas in the Horn formula. For example, once we mark p2 because of one of the criteria above, then all
other occurrences of p2 are marked as well. We use pseudo code to specify this algorithm formally:
1.5 Normal forms
function HORN (φ): /* precondition: φ is a Horn formula */ /* postcondition: HORN (φ) decides the satisfiability for φ */ begin function mark all occurrences of in φ; while there is a conjunct P1 ∧ P2
∧ · · · ∧ Pki → P of φ such that all Pj are marked but P isn’t do mark P end while if ⊥ is marked then return ‘unsatisfiable’ else return ‘satisfiable’ end function We need to make sure that this
algorithm terminates on all Horn formulas φ as input and that its output (= its decision) is always correct. Theorem 1.47 The algorithm HORN is correct for the satisfiability decision problem of Horn
formulas and has no more than n + 1 cycles in its whilestatement if n is the number of atoms in φ. In particular, HORN always terminates on correct input. Proof: Let us first consider the question of
program termination. Notice that entering the body of the while-statement has the effect of marking an unmarked P which is not . Since this marking applies to all occurrences of P in φ, the
while-statement can have at most one more cycle than there are atoms in φ. Since we guaranteed termination, it suffices to show that the answers given by the algorithm HORN are always correct. To that
end, it helps to reveal the functional role of those markings. Essentially, marking a P means that that P has got to be true if the formula φ is ever going to be satisfiable. We use mathematical
induction to show that ‘All marked P are true for all valuations in which φ evaluates to T.’ (1.8) holds after any number of executions of the body of the while-statement above. The base case, zero
executions, is when the while-statement has not yet been entered but we already and only marked all occurrences of . Since must be true in all valuations, (1.8) follows. In the inductive step, we
assume that (1.8) holds after k cycles of the while-statement. Then we need to show that same assertion for all marked P after k + 1 cycles. If we enter the (k + 1)th cycle, the condition of the
while-statement is certainly true. Thus, there exists a conjunct P1 ∧ P2 ∧ · · · ∧ Pki → P of φ such that all Pj are marked. Let v be any valuation
1 Propositional logic
in which φ is true. By our induction hypothesis, we know that all Pj and therefore P1 ∧ P2 ∧ · · · ∧ Pki have to be true in v as well. The conjunct P1 ∧ P2 ∧ · · · ∧ Pki → P of φ has be to true in v,
too, from which we infer that P has to be true in v. By mathematical induction, we therefore secured that (1.8) holds no matter how many cycles that while-statement went through. Finally, we need to
make sure that the if-statement above always renders correct replies. First, if ⊥ is marked, then there has to be some conjunct P1 ∧ P2 ∧ · · · ∧ Pki → ⊥ of φ such that all Pi are marked as well. By
(1.8) that conjunct of φ evaluates to T → F = F whenever φ is true. As this is impossible the reply ‘unsatisfiable’ is correct. Second, if ⊥ is not marked, we simply assign T to all marked atoms and F
to all unmarked atoms and use proof by contradiction to show that φ has to be true with respect to that valuation. If φ is not true under that valuation, it must make one of its principal conjuncts
P1 ∧ P2 ∧ · · · ∧ Pki → P false. By the semantics of implication this can only mean that all Pj are true and P is false. By the definition of our valuation, we then infer that all Pj are marked, so P1
∧ P2 ∧ · · · ∧ Pki → P is a conjunct of φ that would have been dealt with in one of the cycles of the while-statement and so P is marked, too. Since ⊥ is not marked, P has to be or some atom q. In
any event, the conjunct is then true by (1.8), a contradiction 2 Note that the proof by contradiction employed in the last proof was not really needed. It just made the argument seem more natural to
us. The literature is full of such examples where one uses proof by contradiction more out of psychological than proof-theoretical necessity.
1.6 SAT solvers The marking algorithm for Horn formulas computes marks as constraints on all valuations that can make a formule true. By (1.8), all marked atoms have to be true for any such
valuation. We can extend this idea to general formulas φ by computing constraints saying which subformulas of φ require a certain truth value for all valuations that make φ true: ‘All marked
subformulas evaluate to their mark value for all valuations in which φ evaluates to T.’
In that way, marking atomic formulas generalizes to marking subformulas; and ‘true’ marks generalize into ‘true’ and ‘false’ marks. At the same
1.6 SAT solvers
time, (1.9) serves as a guide for designing an algorithm and as an invariant for proving its correctness.
1.6.1 A linear solver We will execute this marking algorithm on the parse tree of formulas, except that we will translate formulas into the adequate fragment φ ::= p | (¬φ) | (φ ∧ φ)
and then share common subformulas of the resulting parse tree, making the tree into a directed, acyclic graph (DAG). The inductively defined translation T (p) = p T (φ1 ∧ φ2 ) = T (φ1 ) ∧ T (φ2 ) T
(φ1 → φ2 ) = ¬(T (φ1 ) ∧ ¬T (φ2 ))
T (¬φ) = ¬T (φ) T (φ1 ∨ φ2 ) = ¬(¬T (φ1 ) ∧ ¬T (φ2 ))
transforms formulas generated by (1.3) into formulas generated by (1.10) such that φ and T (φ) are semantically equivalent and have the same propositional atoms. Therefore, φ is satisfiable iff T (φ)
is satisfiable; and the set of valuations for which φ is true equals the set of valuations for which T (φ) is true. The latter ensures that the diagnostics of a SAT solver, applied to T (φ), is
meaningful for the original formula φ. In the exercises, you are asked to prove these claims. Example 1.48 For the formula φ being p ∧ ¬(q ∨ ¬p) we compute T (φ) = p ∧ ¬¬(¬q ∧ ¬¬p). The parse tree
and DAG of T (φ) are depicted in Figure 1.12. Any valuation that makes p ∧ ¬¬(¬q ∧ ¬¬p) true has to assign T to the topmost ∧-node in its DAG of Figure 1.12. But that forces the mark T on the p-node
and the topmost ¬-node. In the same manner, we arrive at a complete set of constraints in Figure 1.13, where the time stamps ‘1:’ etc indicate the order in which we applied our intuitive reasoning
about these constraints; this order is generally not unique. The formal set of rules for forcing new constraints from old ones is depicted in Figure 1.14. A small circle indicates any node (¬, ∧ or
atom). The force laws for negation, ¬t and ¬f , indicate that a truth constraint on a ¬-node forces its dual value at its sub-node and vice versa. The law ∧te propagates a T constraint on a ∧-node to
its two sub-nodes; dually, ∧ti forces a T mark on a ∧-node if both its children have that mark. The laws ∧fl and ∧fr force a F constraint on a ∧-node if any of its sub-nodes has a F value. The laws
1 Propositional logic
70 ∧
∧ ¬
¬ ¬
∧ ¬
¬ ¬
¬ p
Figure 1.12. Parse tree (left) and directed acyclic graph (right) of the formula from Example 1.48. The p-node is shared on the right. 1: T ∧ ¬
2: T
3: F
4: T
5: T ¬
2: T
4: T
3: F
6: F q
Figure 1.13. A witness to the satisfiability of the formula represented by this DAG.
and ∧frr are more complex: if an ∧-node has a F constraint and one of its sub-nodes has a T constraint, then the other sub-node obtains a F-constraint. Please check that all constraints depicted in
Figure 1.13 are derivable from these rules. The fact that each node in a DAG obtained a forced marking does not yet show that this is a witness to the satisfiability of the formula
1.6 SAT solvers ¬ ¬t :
F forcing laws for negation
¬f :
T ∧ ∧te :
T ∧ T
∧ti :
true conjunction forces true conjuncts
F ∧ ∧fl :
T T true conjunctions force true conjunction
F ∧ ∧fr :
false conjuncts force false conjunction F
F ∧ ∧fll :
F ∧ F
∧frr :
false conjunction and true conjunct force false conjunction T
Figure 1.14. Rules for flow of constraints in a formula’s DAG. Small circles indicate arbitrary nodes (¬, ∧ or atom). Note that the rules ∧fll , ∧frr and ∧ti require that the source constraints of
both =⇒ are present.
represented by this DAG. A post-processing phase takes the marks for all atoms and re-computes marks of all other nodes in a bottom-up manner, as done in Section 1.4 on parse trees. Only if the
resulting marks match the ones we computed have we found a witness. Please verify that this is the case in Figure 1.13. We can apply SAT solvers to checking whether sequents are valid. For example,
the sequent p ∧ q → r p → q → r is valid iff (p ∧ q → r) → p → q → r is a theorem (why?) iff φ = ¬((p ∧ q → r) → p → q → r) is not satisfiable. The DAG of T (φ) is depicted in Figure 1.15. The
annotations “1” etc indicate which nodes represent which sub-formulas. Notice that such DAGs may be constructed by applying the translation clauses for T to sub-formulas in a bottom-up manner –
sharing equal subgraphs were applicable. The findings of our SAT solver can be seen in Figure 1.16. The solver concludes that the indicated node requires the marks T and F for (1.9) to be met. Such
contradictory constraints therefore imply that all formulas T (φ) whose DAG equals that of this figure are not satisfiable. In particular, all
1 Propositional logic “5” ¬
“5” = entire formula “4”= ”3” → ”2”
“4” ¬
“3” = p ∧ q → r “2” = p → ”1”
“1” = q → r ¬ “2” ¬ ∧ ¬
“1” ¬
∧ ∧ ¬
∧ p
Figure 1.15. The DAG for the translation of ¬((p ∧ q → r) → p → q → r). Labels ‘‘1’’ etc indicate which nodes represent what subformulas.
such φ are unsatisfiable. This SAT solver has a linear running time in the size of the DAG for T (φ). Since that size is a linear function of the length of φ – the translation T causes only a linear
blow-up – our SAT solver has a linear running time in the length of the formula. This linearity came with a price: our linear solver fails for all formulas of the form ¬(φ1 ∧ φ2 ).
1.6.2 A cubic solver When we applied our linear SAT solver, we saw two possible outcomes: we either detected contradictory constraints, meaning that no formula represented by the DAG is satisfiable
(e.g. Fig. 1.16); or we managed to force consistent constraints on all nodes, in which case all formulas represented by this DAG are satisfiable with those constraints as a witness (e.g. Fig. 1.13).
Unfortunately, there is a third possibility: all forced constraints are consistent with each other, but not all nodes are constrained! We already remarked that this occurs for formulas of the form ¬
(φ1 ∧ φ2 ).
1.6 SAT solvers
73 ¬
1: T
2: F
3: T
4: T ¬ 5: F ¬ 6: T ∧ its conjunction parent and ∧frr force F
7: T ¬ 8: F ¬
its children and ∧ti force T – a contradiction
4: T
5: F
9: T ∧
7: T
10: T q
10: T 11: F
Figure 1.16. The forcing rules, applied to the DAG of Figure 1.15, detect contradictory constraints at the indicated node – implying that the initial constraint ‘1:T’ cannot be realized. Thus,
formulas represented by this DAG are not satisfiable.
Recall that checking validity of formulas in CNF is very easy. We already hinted at the fact that checking satisfiability of formulas in CNF is hard. To illustrate, consider the formula ((p ∨ (q ∨ r))
∧ ((p ∨ ¬q) ∧ ((q ∨ ¬r) ∧ ((r ∨ ¬p) ∧ (¬p ∨ (¬q ∨ ¬r)))))) (1.11) in CNF – based on Example 4.2, page 77, in [Pap94]. Intuitively, this formula should not be satisfiable. The first and last clause in
(1.11) ‘say’ that at least one of p, q, and r are false and true (respectively). The remaining three clauses, in their conjunction, ‘say’ that p, q, and r all have the same truth value. This cannot
be satisfiable, and a good SAT solver should discover this without any user intervention. Unfortunately, our linear SAT solver can neither detect inconsistent constraints nor compute constraints for
all nodes. Figure 1.17 depicts the DAG for T (φ), where φ is as in (1.11); and reveals
1 Propositional logic
1: T ∧ 2: T ∧ 3: T ∧ 4: T ∧
5: T
6: F ∧ 3: T ¬
4: F ∧
¬ ¬
5: T ¬ 6: F ∧
4: T ¬
2: T ¬
3: F∧
5: F ∧
Figure 1.17. The DAG for the translation of the formula in (1.11). It has a ∧-spine of length 4 as it is a conjunction of five clauses. Its linear analysis gets stuck: all forced constraints are
consistent with each other but several nodes, including all atoms, are unconstrained.
that our SAT solver got stuck: no inconsistent constraints were found and not all nodes obtained constraints; in particular, no atom received a mark! So how can we improve this analysis? Well, we can
mimic the role of LEM to improve the precision of our SAT solver. For the DAG with marks as in Figure 1.17, pick any node n that is not yet marked. Then test node n by making two independent
computations: 1. determine which temporary marks are forced by adding to the marks in Figure 1.17 the T mark only to n; and 2. determine which temporary marks are forced by adding, again to the marks
in Figure 1.17, the F mark only to n.
1.6 SAT solvers
1: T ∧ 2: T ∧ 3: T ∧ 4: T ∧
5: T
6: F ∧
temporary T mark at test node; explore consequences 5: T ¬ 6: F ∧
3: T ¬
b:F ¬
4: F ∧
c:T ¬ i:F
2: T
d:F ∧
h:T ¬
3: F∧ ¬
5: F ∧ c:T ¬
a:T ¬
b:F ¬
f:T ¬
b:F ¬
c:T p
g:F q
c:T r
4: T ¬
contradictory constraints at conjunction
Figure 1.18. Marking an unmarked node with T and exploring what new constraints would follow from this. The analysis shows that this test marking causes contradictory constraints. We use lowercase
letters ‘a:’ etc to denote temporary marks.
If both runs find contradictory constraints, the algorithm stops and reports that T (φ) is unsatisfiable. Otherwise, all nodes that received the same mark in both of these runs receive that very mark
as a permanent one; that is, we update the mark state of Figure 1.17 with all such shared marks. We test any further unmarked nodes in the same manner until we either find contradictory permanent
marks, a complete witness to satisfiability (all nodes have consistent marks), or we have tested all currently unmarked nodes in this manner without detecting any shared marks. Only in the latter case
does the analysis terminate without knowing whether the formulas represented by that DAG are satisfiable.
1 Propositional logic
Example 1.49 We revisit our stuck analysis of Figure 1.17. We test a ¬node and explore the consequences of setting that ¬-node’s mark to T; Figure 1.18 shows the result of that analysis. Dually,
Figure 1.19 tests the consequences of setting that ¬-node’s mark to F. Since both runs reveal a contradiction, the algorithm terminates, ruling that the formula in (1.11) is not satisfiable. In the
exercises, you are asked to show that the specification of our cubic SAT solver is sound. Its running time is indeed cubic in the size of the DAG (and the length of original formula). One factor stems
from the linear SAT solver used in each test run. A second factor is introduced since each unmarked node has to be tested. The third factor is needed since each new permanent mark causes all unmarked
nodes to be tested again. 1: T ∧ 2: T ∧ 3: T ∧ 4: T ∧
5: T
6: F ∧
temporary F mark at test node; 5: T ¬ explore consequences 6: F ∧
3: T ¬
g:F ¬
4: F ∧
f:T ¬ c:F¬
2: T
e:F ∧
d:T ¬
3: F∧ a: F ¬
4: T ¬
e:F ∧
5: F ∧ e:F ¬
d:T ¬
f:T ¬
contradictory constraints at conjunction
Figure 1.19. Marking the same unmarked node with F and exploring what new constraints would follow from this. The analysis shows that this test marking also causes contradictory constraints.
1.6 SAT solvers
1: T ¬ analysis gets stuck right away
testing this node with T renders a contradiction justifying to mark it with F permanently ¬
2: F ∧ ∧
∧ ¬
¬ ∧
Figure 1.20. Testing the indicated node with T causes contradictory constraints, so we may mark that node with F permanently. However, our algorithm does not seem to be able to decide satisfiability
of this DAG even with that optimization.
We deliberately under-specified our cubic SAT solver, but any implementation or optimization decisions need to secure soundness of the analysis. All replies of the form 1. ‘The input formula is not
satisfiable’ and 2. ‘The input formula is satisfiable under the following valuation . . . ’
have to be correct. The third form of reply ‘Sorry, I could not figure this one out.’ is correct by definition. :-) We briefly discuss two sound modifications to the algorithm that introduce some
overhead, but may cause the algorithm to decide many more instances. Consider the state of a DAG right after we have explored consequences of a temporary mark on a test node. 1. If that state –
permanent plus temporary markings – contains contradictory constraints, we can erase all temporary marks and mark the test node permanently with the dual mark of its test. That is, if marking node n
with v resulted in a contradiction, it will get a permanent mark v, where T = F and F = T; otherwise 2. if that state managed to mark all nodes with consistent constraints, we report these markings
as a witness of satisfiability and terminate the algorithm.
If none of these cases apply, we proceed as specified: promote shared marks of the two test runs to permanent ones, if applicable. Example 1.50 To see how one of these optimizations may make a
difference, consider the DAG in Figure 1.20. If we test the indicated node with
1 Propositional logic
T, contradictory constraints arise. Since any witness of satisfiability has to assign some value to that node, we infer that it cannot be T. Thus, we may permanently assign mark F to that node. For
this DAG, such an optimization does not seem to help. No test of an unmarked node detects a shared mark or a shared contradiction. Our cubic SAT solver fails for this DAG.
1.7 Exercises Exercises 1.1 1. Use ¬, →, ∧ and ∨ to express the following declarative sentences in propositional logic; in each case state what your respective propositional atoms p, q, etc. mean: *
(a) If the sun shines today, then it won’t shine tomorrow. (b) Robert was jealous of Yvonne, or he was not in a good mood. (c) If the barometer falls, then either it will rain or it will snow. * (d)
If a request occurs, then either it will eventually be acknowledged, or the requesting process won’t ever be able to make progress. (e) Cancer will not be cured unless its cause is determined and a
new drug for cancer is found. (f) If interest rates go up, share prices go down. (g) If Smith has installed central heating, then he has sold his car or he has not paid his mortgage. * (h) Today it
will rain or shine, but not both. * (i) If Dick met Jane yesterday, they had a cup of coffee together, or they took a walk in the park. (j) No shoes, no shirt, no service. (k) My sister wants a black
and white cat. 2. The formulas of propositional logic below implicitly assume the binding priorities of the logical connectives put forward in Convention 1.3. Make sure that you fully understand
those conventions by reinserting as many brackets as possible. For example, given p ∧ q → r, change it to (p ∧ q) → r since ∧ binds more tightly than →. * (a) ¬p ∧ q → r (b) (p → q) ∧ ¬(r ∨ p → q) *
(c) (p → q) → (r → s ∨ t) (d) p ∨ (¬q → p ∧ r) * (e) p ∨ q → ¬p ∧ r (f) p ∨ p → ¬q * (g) Why is the expression p ∨ q ∧ r problematic?
Exercises 1.2 1. Prove the validity of the following sequents: (a) (p ∧ q) ∧ r, s ∧ t q ∧ s
1.7 Exercises
p∧q q∧p (p ∧ q) ∧ r p ∧ (q ∧ r) p → (p → q), p q q → (p → r), ¬r, q ¬p (p ∧ q) → p p q → (p ∧ q) p (p → q) → q (p → r) ∧ (q → r) p ∧ q → r q → r (p → q) → (p → r) p → (q → r), p → q p → r p → q, r →
s p ∨ r → q ∨ s p ∨ q r → (p ∨ q) ∧ r (p ∨ (q → p)) ∧ q p p → q, r → s p ∧ r → q ∧ s p → q ((p ∧ q) → p) ∧ (p → (p ∧ q)) q → (p → (p → (q → p))) p → q ∧ r (p → q) ∧ (p → r) (p → q) ∧ (p → r) p → q ∧
r (p → q) → ((r → s) → (p ∧ r → q ∧ s)); here you might be able to ‘recycle’ and augment a proof from a previous exercise. (u) p → q ¬q → ¬p * (v) p ∨ (p ∧ q) p (w) r, p → (r → q) p → (q ∧ r) * (x) p
→ (q ∨ r), q → s, r → s p → s * (y) (p ∧ q) ∨ (p ∧ r) p ∧ (q ∨ r). 2. For the sequents below, show which ones are valid and which ones aren’t: * (a) ¬p → ¬q q → p * (b) ¬p ∨ ¬q ¬(p ∧ q) * (c) ¬p, p ∨
q q * (d) p ∨ q, ¬q ∨ r p ∨ r * (e) p → (q ∨ r), ¬q, ¬r ¬p without using the MT rule * (f) ¬p ∧ ¬q ¬(p ∨ q) * (g) p ∧ ¬p ¬(r → q) ∧ (r → q) (h) p → q, s → t p ∨ s → q ∧ t * (i) ¬(¬p ∨ q) p. 3. Prove
the validity of the sequents below: (a) ¬p → p p (b) ¬p p → q (c) p ∨ q, ¬q p * (d) ¬p → (p → (p → q)) (e) ¬(p → q) q → p (f) p → q ¬p ∨ q (g) ¬p ∨ q → (p → q) (b) * (c) (d) * (e) * (f) (g) * (h) *
(i) * (j) (k) * (l) (m) * (n) * (o) (p) (q) * (r) (s) (t)
1 Propositional logic
(h) p → (q ∨ r), ¬q, ¬r |− ¬p (i) (c ∧ n) → t, h ∧ ¬s, h ∧ ¬(s ∨ c) → p |− (n ∧ ¬t) → p (j) the two sequents implict in (1.2) on page 20 (k) q |− (p ∧ q) ∨ (¬p ∧ q) using LEM (l) ¬(p ∧ q) |− ¬p ∨ ¬q
(m) p ∧ q → r |− (p → r) ∨ (q → r) * (n) p ∧ q ¬(¬p ∨ ¬q) (o) ¬(¬p ∨ ¬q) p ∧ q (p) p → q ¬p ∨ q possibly without using LEM? * (q) (p → q) ∨ (q → r) using LEM (r) p → q, ¬p → r, ¬q → ¬r q (s) p → q, r
→ ¬t, q → r p → ¬t (t) (p → q) → r, s → ¬p, t, ¬s ∧ t → q r (u) (s → p) ∨ (t → q) (s → q) ∨ (t → p) (v) (p ∧ q) → r, r → s, q ∧ ¬s ¬p. 4. Explain why intuitionistic logicians also reject the proof
rule PBC. 5. Prove the following theorems of propositional logic: * (a) ((p → q) → q) → ((q → p) → p) (b) Given a proof for the sequent of the previous item, do you now have a quick argument for ((q
→ p) → p) → ((p → q) → q)? (c) ((p → q) ∧ (q → p)) → ((p ∨ q) → (p ∧ q)) * (d) (p → q) → ((¬p → q) → q). 6. Natural deduction is not the only possible formal framework for proofs in propositional
logic. As an abbreviation, we write Γ to denote any finite sequence of formulas φ1 , φ2 , . . . , φn (n ≥ 0). Thus, any sequent may be written as Γ ψ for an appropriate, possibly empty, Γ. In this
exercise we propose a different notion of proof, which states rules for transforming valid sequents into valid sequents. For example, if we have already a proof for the sequent Γ, φ ψ, then we obtain
a proof of the sequent Γ φ → ψ by augmenting this very proof with one application of the rule →i. The new approach expresses this as an inference rule between sequents: Γ, φ ψ →i. Γφ→ψ The rule
‘assumption’ is written as φφ
i.e. the premise is empty. Such rules are called axioms. (a) Express all remaining proof rules of Figure 1.2 in such a form. (Hint: some of your rules may have more than one premise.) (b) Explain why
proofs of Γ ψ in this new system have a tree-like structure with Γ ψ as root. (c) Prove p ∨ (p ∧ q) p in your new proof system.
1.7 Exercises 81 √ 2 cannot be a rational number. Proceed by proof by contradiction: 7. Show that √ assume that 2 is a fraction k/l with integers k and l = 0. On squaring both sides we get 2 = k 2 /
l2 , or equivalently 2l2 = k 2 . We may assume that any common 2 factors of k and l have been cancelled. Can you now argue that 2l2 has a different number of 2 factors from k 2 ? Why would that be a
contradiction and to what? 8. There is an alternative approach to treating negation. One could simply ban the operator ¬ from propositional logic and think of φ → ⊥ as ‘being’ ¬φ. Naturally, such a
logic cannot rely on the natural deduction rules for negation. Which of the rules ¬i, ¬e, ¬¬e and ¬¬i can you simulate with the remaining proof rules by letting ¬φ be φ → ⊥? 9. Let us introduce a new
connective φ ↔ ψ which should abbreviate (φ → ψ) ∧ (ψ → φ). Design introduction and elimination rules for ↔ and show that they are derived rules if φ ↔ ψ is interpreted as (φ → ψ) ∧ (ψ → φ).
Exercises 1.3 In order to facilitate reading these exercises we assume below the usual conventions about binding priorities agreed upon in Convention 1.3. 1. Given the following formulas, draw their
corresponding parse tree: (a) p * (b) p ∧ q (c) p ∧ ¬q → ¬p * (d) p ∧ (¬q → ¬p) (e) p → (¬q ∨ (q → p)) * (f) ¬((¬q ∧ (p → r)) ∧ (r → q)) (g) ¬p ∨ (p → q) (h) (p ∧ q) → (¬r ∨ (q → r)) (i) ((s ∨ (¬p))
→ (¬p)) (j) (s ∨ ((¬p) → (¬p))) (k) (((s → (r ∨ l)) ∨ ((¬q) ∧ r)) → ((¬(p → s)) → r)) (l) (p → q) ∧ (¬r → (q ∨ (¬p ∧ r))). 2. For each formula below, list all its subformulas: * (a) p → (¬p ∨ (¬¬q →
(p ∧ q))) (b) (s → r ∨ l) ∨ (¬q ∧ r) → (¬(p → s) → r) (c) (p → q) ∧ (¬r → (q ∨ (¬p ∧ r))). 3. Draw the parse tree of a formula φ of propositional logic which is * (a) a negation of an implication (b)
a disjunction whose disjuncts are both conjunctions * (c) a conjunction of conjunctions. 4. For each formula below, draw its parse tree and list all subformulas: * (a) ¬(s → (¬(p → (q ∨ ¬s)))) (b)
((p → ¬q) ∨ (p ∧ r) → s) ∨ ¬r.
1 Propositional logic
Figure 1.21. A tree that represents an ill-formed formula. * 5. For the parse tree in Figure 1.22 find the logical formula it represents. 6. For the trees below, find their linear representations and
check whether they correspond to well-formed formulas: (a) the tree in Figure 1.10 on page 44 (b) the tree in Figure 1.23. * 7. Draw a parse tree that represents an ill-formed formula such that (a)
one can extend it by adding one or several subtrees to obtain a tree that represents a well-formed formula; (b) it is inherently ill-formed; i.e. any extension of it could not correspond to a
well-formed formula. 8. Determine, by trying to draw parse trees, which of the following formulas are well-formed: (a) p ∧ ¬(p ∨ ¬q) → (r → s) (b) p ∧ ¬(p ∨ q ∧ s) → (r → s) (c) p ∧ ¬(p ∨ ∧s) → (r →
s). Among the ill-formed formulas above which ones, and in how many ways, could you ‘fix’ by the insertion of brackets only?
Exercises 1.4 * 1. Construct the truth table for ¬p ∨ q and verify that it coincides with the one for p → q. (By ‘coincide’ we mean that the respective columns of T and F values are the same.) 2.
Compute the complete truth table of the formula * (a) ((p → q) → p) → p (b) represented by the parse tree in Figure 1.3 on page 34
1.7 Exercises
p Figure 1.22. A parse tree of a negated implication.
1 Propositional logic
Figure 1.23. Another parse tree of a negated implication. * (c) p ∨ (¬(q ∧ (r → q))) (d) (p ∧ q) → (p ∨ q) (e) ((p → ¬q) → ¬p) → q (f) (p → q) ∨ (p → ¬q) (g) ((p → q) → p) → p (h) ((p ∨ q) → r) → ((p
→ r) ∨ (q → r)) (i) (p → q) → (¬p → ¬q). 3. Given a valuation and a parsetree of a formula, compute the truth value of the formula for that valuation (as done in a bottom-up fashion in Figure 1.7 on
page 40) with the parse tree in * (a) Figure 1.10 on page 44 and the valuation in which q and r evaluate to T and p to F; (b) Figure 1.4 on page 36 and the valuation in which q evaluates to T and p
and r evaluate to F; (c) Figure 1.23 where we let p be T, q be F and r be T; and (d) Figure 1.23 where we let p be F, q be T and r be F. 4. Compute the truth value on the formula’s parse tree, or
specify the corresponding line of a truth table where * (a) p evaluates to F, q to T and the formula is p → (¬q ∨ (q → p)) * (b) the formula is ¬((¬q ∧ (p → r)) ∧ (r → q)), p evaluates to F, q to T
and r evaluates to T.
1.7 Exercises
* 5. A formula is valid iff it computes T for all its valuations; it is satisfiable iff it computes T for at least one of its valuations. Is the formula of the parse tree in Figure 1.10 on page 44
valid? Is it satisfiable? 6. Let ∗ be a new logical connective such that p ∗ q does not hold iff p and q are either both false or both true. (a) Write down the truth table for p ∗ q. (b) Write down the
truth table for (p ∗ p) ∗ (q ∗ q). (c) Does the table in (b) coincide with a table in Figure 1.6 (page 38)? If so, which one? (d) Do you know ∗ already as a logic gate in circuit design? If so, what
is it called? 7. These exercises let you practice proofs using mathematical induction. Make sure that you state your base case and inductive step clearly. You should also indicate where you apply the
induction hypothesis. (a) Prove that (2 · 1 − 1) + (2 · 2 − 1) + (2 · 3 − 1) + · · · + (2 · n − 1) = n2 by mathematical induction on n ≥ 1. (b) Let k and l be natural numbers. We say that k is
divisible by l if there exists a natural number p such that k = p · l. For example, 15 is divisible by 3 because 15 = 5 · 3. Use mathematical induction to show that 11n − 4n is divisible by 7 for all
natural numbers n ≥ 1. * (c) Use mathematical induction to show that 12 + 22 + 32 + · · · + n2 =
n · (n + 1) · (2n + 1) 6
for all natural numbers n ≥ 1. * (d) Prove that 2n ≥ n + 12 for all natural numbers n ≥ 4. Here the base case is n = 4. Is the statement true for any n < 4? (e) Suppose a post office sells only 2c| and
3c| stamps. Show that any postage of 2c| , or over, can be paid for using only these stamps. Hint: use mathematical induction on n, where nc| is the postage. In the inductive step consider two
possibilities: first, nc| can be paid for using only 2c| stamps. Second, paying nc| requires the use of at least one 3c| stamp. (f) Prove that for every prefix of a well-formed propositional logic
formula the number of left brackets is greater or equal to the number of right brackets. * 8. The Fibonacci numbers are most useful in modelling the growth of populations. def def def We define them
by F1 = 1, F2 = 1 and Fn+1 = Fn + Fn−1 for all n ≥ 2. So def F3 = F1 + F2 = 1 + 1 = 2 etc. Show the assertion ‘F3n is even.’ by mathematical induction on n ≥ 1. Note that this assertion is saying
that the sequence F3 , F6 , F9 , . . . consists of even numbers only.
1 Propositional logic
9. Consider the function rank, defined by rank(p) = 1 def
rank(¬φ) = 1 + rank(φ) def
rank(φ ◦ ψ) = 1 + max(rank(φ), rank(ψ)) def
where p is any atom, ◦ ∈ {→, ∨, ∧} and max(n, m) is n if n ≥ m and m otherwise. Recall the concept of the height of a formula (Definition 1.32 on page 44). Use mathematical induction on the height of
φ to show that rank(φ) is nothing but the height of φ for all formulas φ of propositional logic. * 10. Here is an example of why we need to secure the base case for mathematical induction. Consider
the assertion ‘The number n2 + 5n + 1 is even for all n ≥ 1.’ (a) Prove the inductive step of that assertion. (b) Show that the base case fails to hold. (c) Conclude that the assertion is false. (d)
Use mathematical induction to show that n2 + 5n + 1 is odd for all n ≥ 1. 11. For the soundness proof of Theorem 1.35 on page 46, (a) explain why we could not use mathematical induction but had to
resort to course-of-values induction; (b) give justifications for all inferences that were annotated with ‘why?’ and (c) complete the case analysis ranging over the final proof rule applied; inspect
the summary of natural deduction rules in Figure 1.2 on page 27 to see which cases are still missing. Do you need to include derived rules? 12. Show that the following sequents are not valid by
finding a valuation in which the truth values of the formulas to the left of are T and the truth value of the formula to the right of is F. (a) ¬p ∨ (q → p) ¬p ∧ q (b) ¬r → (p ∨ q), r ∧ ¬q r → q * (c)
p → (q → r) p → (r → q) (d) ¬p, p ∨ q ¬q (e) p → (¬q ∨ r), ¬r ¬q → ¬p. 13. For each of the following invalid sequents, give examples of natural language declarative sentences for the atoms p, q and r
such that the premises are true, but the conclusion false. * (a) p ∨ q p ∧ q * (b) ¬p → ¬q ¬q → ¬p (c) p → q p ∨ q (d) p → (q ∨ r) (p → q) ∧ (p → r). 14. Find a formula of propositional logic φ which
contains only the atoms p, q and r and which is true only when p and q are false, or when ¬q ∧ (p ∨ r) is true.
1.7 Exercises
15. Use mathematical induction on n to prove the theorem ((φ1 ∧ (φ2 ∧ (· · · ∧ φn ) . . . ) → ψ) → (φ1 → (φ2 → (. . . (φn → ψ) . . . )))). 16. Prove the validity of the following sequents needed to
secure the completeness result for propositional logic: (a) φ1 ∧ ¬φ2 ¬(φ1 → φ2 ) (b) ¬φ1 ∧ ¬φ2 φ1 → φ2 (c) ¬φ1 ∧ φ2 φ1 → φ2 (d) φ1 ∧ φ2 φ1 → φ2 (e) ¬φ1 ∧ φ2 ¬(φ1 ∧ φ2 ) (f) ¬φ1 ∧ ¬φ2 ¬(φ1 ∧ φ2 ) (g)
φ1 ∧ ¬φ2 ¬(φ1 ∧ φ2 ) (h) ¬φ1 ∧ ¬φ2 ¬(φ1 ∨ φ2 ) (i) φ1 ∧ φ2 φ1 ∨ φ2 (j) ¬φ1 ∧ φ2 φ1 ∨ φ2 (k) φ1 ∧ ¬φ2 φ1 ∨ φ2 . 17. Does φ hold for the φ below? Please justify your answer. (a) (p → q) ∨ (q → r) * (b)
((q → (p ∨ (q → p))) ∨ ¬(p → q)) → p.
Exercises 1.5 1. Show that a formula φ is valid iff ≡ φ, where is an abbreviation for an instance p ∨ ¬p of LEM. 2. Which of these formulas are semantically equivalent to p → (q ∨ r)? (a) q ∨ (¬p ∨ r)
* (b) q ∧ ¬r → p (c) p ∧ ¬r → q * (d) ¬q ∧ ¬r → ¬p. 3. An adequate set of connectives for propositional logic is a set such that for every formula of propositional logic there is an equivalent
formula with only connectives from that set. For example, the set {¬, ∨} is adequate for propositional logic, because any occurrence of ∧ and → can be removed by using the equivalences φ → ψ ≡ ¬φ ∨ ψ
and φ ∧ ψ ≡ ¬(¬φ ∨ ¬ψ). (a) Show that {¬, ∧}, {¬, →} and {→, ⊥} are adequate sets of connectives for propositional logic. (In the latter case, we are treating ⊥ as a nullary connective.) (b) Show
that, if C ⊆ {¬, ∧, ∨, →, ⊥} is adequate for propositional logic, then ¬ ∈ C or ⊥ ∈ C. (Hint: suppose C contains neither ¬ nor ⊥ and consider the truth value of a formula φ, formed by using only the
connectives in C, for a valuation in which every atom is assigned T.) (c) Is {↔, ¬} adequate? Prove your answer. 4. Use soundness or completeness to show that a sequent φ1 , φ2 , . . . , φn ψ has a
proof iff φ1 → φ2 → . . . φn → ψ is a tautology.
1 Propositional logic
5. Show that the relation ≡ is (a) reflexive: φ ≡ φ holds for all φ (b) symmetric: φ ≡ ψ implies ψ ≡ φ and (c) transitive: φ ≡ ψ and ψ ≡ η imply φ ≡ η. 6. Show that, with respect to ≡, (a) ∧ and ∨ are
idempotent: i. φ ∧ φ ≡ φ ii. φ ∨ φ ≡ φ (b) ∧ and ∨ are commutative: i. φ ∧ ψ ≡ ψ ∧ φ ii. φ ∨ ψ ≡ ψ ∨ φ (c) ∧ and ∨ are associative: i. φ ∧ (ψ ∧ η) ≡ (φ ∧ ψ) ∧ η ii. φ ∨ (ψ ∨ η) ≡ (φ ∨ ψ) ∨ η (d) ∧
and ∨ are absorptive: * i. φ ∧ (φ ∨ η) ≡ φ ii. φ ∨ (φ ∧ η) ≡ φ (e) ∧ and ∨ are distributive: i. φ ∧ (ψ ∨ η) ≡ (φ ∧ ψ) ∨ (φ ∧ η) * ii. φ ∨ (ψ ∧ η) ≡ (φ ∨ ψ) ∧ (φ ∨ η) (f) ≡ allows for double negation:
φ ≡ ¬¬φ and (g) ∧ and ∨ satisfies the de Morgan rules: i. ¬(φ ∧ ψ) ≡ ¬φ ∨ ¬ψ * ii. ¬(φ ∨ ψ) ≡ ¬φ ∧ ¬ψ. 7. Construct a formula in CNF based on each of the following truth tables: * (a) q φ1 T F T F F F
F T
p T F T F * (b) p T T T F T F F F
q T T F T F T F F
r φ2 T T F F T F T T F F F F T T F F
1.7 Exercises
(c) r T T T F T F F F
s T T F T F T F F
q φ3 T F F T T F T F F T F F T F F T
* 8. Write a recursive function IMPL FREE which requires a (parse tree of a) propositional formula as input and produces an equivalent implication-free formula as output. How many clauses does your
case statement need? Recall Definition 1.27 on page 32. * 9. Compute CNF (NNF (IMPL FREE ¬(p → (¬(q ∧ (¬p → q)))))). 10. Use structural induction on the grammar of formulas in CNF to show that the
‘otherwise’ case in calls to DISTR applies iff both η1 and η2 are of type D in (1.6) on page 55. 11. Use mathematical induction on the height of φ to show that the call CNF (NNF (IMPL FREE φ))
returns, up to associativity, φ if the latter is already in CNF. 12. Why do the functions CNF and DISTR preserve NNF and why is this important? 13. For the call CNF (NNF (IMPL FREE (φ))) on a formula
φ of propositional logic, explain why (a) its output is always a formula in CNF (b) its output is semantically equivalent to φ (c) that call always terminates. 14. Show that all the algorithms
presented in Section 1.5.2 terminate on any input meeting their precondition. Can you formalise some of your arguments? Note that algorithms might not call themselves again on formulas with smaller
height. E.g. the call of CNF (φ1 ∨ φ2 ) results in a call DISTR (CNF(φ1 ), CNF(φ2 )), where CNF(φi ) may have greater height than φi . Why is this not a problem? 15. Apply algorithm HORN from page 66
to each of these Horn formulas: * (a) (p ∧ q ∧ w → ⊥) ∧ (t → ⊥) ∧ (r → p) ∧ ( → r) ∧ ( → q) ∧ (u → s) ∧ ( → u) (b) (p ∧ q ∧ w → ⊥) ∧ (t → ⊥) ∧ (r → p) ∧ ( → r) ∧ ( → q) ∧ (r ∧ u → w) ∧ (u → s) ∧ ( →
u) (c) (p ∧ q ∧ s → p) ∧ (q ∧ r → p) ∧ (p ∧ s → s) (d) (p ∧ q ∧ s → ⊥) ∧ (q ∧ r → p) ∧ ( → s) (e) (p5 → p11 ) ∧ (p2 ∧ p3 ∧ p5 → p13 ) ∧ ( → p5 ) ∧ (p5 ∧ p11 → ⊥) (f) ( → q) ∧ ( → s) ∧ (w → ⊥) ∧ (p ∧
q ∧ s → ⊥) ∧ (v → s) ∧ ( → r) ∧ (r → p)
1 Propositional logic
* (g) ( → q) ∧ ( → s) ∧ (w → ⊥) ∧ (p ∧ q ∧ s → v) ∧ (v → s) ∧ ( → r) ∧ (r → p). 16. Explain why the algorithm HORN fails to work correctly if we change the concept of Horn formulas by extending the
clause for P on page 65 to P ::= ⊥ | | p | ¬p? 17. What can you say about the CNF of Horn formulas. More precisely, can you specify syntactic criteria for a CNF that ensure that there is an
equivalent Horn formula? Can you describe informally programs which would translate from one form of representation into another?
Exercises 1.6 1. Use mathematical induction to show that, for all φ of (1.3) on page 33, (a) T (φ) can be generated by (1.10) on page 69, (b) T (φ) has the same set of valuations as φ, and (c) the
set of valuations in which φ is true equals the set of valuations in which T (φ) is true. * 2. Show that all rules of Figure 1.14 (page 71) are sound: if all current marks satisfy the invariant (1.9)
from page 68, then this invariant still holds if the derived constraint of that rule becomes an additional mark. 3. In Figure 1.16 on page 73 we detected a contradiction which secured the validity of
the sequent p ∧ q → r p → q → r. Use the same method with the linear SAT solver to show that the sequent (p → q) ∨ (r → p) is valid. (This is interesting since we proved this validity in natural
deduction with a judicious choice of the proof rule LEM; and the linear SAT solver does not employ any case analysis.) * 4. Consider the sequent p ∨ q, p → r r. Determine a DAG which is not
satisfiable iff this sequent is valid. Tag the DAG’s root node with ‘1: T,’ apply the forcing laws to it, and extract a witness to the DAG’s satisfiability. Explain in what sense this witness serves as
an explanation for the fact that p ∨ q, p → r r is not valid. 5. Explain in what sense the SAT solving technique, as presented in this chapter, can be used to check whether formulas are tautologies.
6. For φ from (1.10), can one reverse engineer φ from the DAG of T (φ)? 7. Consider a modification of our method which initially tags a DAG’s root node with ‘1: F.’ In that case, (a) are the forcing
laws still sound? If so, state the invariant. (b) what can we say about the formula(s) a DAG represents if i. we detect contradictory constraints? ii. we compute consistent forced constraints for
each node? 8. Given an arbitrary Horn formula φ, compare our linear SAT solver – applied to T (φ) – to the marking algorithm – applied to φ. Discuss similarities and differences of these approaches.
1.8 Bibliographic notes
9. Consider Figure 1.20 on page 77. Verify that (a) its test produces contradictory constraints (b) its cubic analysis does not decide satisfiability, regardless of whether the two optimizations we
described are present. 10. Verify that the DAG of Figure 1.17 (page 74) is indeed the one obtained for T (φ), where φ is the formula in (1.11) on page 73. * 11. An implementor may be concerned with
the possibility that the answers to the cubic SAT solver may depend on a particular order in which we test unmarked nodes or use the rules in Figure 1.14. Give a semi-formal argument for why the
analysis results don’t depend on such an order. 12. Find a formula φ such that our cubic SAT solver cannot decide the satisfiability of T (φ). 13. Advanced Project: Write a complete implementation of
the cubic SAT solver described in Section 1.6.2. It should read formulas from the keyboard or a file; should assume right-associativity of ∨, ∧, and → (respectively); compute the DAG of T (φ); perform
the cubic SAT solver next. Think also about including appropriate user output, diagnostics, and optimizations. 14. Show that our cubic SAT solver specified in this section (a) terminates on all
syntactically correct input; (b) satisfies the invariant (1.9) after the first permanent marking; (c) preserves (1.9) for all permanent markings it makes; (d) computes only correct satisfiability
witnesses; (e) computes only correct ‘not satisfiable’ replies; and (f) remains to be correct under the two modifications described on page 77 for handling results of a node’s two test runs.
1.8 Bibliographic notes Logic has a long history stretching back at least 2000 years, but the truthvalue semantics of propositional logic presented in this and every logic textbook today was invented
only about 160 years ago, by G. Boole [Boo54]. Boole used the symbols + and · for disjunction and conjunction. Natural deduction was invented by G. Gentzen [Gen69], and further developed by D.
Prawitz [Pra65]. Other proof systems existed before then, notably axiomatic systems which present a small number of axioms together with the rule modus ponens (which we call →e). Proof systems often
present as small a number of axioms as possible; and only for an adequate set of connectives such as → and ¬. This makes them hard to use in practice. Gentzen improved the situation by inventing the
idea of working with assumptions (used by the rules →i, ¬i and ∨e) and by treating all the connectives separately.
1 Propositional logic
Our linear and cubic SAT solvers are variants of St˚ almarck’s method [SS90], a SAT solver which is patented in Sweden and in the United States of America. Further historical remarks, and also
pointers to other contemporary books about propositional and predicate logic, can be found in the bibliographic remarks at the end of Chapter 2. For an introduction to algorithms and data structures
see e.g. [Wei98].
2 Predicate logic
2.1 The need for a richer language In the first chapter, we developed propositional logic by examining it from three different angles: its proof theory (the natural deduction calculus), its syntax (the
tree-like nature of formulas) and its semantics (what these formulas actually mean). From the outset, this enterprise was guided by the study of declarative sentences, statements about the world
which can, for every valuation or model, be given a truth value. We begin this second chapter by pointing out the limitations of propositional logic with respect to encoding declarative sentences.
Propositional logic dealt quite satisfactorily with sentence components like not, and, or and if . . . then, but the logical aspects of natural and artificial languages are much richer than that. What
can we do with modifiers like there exists . . . , all . . . , among . . . and only . . . ? Here, propositional logic shows clear limitations and the desire to express more subtle declarative
sentences led to the design of predicate logic, which is also called first-order logic. Let us consider the declarative sentence Every student is younger than some instructor.
In propositional logic, we could identify this assertion with a propositional atom p. However, that fails to reflect the finer logical structure of this sentence. What is this statement about? Well, it
is about being a student, being an instructor and being younger than somebody else. These are all properties of some sort, so we would like to have a mechanism for expressing them together with their
logical relationships and dependences. We now use predicates for that purpose. For example, we could write S(andy) to denote that Andy is a student and I(paul) to say that Paul is an instructor.
Likewise, Y (andy, paul) could mean that Andy is younger than 93
2 Predicate logic
Paul. The symbols S, I and Y are called predicates. Of course, we have to be clear about their meaning. The predicate Y could have meant that the second person is younger than the first one, so we
need to specify exactly what these symbols refer to. Having such predicates at our disposal, we still need to formalise those parts of the sentence above which speak of every and some. Obviously,
this sentence refers to the individuals that make up some academic community (left implicit by the sentence), like Kansas State University or the University of Birmingham, and it says that for each
student among them there is an instructor among them such that the student is younger than the instructor. These predicates are not yet enough to allow us to express the sentence in (2.1). We don’t
really want to write down all instances of S(·) where · is replaced by every student’s name in turn. Similarly, when trying to codify a sentence having to do with the execution of a program, it would
be rather laborious to have to write down every state of the computer. Therefore, we employ the concept of a variable. Variables are written u, v, w, x, y, z, . . . or x1 , y3 , u5 , . . . and can be
thought of as place holders for concrete values (like a student, or a program state). Using variables, we can now specify the meanings of S, I and Y more formally: S(x) :
x is a student
I(x) :
x is an instructor
Y (x, y) :
x is younger than y.
Note that the names of the variables are not important, provided that we use them consistently. We can state the intended meaning of I by writing I(y) :
y is an instructor
or, equivalently, by writing I(z) :
z is an instructor.
Variables are mere place holders for objects. The availability of variables is still not sufficient for capturing the essence of the example sentence above. We need to convey the meaning of ‘Every
student x is younger than some instructor y.’ This is where we need to introduce quantifiers ∀ (read: ‘for all’) and ∃ (read: ‘there exists’ or ‘for some’) which always come attached to a variable,
as in ∀x (‘for all x’) or in ∃z (‘there exists z’, or ‘there is some z’). Now we can write the example sentence in an entirely symbolic way as ∀x (S(x) → (∃y (I(y) ∧ Y (x, y)))).
2.1 The need for a richer language
Actually, this encoding is rather a paraphrase of the original sentence. In our example, the re-translation results in For every x, if x is a student, then there is some y which is an instructor such
that x is younger than y.
Different predicates can have a different number of arguments. The predicates S and I have just one (they are called unary predicates), but predicate Y requires two arguments (it is called a binary
predicate). Predicates with any finite number of arguments are possible in predicate logic. Another example is the sentence Not all birds can fly. For that we choose the predicates B and F which have
one argument expressing B(x) :
x is a bird
F (x) :
x can fly.
The sentence ‘Not all birds can fly’ can now be coded as ¬(∀x (B(x) → F (x))) saying: ‘It is not the case that all things which are birds can fly.’ Alternatively, we could code this as ∃x (B(x) ∧ ¬F
(x)) meaning: ‘There is some x which is a bird and cannot fly.’ Note that the first version is closer to the linguistic structure of the sentence above. These two formulas should evaluate to T in the
world we currently live in since, for example, penguins are birds which cannot fly. Shortly, we address how such formulas can be given their meaning in general. We will also explain why formulas like
the two above are indeed equivalent semantically. Coding up complex facts expressed in English sentences as logical formulas in predicate logic is important – e.g. in software design with UML or in
formal specification of safety-critical systems – and much more care must be taken than in the case of propositional logic. However, once this translation has been accomplished our main objective is
to reason symbolically () or semantically () about the information expressed in those formulas. In Section 2.3, we extend our natural deduction calculus of propositional logic so that it covers
logical formulas of predicate logic as well. In this way we are able to prove the validity of sequents φ1 , φ2 , . . . , φn ψ in a similar way to that in the first chapter.
2 Predicate logic
In Section 2.4, we generalize the valuations of Chapter 1 to a proper notion of models, real or artificial worlds in which formulas of predicate logic can be true or false, which allows us to define
semantic entailment φ1 , φ2 , . . . , φn ψ. The latter expresses that, given any such model in which all φ1 , φ2 , . . . , φn hold, it is the case that ψ holds in that model as well. In that case,
one also says that ψ is semantically entailed by φ1 , φ2 , . . . , φn . Although this definition of semantic entailment closely matches the one for propositional logic in Definition 1.34, the process
of evaluating a predicate formula differs from the computation of truth values for propositional logic in the treatment of predicates (and functions). We discuss it in detail in Section 2.4. It is
outside the scope of this book to show that the natural deduction calculus for predicate logic is sound and complete with respect to semantic entailment; but it is indeed the case that φ1 , φ2 , . .
. , φn ψ
φ1 , φ2 , . . . , φn ψ
for formulas of the predicate calculus. The first proof of this was done by the mathematician K. G¨ odel. What kind of reasoning must predicate logic be able to support? To get a feel for that, let us
consider the following argument: No books are gaseous. Dictionaries are books. Therefore, no dictionary is gaseous.
The predicates we choose are B(x) :
x is a book
G(x) :
x is gaseous
D(x) :
x is a dictionary.
Evidently, we need to build a proof theory and semantics that allow us to derive the validity and semantic entailment, respectively, of ¬∃x (B(x) ∧ G(x)), ∀x (D(x) → B(x)) ¬∃x (D(x) ∧ G(x)) ¬∃x (B(x)
∧ G(x)), ∀x (D(x) → B(x)) ¬∃x (D(x) ∧ G(x)). Verify that these sequents express the argument above in a symbolic form. Predicate logic extends propositional logic not only with quantifiers but with
one more concept, that of function symbols. Consider the declarative sentence Every child is younger than its mother.
2.1 The need for a richer language
Using predicates, we could express this sentence as ∀x ∀y (C(x) ∧ M (y, x) → Y (x, y)) where C(x) means that x is a child, M (x, y) means that x is y’s mother and Y (x, y) means that x is younger
than y. (Note that we actually used M (y, x) (y is x’s mother), not M (x, y).) As we have coded it, the sentence says that, for all children x and any mother y of theirs, x is younger than y. It is
not very elegant to say ‘any of x’s mothers’, since we know that every individual has one and only one mother1 . The inelegance of coding ‘mother’ as a predicate is even more apparent if we consider
the sentence Andy and Paul have the same maternal grandmother.
which, using ‘variables’ a and p for Andy and Paul and a binary predicate M for mother as before, becomes ∀x ∀y ∀u ∀v (M (x, y) ∧ M (y, a) ∧ M (u, v) ∧ M (v, p) → x = u). This formula says that, if y
and v are Andy’s and Paul’s mothers, respectively, and x and u are their mothers (i.e. Andy’s and Paul’s maternal grandmothers, respectively), then x and u are the same person. Notice that we used a
special predicate in predicate logic, equality; it is a binary predicate, i.e. it takes two arguments, and is written =. Unlike other predicates, it is usually written in between its arguments rather
than before them; that is, we write x = y instead of = (x, y) to say that x and y are equal. The function symbols of predicate logic give us a way of avoiding this ugly encoding, for they allow us to
represent y’s mother in a more direct way. Instead of writing M (x, y) to mean that x is y’s mother, we simply write m(y) to mean y’s mother. The symbol m is a function symbol: it takes one argument
and returns the mother of that argument. Using m, the two sentences above have simpler encodings than they had using M : ∀x (C(x) → Y (x, m(x))) now expresses that every child is younger than its
mother. Note that we need only one variable rather than two. Representing that Andy and Paul have the same maternal grandmother is even simpler; it is written m(m(a)) = m(m(p)) quite directly saying
that Andy’s maternal grandmother is the same person as Paul’s maternal grandmother. 1
We assume that we are talking about genetic mothers, not adopted mothers, step mothers etc.
2 Predicate logic
One can always do without function symbols, by using a predicate symbol instead. However, it is usually neater to use function symbols whenever possible, because we get more compact encodings.
However, function symbols can be used only in situations in which we want to denote a single object. Above, we rely on the fact that every individual has a uniquely defined mother, so that we can talk
about x’s mother without risking any ambiguity (for example, if x had no mother, or two mothers). For this reason, we cannot have a function symbol b(·) for ‘brother’. It might not make sense to talk
about x’s brother, for x might not have any brothers, or he might have several. ‘Brother’ must be coded as a binary predicate. To exemplify this point further, if Mary has several brothers, then the
claim that ‘Ann likes Mary’s brother’ is ambiguous. It might be that Ann likes one of Mary’s brothers, which we would write as ∃x (B(x, m) ∧ L(a, x)) where B and L mean ‘is brother of’ and ‘likes,’
and a and m mean Ann and Mary. This sentence says that there exists an x which is a brother of Mary and is liked by Ann. Alternatively, if Ann likes all of Mary’s brothers, we write it as ∀x (B(x, m)
→ L(a, x)) saying that any x which is a brother of Mary is liked by Ann. Predicates should be used if a ‘function’ such as ‘your youngest brother’ does not always have a value. Different function
symbols may take different numbers of arguments. Functions may take zero arguments and are then called constants: a and p above are constants for Andy and Paul, respectively. In a domain involving
students and the grades they get in different courses, one might have the binary function symbol g(·, ·) taking two arguments: g(x, y) refers to the grade obtained by student x in course y.
2.2 Predicate logic as a formal language The discussion of the preceding section was intended to give an impression of how we code up sentences as formulas of predicate logic. In this section, we
will be more precise about it, giving syntactic rules for the formation of predicate logic formulas. Because of the power of predicate logic, the language is much more complex than that of
propositional logic. The first thing to note is that there are two sorts of things involved in a predicate logic formula. The first sort denotes the objects that we are
2.2 Predicate logic as a formal language
talking about: individuals such as a and p (referring to Andy and Paul) are examples, as are variables such as x and v. Function symbols also allow us to refer to objects: thus, m(a) and g(x, y) are
also objects. Expressions in predicate logic which denote objects are called terms. The other sort of things in predicate logic denotes truth values; expressions of this kind are formulas: Y (x, m
(x)) is a formula, though x and m(x) are terms. A predicate vocabulary consists of three sets: a set of predicate symbols P, a set of function symbols F and a set of constant symbols C. Each
predicate symbol and each function symbol comes with an arity, the number of arguments it expects. In fact, constants can be thought of as functions which don’t take any arguments (and we even drop
the argument brackets) – therefore, constants live in the set F together with the ‘true’ functions which do take arguments. From now on, we will drop the set C, since it is convenient to do so, and
stipulate that constants are 0-arity, so-called nullary, functions.
2.2.1 Terms The terms of our language are made up of variables, constant symbols and functions applied to those. Functions may be nested, as in m(m(x)) or g(m(a), c): the grade obtained by Andy’s
mother in the course c. Definition 2.1 Terms are defined as follows. r Any variable is a term. r If c ∈ F is a nullary function, then c is a term. r If t1 , t2 , . . . , tn are terms and f ∈ F has
arity n > 0, then f (t1 , t2 , . . . , tn ) is a term. r Nothing else is a term.
In Backus Naur form we may write t ::= x | c | f (t, . . . , t) where x ranges over a set of variables var, c over nullary function symbols in F, and f over those elements of F with arity n > 0. It
is important to note that r the first building blocks of terms are constants (nullary functions) and variables; r more complex terms are built from function symbols using as many previously built
terms as required by such function symbols; and r the notion of terms is dependent on the set F. If you change it, you change the set of terms.
2 Predicate logic
Example 2.2 Suppose n, f and g are function symbols, respectively nullary, unary and binary. Then g(f (n), n) and f (g(n, f (n))) are terms, but g(n) and f (f (n), n) are not (they violate the
arities). Suppose 0, 1, . . . are nullary, s is unary, and +, −, and ∗ are binary. Then ∗(−(2, +(s(x), y)), x) is a term, whose parse tree is illustrated in Figure 2.14 (page 159). Usually, the
binary symbols are written infix rather than prefix; thus, the term is usually written (2 − (s(x) + y)) ∗ x.
2.2.2 Formulas The choice of sets P and F for predicate and function symbols, respectively, is driven by what we intend to describe. For example, if we work on a database representing relations
between our kin we might want to consider P = {M, F, S, D}, referring to being male, being female, being a son of . . . and being a daughter of . . . . Naturally, F and M are unary predicates (they
take one argument) whereas D and S are binary (taking two). Similarly, we may define F = {mother-of, f ather-of }. We already know what the terms over F are. Given that knowledge, we can now proceed
to define the formulas of predicate logic. Definition 2.3 We define the set of formulas over (F, P) inductively, using the already defined set of terms over F: r If P ∈ P is a predicate symbol of arity
n ≥ 1, and if t , t , . . . , t are terms over 1 2 n F, then P (t1 , t2 , . . . , tn ) is a formula. r If φ is a formula, then so is (¬φ). r If φ and ψ are formulas, then so are (φ ∧ ψ), (φ ∨ ψ) and
(φ → ψ). r If φ is a formula and x is a variable, then (∀x φ) and (∃x φ) are formulas. r Nothing else is a formula.
Note how the arguments given to predicates are always terms. This can also be seen in the Backus Naur form (BNF) for predicate logic: φ ::= P (t1 , t2 , . . . , tn ) | (¬φ) | (φ ∧ φ) | (φ ∨ φ) | (φ →
φ) | (∀x φ) | (∃x φ) (2.2) where P ∈ P is a predicate symbol of arity n ≥ 1, ti are terms over F and x is a variable. Recall that each occurrence of φ on the right-hand side of the ::= stands for any
formula already constructed by these rules. (What role could predicate symbols of arity 0 play?)
2.2 Predicate logic as a formal language
Figure 2.1. A parse tree of a predicate logic formula.
Convention 2.4 For convenience, we retain the usual binding priorities agreed upon in Convention 1.3 and add that ∀y and ∃y bind like ¬. Thus, the order is: r ¬, ∀y and ∃y bind most tightly; r then ∨
and ∧; r then →, which is right-associative.
We also often omit brackets around quantifiers, provided that doing so introduces no ambiguities. Predicate logic formulas can be represented by parse trees. For example, the parse tree in Figure 2.1
represents the formula ∀x ((P (x) → Q(x)) ∧ S(x, y)).
Example 2.5 Consider translating the sentence Every son of my father is my brother.
into predicate logic. As before, the design choice is whether we represent ‘father’ as a predicate or as a function symbol. 1. As a predicate. We choose a constant m for ‘me’ or ‘I,’ so m is a term,
and we choose further {S, F, B} as the set of predicates with meanings
2 Predicate logic S(x, y) :
x is a son of y
F (x, y) :
x is the father of y
B(x, y) :
x is a brother of y.
Then the symbolic encoding of the sentence above is ∀x ∀y (F (x, m) ∧ S(y, x) → B(y, m))
saying: ‘For all x and all y, if x is a father of m and if y is a son of x, then y is a brother of m.’ 2. As a function. We keep m, S and B as above and write f for the function which, given an
argument, returns the corresponding father. Note that this works only because fathers are unique and always defined, so f really is a function as opposed to a mere relation. The symbolic encoding of
the sentence above is now ∀x (S(x, f (m)) → B(x, m))
meaning: ‘For all x, if x is a son of the father of m, then x is a brother of m;’ it is less complex because it involves only one quantifier.
Formal specifications require domain-specific knowledge. Domain-experts often don’t make some of this knowledge explicit, so a specifier may miss important constraints for a model or implementation.
For example, the specification in (2.3) and (2.4) may seem right, but what about the case when the values of x and m are equal? If the domain of kinship is not common knowledge, then a specifier may
not realize that a man cannot be his own brother. Thus, (2.3) and (2.4) are not completely correct!
2.2.3 Free and bound variables The introduction of variables and quantifiers allows us to express the notions of all . . . and some . . . Intuitively, to verify that ∀x Q(x) is true amounts to
replacing x by any of its possible values and checking that Q holds for each one of them. There are two important and different senses in which such formulas can be ‘true.’ First, if we give concrete
meanings to all predicate and function symbols involved we have a model and can check whether a formula is true for this particular model. For example, if a formula encodes a required behaviour of a
hardware circuit, then we would want to know whether it is true for the model of the circuit. Second, one sometimes would like to ensure that certain formulas are true for all models. Consider P (c)
∧ ∀y(P (y) → Q(y)) → Q(c) for a constant c; clearly, this formula should be true no matter what model we are looking at. It is this second kind of truth which is the primary focus of Section 2.3.
2.2 Predicate logic as a formal language
Unfortunately, things are more complicated if we want to define formally what it means for a formula to be true in a given model. Ideally, we seek a definition that we could use to write a computer
program verifying that a formula holds in a given model. To begin with, we need to understand that variables occur in different ways. Consider the formula ∀x ((P (x) → Q(x)) ∧ S(x, y)). We draw its
parse tree in the same way as for propositional formulas, but with two additional sorts of nodes: r The quantifiers ∀x and ∃y form nodes and have, like negation, just one subtree. r Predicate
expressions, which are generally of the form P (t1 , t2 , . . . , tn ), have the symbol P as a node, but now P has n many subtrees, namely the parse trees of the terms t1 , t2 , . . . , tn .
So in our particular case above we arrive at the parse tree in Figure 2.1. You can see that variables occur at two different sorts of places. First, they appear next to quantifiers ∀ and ∃ in nodes
like ∀x and ∃z; such nodes always have one subtree, subsuming their scope to which the respective quantifier applies. The other sort of occurrence of variables is leaf nodes containing variables. If
variables are leaf nodes, then they stand for values that still have to be made concrete. There are two principal such occurrences: 1. In our example in Figure 2.1, we have three leaf nodes x. If we
walk up the tree beginning at any one of these x leaves, we run into the quantifier ∀x. This means that those occurrences of x are actually bound to ∀x so they represent, or stand for, any possible
value of x. 2. In walking upwards, the only quantifier that the leaf node y runs into is ∀x but that x has nothing to do with y; x and y are different place holders. So y is free in this formula. This
means that its value has to be specified by some additional information, for example, the contents of a location in memory.
Definition 2.6 Let φ be a formula in predicate logic. An occurrence of x in φ is free in φ if it is a leaf node in the parse tree of φ such that there is no path upwards from that node x to a node ∀x
or ∃x. Otherwise, that occurrence of x is called bound. For ∀x φ, or ∃x φ, we say that φ – minus any of φ’s subformulas ∃x ψ, or ∀x ψ – is the scope of ∀x, respectively ∃x. Thus, if x occurs in φ,
then it is bound if, and only if, it is in the scope of some ∃x or some ∀x; otherwise it is free. In terms of parse trees, the scope of a quantifier is just its subtree, minus any subtrees which
re-introduce a
2 Predicate logic
y free
x bound
x bound
x free
Figure 2.2. A parse tree of a predicate logic formula illustrating free and bound occurrences of variables.
quantifier for x; e.g. the scope of ∀x in ∀x (P (x) → ∃x Q(x)) is P (x). It is quite possible, and common, that a variable is bound and free in a formula. Consider the formula (∀x (P (x) ∧ Q(x))) →
(¬P (x) ∨ Q(y)) and its parse tree in Figure 2.2. The two x leaves in the subtree of ∀x are bound since they are in the scope of ∀x, but the leaf x in the right subtree of → is free since it is not
in the scope of any quantifier ∀x or ∃x. Note, however, that a single leaf either is under the scope of a quantifier, or it isn’t. Hence individual occurrences of variables are either free or bound,
never both at the same time.
2.2.4 Substitution Variables are place holders so we must have some means of replacing them with more concrete information. On the syntactic side, we often need to replace a leaf node x by the parse
tree of an entire term t. Recall from the definition of formulas that any replacement of x may only be a term; it could not be a predicate expression, or a more complex formula, for x serves as a term
to a predicate symbol one step higher up in the parse tree (see Definition 2.1 and the grammar in (2.2)). In substituting t for x we have to
2.2 Predicate logic as a formal language
leave untouched the bound leaves x since they are in the scope of some ∃x or ∀x, i.e. they stand for some unspecified or all values respectively. Definition 2.7 Given a variable x, a term t and a
formula φ we define φ[t/x] to be the formula obtained by replacing each free occurrence of variable x in φ with t. Substitutions are easily understood by looking at some examples. Let f be a function
symbol with two arguments and φ the formula with the parse tree in Figure 2.1. Then f (x, y) is a term and φ[f (x, y)/x] is just φ again. This is true because all occurrences of x are bound in φ, so
none of them gets substituted. Now consider φ to be the formula with the parse tree in Figure 2.2. Here we have one free occurrence of x in φ, so we substitute the parse tree of f (x, y) for that
free leaf node x and obtain the parse tree in Figure 2.3. Note that the bound x leaves are unaffected by this operation. You can see that the process of substitution is straightforward, but requires
that it be applied only to the free occurrences of the variable to be substituted. A word on notation: in writing φ[t/x], we really mean this to be the formula obtained by performing the operation [t
/x] on φ. Strictly speaking, the chain of symbols φ[t/x] is not a logical formula, but its result will be a formula, provided that φ was one in the first place.
x replaced by the term f (x, y)
Figure 2.3. A parse tree of a formula resulting from substitution.
2 Predicate logic
Unfortunately, substitutions can give rise to undesired side effects. In performing a substitution φ[t/x], the term t may contain a variable y, where free occurrences of x in φ are under the scope of
∃y or ∀y in φ. By carrying out this substitution φ[t/x], the value y, which might have been fixed by a concrete context, gets caught in the scope of ∃y or ∀y. This binding capture overrides the
context specification of the concrete value of y, for it will now stand for ‘some unspecified’ or ‘all ,’ respectively. Such undesired variable captures are to be avoided at all costs. Definition 2.8
Given a term t, a variable x and a formula φ, we say that t is free for x in φ if no free x leaf in φ occurs in the scope of ∀y or ∃y for any variable y occurring in t. This definition is maybe hard
to swallow. Let us think of it in terms of parse trees. Given the parse tree of φ and the parse tree of t, we can perform the substitution [t/x] on φ to obtain the formula φ[t/x]. The latter has a
parse tree where all free x leaves of the parse tree of φ are replaced by the parse tree of t. What ‘t is free for x in φ’ means is that the variable leaves of the parse tree of t won’t become bound
if placed into the bigger parse tree of φ[t/x]. For example, if we consider x, t and φ in Figure 2.3, then t is free for x in φ since the new leaf variables x and y of t are not under the scope of
any quantifiers involving x or y. Example 2.9 Consider the φ with parse tree in Figure 2.4 and let t be f (y, y). All two occurrences of x in φ are free. The leftmost occurrence of x could be
substituted since it is not in the scope of any quantifier, but substituting the rightmost x leaf introduces a new variable y in t which becomes bound by ∀y. Therefore, f (y, y) is not free for x in
φ. What if there are no free occurrences of x in φ? Inspecting the definition of ‘t is free for x in φ,’ we see that every term t is free for x in φ in that case, since no free variable x of φ is
below some quantifier in the parse tree of φ. So the problematic situation of variable capture in performing φ[t/x] cannot occur. Of course, in that case φ[t/x] is just φ again. It might be helpful to
compare ‘t is free for x in φ’ with a precondition of calling a procedure for substitution. If you are asked to compute φ[t/x] in your exercises or exams, then that is what you should do; but any
reasonable implementation of substitution used in a theorem prover would have to check whether t is free for x in φ and, if not, rename some variables with fresh ones to avoid the undesirable capture
of variables.
2.3 Proof theory of predicate logic
the term f (y, y) is not free for x in this formula
Figure 2.4. A parse tree for which a substitution has dire consequences.
2.3 Proof theory of predicate logic 2.3.1 Natural deduction rules Proofs in the natural deduction calculus for predicate logic are similar to those for propositional logic in Chapter 1, except that
we have new proof rules for dealing with the quantifiers and with the equality symbol. Strictly speaking, we are overloading the previously established proof rules for the propositional connectives ∧,
∨ etc. That simply means that any proof rule of Chapter 1 is still valid for logical formulas of predicate logic (we originally defined those rules for logical formulas of propositional logic). As in
the natural deduction calculus for propositional logic, the additional rules for the quantifiers and equality will come in two flavours: introduction and elimination rules. The proof rules for equality
First, let us state the proof rules for equality. Here equality does not mean syntactic, or intensional, equality, but equality in terms of computation results. In either of these senses, any term t
has to be equal to itself. This is expressed by the introduction rule for equality: t=t
which is an axiom (as it does not depend on any premises). Notice that it
2 Predicate logic
may be invoked only if t is a term, our language doesn’t permit us to talk about equality between formulas. This rule is quite evidently sound, but it is not very useful on its own. What we need is a
principle that allows us to substitute equals for equals repeatedly. For example, suppose that y ∗ (w + 2) equals y ∗ w + y ∗ 2; then it certainly must be the case that z ≥ y ∗ (w + 2) implies z ≥ y
∗ w + y ∗ 2 and vice versa. We may now express this substitution principle as the rule =e: t1 = t2 φ[t1 /x] φ[t2 /x]
Note that t1 and t2 have to be free for x in φ, whenever we want to apply the rule =e; this is an example of a side condition of a proof rule.
Convention 2.10 Throughout this section, when we write a substitution in the form φ[t/x], we implicitly assume that t is free for x in φ; for, as we saw in the last section, a substitution doesn’t
make sense otherwise. We obtain proof 1
(x + 1) = (1 + x)
(x + 1 > 1) → (x + 1 > 0)
(1 + x > 1) → (1 + x > 0)
=e 1, 2
establishing the validity of the sequent x + 1 = 1 + x, (x + 1 > 1) → (x + 1 > 0) (1 + x) > 1 → (1 + x) > 0. In this particular proof t1 is (x + 1), t2 is (1 + x) and φ is (x > 1) → (x > 0). We used
the name =e since it reflects what this rule is doing to data: it eliminates the equality in t1 = t2 by replacing all t1 in φ[t1 /x] with t2 . This is a sound substitution principle, since the
assumption that t1 equals t2 guarantees that the logical meanings of φ[t1 /x] and φ[t2 /x] match. The principle of substitution, in the guise of the rule =e, is quite powerful. Together with the rule
=i, it allows us to show the sequents t1 = t2 t2 = t1 t 1 = t 2 , t2 = t 3 t 1 = t 3 .
(2.6) (2.7)
2.3 Proof theory of predicate logic
A proof for (2.6) is: 1
t1 = t 2
t1 = t1
t2 = t1
=e 1, 2
where φ is x = t1 . A proof for (2.7) is: 1
t2 = t3
t1 = t2
t1 = t3
=e 1, 2
where φ is t1 = x, so in line 2 we have φ[t2 /x] and in line 3 we obtain φ[t3 /x], as given by the rule =e applied to lines 1 and 2. Notice how we applied the scheme =e with several different
instantiations. Our discussion of the rules =i and =e has shown that they force equality to be reflexive (2.5), symmetric (2.6) and transitive (2.7). These are minimal and necessary requirements for
any sane concept of (extensional) equality. We leave the topic of equality for now to move on to the proof rules for quantifiers. The proof rules for universal quantification The rule for eliminating
∀ is the following: ∀x φ φ[t/x]
∀x e.
It says: If ∀x φ is true, then you could replace the x in φ by any term t (given, as usual, the side condition that t be free for x in φ) and conclude that φ[t/x] is true as well. The intuitive
soundness of this rule is self-evident. Recall that φ[t/x] is obtained by replacing all free occurrences of x in φ by t. You may think of the term t as a more concrete instance of x. Since φ is
assumed to be true for all x, that should also be the case for any term t. Example 2.11 To see the necessity of the proviso that t be free for x in φ, consider the case that φ is ∃y (x < y) and the
term to be substituted for x is y. Let’s suppose we are reasoning about numbers with the usual ‘smaller than’ relation. The statement ∀x φ then says that for all numbers n there is some bigger number
m, which is indeed true of integers or real numbers. However, φ[y/x] is the formula ∃y (y < y) saying that there is a number which is bigger than itself. This is wrong; and we must not allow a proof
rule which derives semantically wrong things from semantically valid
2 Predicate logic
ones. Clearly, what went wrong was that y became bound in the process of substitution; y is not free for x in φ. Thus, in going from ∀x φ to φ[t/x], we have to enforce the side condition that t be
free for x in φ: use a fresh variable for y to change φ to, say, ∃z (x < z) and then apply [y/x] to that formula, rendering ∃z (y < z). The rule ∀x i is a bit more complicated. It employs a proof box
similar to those we have already seen in natural deduction for propositional logic, but this time the box is to stipulate the scope of the ‘dummy variable’ x0 rather than the scope of an assumption.
The rule ∀x i is written x0
.. . φ[x0 /x] ∀x φ
∀x i.
It says: If, starting with a ‘fresh’ variable x0 , you are able to prove some formula φ[x0 /x] with x0 in it, then (because x0 is fresh) you can derive ∀x φ. The important point is that x0 is a new
variable which doesn’t occur anywhere outside its box ; we think of it as an arbitrary term. Since we assumed nothing about this x0 , anything would work in its place; hence the conclusion ∀x φ. It
takes a while to understand this rule, since it seems to be going from the particular case of φ to the general case ∀x φ. The side condition, that x0 does not occur outside the box, is what allows us
to get away with this. To understand this, think of the following analogy. If you want to prove to someone that you can, say, split a tennis ball in your hand by squashing it, you might say ‘OK, give
me a tennis ball and I’ll split it.’ So we give you one and you do it. But how can we be sure that you could split any tennis ball in this way? Of course, we can’t give you all of them, so how could
we be sure that you could split any one? Well, we assume that the one you did split was an arbitrary, or ‘random,’ one, i.e. that it wasn’t special in any way – like a ball which you may have
‘prepared’ beforehand; and that is enough to convince us that you could split any tennis ball. Our rule says that if you can prove φ about an x0 that isn’t special in any way, then you could prove it
for any x whatsoever. To put it another way, the step from φ to ∀x φ is legitimate only if we have arrived at φ in such a way that none of its assumptions contain x as a free variable. Any assumption
which has a free occurrence of x puts constraints
2.3 Proof theory of predicate logic
on such an x. For example, the assumption bird(x) confines x to the realm of birds and anything we can prove about x using this formula will have to be a statement restricted to birds and not about
anything else we might have had in mind. It is time we looked at an example of these proof rules at work. Here is a proof of the sequent ∀x (P (x) → Q(x)), ∀x P (x) ∀x Q(x): 1
∀x (P (x) → Q(x))
∀x P (x)
P (x0 ) → Q(x0 )
∀x e 1
P (x0 )
∀x e 2
Q(x0 )
→e 3, 4
∀x Q(x)
∀x i 3−5
The structure of this proof is guided by the fact that the conclusion is a ∀ formula. To arrive at this, we will need an application of ∀x i, so we set up the box controlling the scope of x0 . The
rest is now mechanical: we prove ∀x Q(x) by proving Q(x0 ); but the latter we can prove as soon as we can prove P (x0 ) and P (x0 ) → Q(x0 ), which themselves are instances of the premises (obtained
by ∀e with the term x0 ). Note that we wrote the name of the dummy variable to the left of the first proof line in its scope box. Here is a simpler example which uses only ∀x e: we show the validity
of the sequent P (t), ∀x (P (x) → ¬Q(x)) ¬Q(t) for any term t: 1
P (t)
∀x (P (x) → ¬Q(x)) premise
P (t) → ¬Q(t)
∀x e 2
→e 3, 1
Note that we invoked ∀x e with the same instance t as in the assumption P (t). If we had invoked ∀x e with y, say, and obtained P (y) → ¬Q(y), then that would have been valid, but it would not have
been helpful in the case that y was different from t. Thus, ∀x e is really a scheme of rules, one for each term t (free for x in φ), and we should make our choice on the basis of consistent pattern
matching. Further, note that we have rules ∀x i and ∀x e for each variable x. In particular, there are rules ∀y i, ∀y e and so on. We
2 Predicate logic
will write ∀i and ∀e when we speak about such rules without concern for the actual quantifier variable. Notice also that, although the square brackets representing substitution appear in the rules ∀i
and ∀e, they do not appear when we use those rules. The reason for this is that we actually carry out the substitution that is asked for. In the rules, the expression φ[t/x] means: ‘φ, but with free
occurrences of x replaced by t.’ Thus, if φ is P (x, y) → Q(y, z) and the rule refers to φ[a/y], we carry out the substitution and write P (x, a) → Q(a, z) in the proof. A helpful way of
understanding the universal quantifier rules is to compare the rules for ∀ with those for ∧. The rules for ∀ are in some sense generalisations of those for ∧; whereas ∧ has just two conjuncts, ∀ acts
like it conjoins lots of formulas (one for each substitution instance of its variable). Thus, whereas ∧i has two premises, ∀x i has a premise φ[x0 /x] for each possible ‘value’ of x0 . Similarly,
where and-elimination allows you to deduce from φ ∧ ψ whichever of φ and ψ you like, forall-elimination allows you to deduce φ[t/x] from ∀x φ, for whichever t you (and the side condition) like. To
say the same thing another way: think of ∀x i as saying: to prove ∀x φ, you have to prove φ[x0 /x] for every possible value x0 ; while ∧i says that to prove φ1 ∧ φ2 you have to prove φi for every i =
1, 2. The proof rules for existential quantification The analogy between ∀ and ∧ extends also to ∃ and ∨; and you could even try to guess the rules for ∃ by starting from the rules for ∨ and applying
the same ideas as those that related ∧ to ∀. For example, we saw that the rules for or-introduction were a sort of dual of those for and-elimination; to emphasise this point, we could write them as
φk φ1 ∧ φ2 ∨ik ∧ek φk φ1 ∨ φ2 where k can be chosen to be either 1 or 2. Therefore, given the form of forall-elimination, we can infer that exists-introduction must be simply φ[t/x] ∃xφ
∃x i.
Indeed, this is correct: it simply says that we can deduce ∃x φ whenever we have φ[t/x] for some term t (naturally, we impose the side condition that t be free for x in φ). In the rule ∃i, we see
that the formula φ[t/x] contains, from a computational point of view, more information than ∃x φ. The latter merely says
2.3 Proof theory of predicate logic
that φ holds for some, unspecified, value of x; whereas φ[t/x] has a witness t at its disposal. Recall that the square-bracket notation asks us actually to carry out the substitution. However, the
notation φ[t/x] is somewhat misleading since it suggests not only the right witness t but also the formula φ itself. For example, consider the situation in which t equals y such that φ[y/x] is y = y.
Then you can check for yourself that φ could be a number of things, like x = x or x = y. Thus, ∃x φ will depend on which of these φ you were thinking of. Extending the analogy between ∃ and ∨, the
rule ∨e leads us to the following formulation of ∃e: x0 φ[x0 /x] .. . χ
∃x φ χ
Like ∨e, it involves a case analysis. The reasoning goes: We know ∃x φ is true, so φ is true for at least one ‘value’ of x. So we do a case analysis over all those possible values, writing x0 as a
generic value representing them all. If assuming φ[x0 /x] allows us to prove some χ which doesn’t mention x0 , then this χ must be true whichever x0 makes φ[x0 /x] true. And that’s precisely what the
rule ∃e allows us to deduce. Of course, we impose the side condition that x0 can’t occur outside its box (therefore, in particular, it cannot occur in χ). The box is controlling two things: the scope
of x0 and also the scope of the assumption φ[x0 /x]. Just as ∨e says that to use φ1 ∨ φ2 , you have to be prepared for either of the φi , so ∃e says that to use ∃x φ you have to be prepared for any
possible φ[x0 /x]. Another way of thinking about ∃e goes like this: If you know ∃x φ and you can derive some χ from φ[x0 /x], i.e. by giving a name to the thing you know exists, then you can derive χ
even without giving that thing a name (provided that χ does not refer to the name x0 ). The rule ∃x e is also similar to ∨e in the sense that both of them are elimination rules which don’t have to
conclude a subformula of the formula they are about to eliminate. Please verify that all other elimination rules introduced so far have this subformula property.2 This property is computationally
very pleasant, for it allows us to narrow down the search space for a proof dramatically. Unfortunately, ∃x e, like its cousin ∨e, is not of that computationally benign kind. 2
For ∀x e we perform a substitution [t/x], but it preserves the logical structure of φ.
2 Predicate logic
Let us practice these rules on a couple of examples. Certainly, we should be able to prove the validity of the sequent ∀x φ ∃x φ. The proof 1
∀x φ
φ[x/x] ∀x e 1
∃x φ
premise ∃x i 2
demonstrates that, where we chose t to be x with respect to both ∀x e and to ∃x i (and note that x is free for x in φ and that φ[x/x] is simply φ again). Proving the validity of the sequent ∀x (P (x)
→ Q(x)), ∃x P (x) ∃x Q(x) is more complicated: 1
∀x (P (x) → Q(x)) premise
∃x P (x)
P (x0 )
P (x0 ) → Q(x0 )
∀x e 1
Q(x0 )
→e 4, 3
∃x Q(x)
∃x i 5
∃x Q(x)
∃x e 2, 3−6
The motivation for introducing the box in line 3 of this proof is the existential quantifier in the premise ∃x P (x) which has to be eliminated. Notice that the ∃ in the conclusion has to be
introduced within the box and observe the nesting of these two steps. The formula ∃x Q(x) in line 6 is the instantiation of χ in the rule ∃e and does not contain an occurrence of x0 , so it is
allowed to leave the box to line 7. The almost identical ‘proof’ 1
∀x (P (x) → Q(x)) premise
∃x P (x)
P (x0 )
P (x0 ) → Q(x0 )
∀x e 1
Q(x0 )
→e 4, 3
Q(x0 )
∃x e 2, 3−5
∃x Q(x)
∃x i 6
is illegal! Line 6 allows the fresh parameter x0 to escape the scope of the box which declares it. This is not permissible and we will see on page 116 an example where such illicit use of proof rules
results in unsound arguments.
2.3 Proof theory of predicate logic
A sequent with a slightly more complex proof is ∀x (Q(x) → R(x)), ∃x (P (x) ∧ Q(x)) ∃x (P (x) ∧ R(x)) and could model some argument such as If all quakers are reformists and if there is a protestant
who is also a quaker, then there must be a protestant who is also a reformist.
One possible proof strategy is to assume P (x0 ) ∧ Q(x0 ), get the instance Q(x0 ) → R(x0 ) from ∀x (Q(x) → R(x)) and use ∧e2 to get our hands on Q(x0 ), which gives us R(x0 ) via →e . . . : 1
∀x (Q(x) → R(x)) premise
∃x (P (x) ∧ Q(x))
P (x0 ) ∧ Q(x0 )
Q(x0 ) → R(x0 )
∀x e 1
Q(x0 )
∧e2 3
R(x0 )
→e 4, 5
P (x0 )
∧e1 3
P (x0 ) ∧ R(x0 )
∧i 7, 6
∃x (P (x) ∧ R(x))
∃x i 8
∃x (P (x) ∧ R(x))
∃x e 2, 3−9
Note the strategy of this proof: We list the two premises. The second premise is of use here only if we apply ∃x e to it. This sets up the proof box in lines 3−9 as well as the fresh parameter name
x0 . Since we want to prove ∃x (P (x) ∧ R(x)), this formula has to be the last one in the box (our goal) and the rest involves ∀x e and ∃x i. The rules ∀i and ∃e both have the side condition that the
dummy variable cannot occur outside the box in the rule. Of course, these rules may still be nested, by choosing another fresh name (e.g. y0 ) for the dummy variable. For example, consider the
sequent ∃x P (x), ∀x ∀y (P (x) → Q(y)) ∀y Q(y). (Look how strong the second premise is, by the way: given any x, y, if P (x), then Q(y). This means that, if there is any object with the property P ,
then all objects shall have the property Q.) Its proof goes as follows: We take an arbitrary y0 and prove Q(y0 ); this we do by observing that, since some x
2 Predicate logic
satisfies P , so by the second premise any y satisfies Q: 1
∃x P (x)
∀x∀y (P (x) → Q(y))
P (x0 )
∀y (P (x0 ) → Q(y))
∀x e 2
P (x0 ) → Q(y0 )
∀y e 5
Q(y0 )
→e 6, 4
Q(y0 )
∃x e 1, 4−7
∀y Q(y)
∀y i 3−8
There is no special reason for picking x0 as a name for the dummy variable we use for ∀x and ∃x and y0 as a name for ∀y and ∃y. We do this only because it makes it easier for us humans. Again, study
the strategy of this proof. We ultimately have to show a ∀y formula which requires us to use ∀y i, i.e. we need to open up a proof box (lines 3−8) whose subgoal is to prove a generic instance Q(y0 ).
Within that box we want to make use of the premise ∃x P (x) which results in the proof box set-up of lines 4−7. Notice that, in line 8, we may well move Q(y0 ) out of the box controlled by x0 . We
have repeatedly emphasised the point that the dummy variables in the rules ∃e and ∀i must not occur outside their boxes. Here is an example which shows how things would go wrong if we didn’t have
this side condition. Consider the invalid sequent ∃x P (x), ∀x (P (x) → Q(x)) ∀y Q(y). (Compare it with the previous sequent; the second premise is now much weaker, allowing us to conclude Q only for
those objects for which we know P .) Here is an alleged ‘proof’ of its validity: 1
∃x P (x)
∀x (P (x) → Q(x)) premise
P (x0 )
P (x0 ) → Q(x0 )
∀x e 2
Q(x0 )
→e 5, 4
Q(x0 )
∃x e 1, 4−6
∀y Q(y)
∀y i 3−7
2.3 Proof theory of predicate logic
The last step introducing ∀y is not the bad one; that step is fine. The bad one is the second from last one, concluding Q(x0 ) by ∃x e and violating the side condition that x0 may not leave the scope
of its box. You can try a few other ways of ‘proving’ this sequent, but none of them should work (assuming that our proof system is sound with respect to semantic entailment, which we define in the
next section). Without this side condition, we would also be able to prove that ‘all x satisfy the property P as soon as one of them does so,’ a semantic disaster of biblical proportions!
2.3.2 Quantifier equivalences We have already hinted at semantic equivalences between certain forms of quantification. Now we want to provide formal proofs for some of the most commonly used quantifier
equivalences. Quite a few of them involve several quantifications over more than just one variable. Thus, this topic is also good practice for using the proof rules for quantifiers in a nested fashion.
For example, the formula ∀x ∀y φ should be equivalent to ∀y ∀x φ since both say that φ should hold for all values of x and y. What about (∀x φ) ∧ (∀x ψ) versus ∀x (φ ∧ ψ)? A moment’s thought reveals
that they should have the same meaning as well. But what if the second conjunct does not start with ∀x? So what if we are looking at (∀x φ) ∧ ψ in general and want to compare it with ∀x (φ ∧ ψ)? Here
we need to be careful, since x might be free in ψ and would then become bound in the formula ∀x (φ ∧ ψ). Example 2.12 We may specify ‘Not all birds can fly.’ as ¬∀x (B(x) → F (x)) or as ∃x (B(x) ∧ ¬F
(x)). The former formal specification is closer to the structure of the English specification, but the latter is logically equivalent to the former. Quantifier equivalences help us in establishing that
specifications that ‘look’ different are really saying the same thing. Here are some quantifier equivalences which you should become familiar with. As in Chapter 1, we write φ1 φ2 as an abbreviation for
the validity of φ1 φ2 and φ2 φ1 . Theorem 2.13 Let φ and ψ be formulas of predicate logic. Then we have the following equivalences: 1. (a) ¬∀x φ ∃x ¬φ (b) ¬∃x φ ∀x ¬φ. 2. Assuming that x is not free
in ψ:
2 Predicate logic
118 (a) (b) (c) (d) (e) (f) (g) (h) 3. (a) (b) 4. (a) (b)
∀x φ ∧ ψ ∀x (φ ∧ ψ)3 ∀x φ ∨ ψ ∀x (φ ∨ ψ) ∃x φ ∧ ψ ∃x (φ ∧ ψ) ∃x φ ∨ ψ ∃x (φ ∨ ψ) ∀x (ψ → φ) ψ → ∀x φ ∃x (φ → ψ) ∀x φ → ψ ∀x (φ → ψ) ∃x φ → ψ ∃x (ψ → φ) ψ → ∃x φ. ∀x φ ∧ ∀x ψ ∀x (φ ∧ ψ) ∃x φ ∨ ∃x ψ ∃x
(φ ∨ ψ). ∀x ∀y φ ∀y ∀x φ ∃x ∃y φ ∃y ∃x φ.
PROOF: We will prove most of these sequents; the proofs for the remaining ones are straightforward adaptations and are left as exercises. Recall that we sometimes write ⊥ to denote any contradiction.
1. (a) We will lead up to this by proving the validity of two simpler sequents first: ¬(p1 ∧ p2 ) ¬p1 ∨ ¬p2 and then ¬∀x P (x) ∃x ¬P (x). The reason for proving the first of these is to illustrate the
close relationship between ∧ and ∨ on the one hand and ∀ and ∃ on the other – think of a model with just two elements 1 and 2 such that pi (i = 1, 2) stands for P (x) evaluated at i. The idea is that
proving this propositional sequent should give us inspiration for proving the second one of predicate logic. The reason for proving the latter sequent is that it is a special case (in which φ equals
P (x)) of the one we are really after, so again it should be simpler while providing some inspiration. So, let’s go.
¬(p1 ∧ p2 )
¬(¬p1 ∨ ¬p2 )
¬p1 ∨ ¬p2
∨i1 3
¬p1 ∨ ¬p2
∨i2 3
¬e 4, 2
¬e 4, 2
PBC 3−5
PBC 3−5
p1 ∧ p 2
∧i 6, 6
¬e 7, 1
¬p1 ∨ ¬p2
PBC 2−8
Remember that ∀x φ ∧ ψ is implicitly bracketed as (∀x φ) ∧ ψ, by virtue of the binding priorities.
2.3 Proof theory of predicate logic
You have seen this sort of proof before, in Chapter 1. It is an example of something which requires proof by contradiction, or ¬¬e, or LEM (meaning that it simply cannot be proved in the reduced
natural deduction system which discards these three rules) – in fact, the proof above used the rule PBC three times. Now we prove the validity of ¬∀x P (x) ∃x ¬P (x) similarly, except that where the
rules for ∧ and ∨ were used we now use those for ∀ and ∃: 1
¬∀x P (x)
¬∃x ¬P (x) assumption
¬P (x0 )
∃x¬P (x)
∃x i 4
¬e 5, 2
P (x0 )
PBC 4−6
∀x P (x)
∀x i 3−7
¬e 8, 1
∃x ¬P (x)
PBC 2−9
You will really benefit by spending time understanding the way this proof mimics the one above it. This insight is very useful for constructing predicate logic proofs: you first construct a similar
propositional proof and then mimic it. Next we prove that ¬∀x φ ∃x ¬φ is valid: 1
¬∀x φ
¬∃x ¬φ
¬φ[x0 /x]
∃x i 4
¬e 5, 2
φ[x0 /x]
PBC 4−6
∀x φ
∀x i 3−7
¬e 8, 1
∃x ¬φ
PBC 2−9
2 Predicate logic
Proving that the reverse ∃x ¬φ ¬∀x φ is valid is more straightforward, for it does not involve proof by contradiction, ¬¬e, or LEM. Unlike its converse, it has a constructive proof which the
intuitionists do accept. We could again prove the corresponding propositional sequent, but we leave that as an exercise. 1
∃x ¬φ
∀x φ
¬φ[x0 /x]
φ[x0 /x]
∀x e 2
¬e 5, 4
∃x e 1, 3−6
¬∀x φ
¬i 2−7
2. (a) Validity of ∀x φ ∧ ψ ∀x (φ ∧ ψ) can be proved thus: 1
(∀x φ) ∧ ψ
∀x φ
∧e1 1
∧e2 1
φ[x0 /x]
∀x e 2
φ[x0 /x] ∧ ψ
∧i 5, 3
(φ ∧ ψ)[x0 /x]
identical to 6, since x not free in ψ
∀x (φ ∧ ψ)
∀x i 4−7
The argument for the reverse validity can go like this: ∀x (φ ∧ ψ)
(φ ∧ ψ)[x0 /x]
∀x e 1
φ[x0 /x] ∧ ψ
identical to 3, since x not free in ψ
∧e2 3
φ[x0 /x]
∧e1 3
∀x φ
∀x i 2−6
(∀x φ) ∧ ψ
∧i 7, 5
2.3 Proof theory of predicate logic
Notice that the use of ∧i in the last line is permissible, because ψ was obtained for any instantiation of the formula in line 1; although a formal tool for proof support may complain about such
practice. 3. (b) The sequent (∃x φ) ∨ (∃x ψ) ∃x (φ ∨ ψ) is proved valid using the rule ∨e; so we have two principal cases, each of which requires the rule ∃x i: 1
(∃x φ) ∨ (∃x ψ)
∃x φ
φ[x0 /x]
∃x ψ
ψ[x0 /x]
φ[x0 /x]∨ψ[x0 /x]
φ[x0 /x]∨ψ[x0 /x] ∨i 3
(φ ∨ ψ)[x0 /x]
(φ ∨ ψ)[x0 /x]
∃x (φ ∨ ψ)
∃x (φ ∨ ψ)
∃x i 5
∃x (φ ∨ ψ)
∃x (φ ∨ ψ)
∃x e 2, 3−6
∃x (φ ∨ ψ)
∨e 1, 2−7
The converse sequent has ∃x (φ ∨ ψ) as premise, so its proof has to use ∃x e as its last rule; for that rule, we need φ ∨ ψ as a temporary assumption and need to conclude (∃x φ) ∨ (∃x ψ) from those
data; of course, the assumption φ ∨ ψ requires the usual case analysis:
∃x (φ ∨ ψ)
(φ ∨ ψ)[x0 /x]
φ[x0 /x] ∨ ψ[x0 /x]
φ[x0 /x]
ψ[x0 /x]
∃x φ
∃x ψ
∃x i 4
∃x φ ∨ ∃x ψ
∃x φ ∨ ∃x ψ
∨i 5
∃x φ ∨ ∃x ψ
∨e 3, 4−6
∃x φ ∨ ∃x ψ
∃x e 1, 2−7
4. (b) Given the premise ∃x ∃y φ, we have to nest ∃x e and ∃y e to conclude ∃y ∃x φ. Of course, we have to obey the format of these elimination rules as done below:
2 Predicate logic
∃x ∃y φ
(∃y φ)[x0 /x]
∃y (φ[x0 /x])
identical, since x, y different variables
φ[x0 /x][y0 /y] assumption
φ[y0 /y][x0 /x]
identical, since x, y, x0 , y0 different variables
∃x φ[y0 /y]
∀x i 5
∃y ∃x φ
∀y i 6
∃y ∃x φ
∃y e3, 4−7
∃y ∃x φ
∃x e1, 2−8
The validity of the converse sequent is proved in the same way by swapping 2 the roles of x and y.
2.4 Semantics of predicate logic Having seen how natural deduction of propositional logic can be extended to predicate logic, let’s now look at how the semantics of predicate logic works. Just like
in the propositional case, the semantics should provide a separate, but ultimately equivalent, characterisation of the logic. By ‘separate,’ we mean that the meaning of the connectives is defined in a
different way; in proof theory, they were defined by proof rules providing an operative explanation. In semantics, we expect something like truth tables. By ‘equivalent,’ we mean that we should be able
to prove soundness and completeness, as we did for propositional logic – although a fully fledged proof of soundness and completeness for predicate logic is beyond the scope of this book. Before we
begin describing the semantics of predicate logic, let us look more closely at the real difference between a semantic and a proof-theoretic account. In proof theory, the basic object which is
constructed is a proof. Let us write Γ as a shorthand for lists of formulas φ1 , φ2 , . . . , φn . Thus, to show that Γ ψ is valid, we need to provide a proof of ψ from Γ. Yet, how can we show that ψ
is not a consequence of Γ? Intuitively, this is harder; how can you possibly show that there is no proof of something? You would have to consider every ‘candidate’ proof and show it is not one. Thus,
proof theory gives a ‘positive’ characterisation of the logic; it provides convincing evidence for assertions like ‘Γ ψ is valid,’ but it is not very useful for establishing evidence for assertions
of the form ‘Γ φ is not valid.’
2.4 Semantics of predicate logic
Semantics, on the other hand, works in the opposite way. To show that ψ is not a consequence of Γ is the ‘easy’ bit: find a model in which all φi are true, but ψ isn’t. Showing that ψ is a consequence
of Γ, on the other hand, is harder in principle. For propositional logic, you need to show that every valuation (an assignment of truth values to all atoms involved) that makes all φi true also makes
ψ true. If there is a small number of valuations, this is not so bad. However, when we look at predicate logic, we will find that there are infinitely many valuations, called models from hereon, to
consider. Thus, in semantics we have a ‘negative’ characterisation of the logic. We find establishing assertions of the form ‘Γ ψ’ (ψ is not a semantic entailment of all formulas in Γ) easier than
establishing ‘Γ ψ’ (ψ is a semantic entailment of Γ), for in the former case we need only talk about one model, whereas in the latter we potentially have to talk about infinitely many. All this goes
to show that it is important to study both proof theory and semantics. For example, if you are trying to show that ψ is not a consequence of Γ and you have a hard time doing that, you might want to
change your strategy for a while by trying to prove the validity of Γ ψ. If you find a proof, you know for sure that ψ is a consequence of Γ. If you can’t find a proof, then your attempts at proving it
often provide insights which lead you to the construction of a counter example. The fact that proof theory and semantics for predicate logic are equivalent is amazing, but it does not stop them
having separate roles in logic, each meriting close study.
2.4.1 Models Recall how we evaluated formulas in propositional logic. For example, the formula (p ∨ ¬q) → (q → p) is evaluated by computing a truth value (T or F) for it, based on a given valuation
(assumed truth values for p and q). This activity is essentially the construction of one line in the truth table of (p ∨ ¬q) → (q → p). How can we evaluate formulas in predicate logic, e.g. ∀x ∃y ((P
(x) ∨ ¬Q(y)) → (Q(x) → P (y))) which ‘enriches’ the formula of propositional logic above? Could we simply assume truth values for P (x), Q(y), Q(x) and P (y) and compute a truth value as before? Not
quite, since we have to reflect the meaning of the quantifiers ∀x and ∃y, their dependences and the actual parameters of P and Q – a formula ∀x ∃y R(x, y) generally means something else other than ∃y
∀x R(x, y); why? The problem is that variables are place holders for any, or some, unspecified concrete values. Such values can be of almost any kind: students, birds, numbers, data structures,
programs and so on.
2 Predicate logic
Thus, if we encounter a formula ∃y ψ, we try to find some instance of y (some concrete value) such that ψ holds for that particular instance of y. If this succeeds (i.e. there is such a value of y for
which ψ holds), then ∃y ψ evaluates to T; otherwise (i.e. there is no concrete value of y which realises ψ) it returns F. Dually, evaluating ∀x ψ amounts to showing that ψ evaluates to T for all
possible values of x; if this is successful, we know that ∀x ψ evaluates to T; otherwise (i.e. there is some value of x such that ψ computes F) it returns F. Of course, such evaluations of formulas
require a fixed universe of concrete values, the things we are, so to speak, talking about. Thus, the truth value of a formula in predicate logic depends on, and varies with, the actual choice of
values and the meaning of the predicate and function symbols involved. If variables can take on only finitely many values, we can write a program that evaluates formulas in a compositional way. If the
root node of φ is ∧, ∨, → or ¬, we can compute the truth value of φ by using the truth table of the respective logical connective and by computing the truth values of the subtree(s) of that root, as
discussed in Chapter 1. If the root is a quantifier, we have sketched above how to proceed. This leaves us with the case of the root node being a predicate symbol P (in propositional logic this was an
atom and we were done already). Such a predicate requires n arguments which have to be terms t1 , t2 , . . . , tn . Therefore, we need to be able to assign truth values to formulas of the form P (t1
, t2 , . . . , tn ). For formulas P (t1 , t2 , . . . , tn ), there is more going on than in the case of propositional logic. For n = 2, the predicate P could stand for something like ‘the number
computed by t1 is less than, or equal to, the number computed by t2 .’ Therefore, we cannot just assign truth values to P directly without knowing the meaning of terms. We require a model of all
function and predicate symbols involved. For example, terms could denote real numbers and P could denote the relation ‘less than or equal to’ on the set of real numbers. Definition 2.14 Let F be a
set of function symbols and P a set of predicate symbols, each symbol with a fixed number of required arguments. A model M of the pair (F, P) consists of the following set of data: 1. A non-empty set
A, the universe of concrete values; 2. for each nullary function symbol f ∈ F, a concrete element f M of A 3. for each f ∈ F with arity n > 0, a concrete function f M : An → A from An , the set of
n-tuples over A, to A; and 4. for each P ∈ P with arity n > 0, a subset P M ⊆ An of n-tuples over A.
2.4 Semantics of predicate logic
The distinction between f and f M and between P and P M is most important. The symbols f and P are just that: symbols, whereas f M and P M denote a concrete function (or element) and relation in a
model M, respectively. def
Example 2.15 Let F = {i} and P = {R, F }; where i is a constant, F a predicate symbol with one argument and R a predicate symbol with two arguments. A model M contains a set of concrete elements A –
which may be a set of states of a computer program. The interpretations iM , RM , and F M may then be a designated initial state, a state transition relation, and a set def def of final (accepting)
states, respectively. For example, let A = {a, b, c}, iM = def def a, RM = {(a, a), (a, b), (a, c), (b, c), (c, c)}, and F M = {b, c}. We informally check some formulas of predicate logic for this
model: 1. The formula ∃y R(i, y) says that there is a transition from the initial state to some state; this is true in our model, as there are transitions from the initial state a to a, b, and c. 2.
The formula ¬F (i) states that the initial state is not a final, accepting state. This is true in our model as b and c are the only final states and a is the intitial one. 3. The formula ∀x∀y∀z (R(x,
y) ∧ R(x, z) → y = z) makes use of the equality predicate and states that the transition relation is deterministic: all transitions from any state can go to at most one state (there may be no
transitions from a state as well). This is false in our model since state a has transitions to b and c. 4. The formula ∀x∃y R(x, y) states that the model is free of states that deadlock: all states
have a transition to some state. This is true in our model: a can move to a, b or c; and b and c can move to c. def
Example 2.16 Let F = {e, ·} and P = {≤}, where e is a constant, · is a function of two arguments and ≤ is a predicate in need of two arguments as well. Again, we write · and ≤ in infix notation as in
(t1 · t2 ) ≤ (t · t).
2 Predicate logic
The model M we have in mind has as set A all binary strings, finite words over the alphabet {0, 1}, including the empty string denoted by . The interpretation eM of e is just the empty word . The
interpretation ·M of · is the concatenation of words. For example, 0110 ·M 1110 equals 01101110. In general, if a1 a2 . . . ak and b1 b2 . . . bn are such words with ai , bj ∈ {0, 1}, then a1 a2 . .
. ak ·M b1 b2 . . . bn equals a1 a2 . . . ak b1 b2 . . . bn . Finally, we interpret ≤ as the prefix ordering of words. We say that s1 is a prefix of s2 if there is a binary word s3 such that s1 ·M s3
equals s2 . For example, 011 is a prefix of 011001 and of 011, but 010 is neither. Thus, ≤M is the set {(s1 , s2 ) | s1 is a prefix of s2 }. Here are again some informal model checks: 1. In our model,
the formula ∀x ((x ≤ x · e) ∧ (x · e ≤ x)) says that every word is a prefix of itself concatenated with the empty word and conversely. Clearly, this holds in our model, for s ·M is just s and every
word is a prefix of itself. 2. In our model, the formula ∃y ∀x (y ≤ x) says that there exists a word s that is a prefix of every other word. This is true, for we may chose as such a word (there is no
other choice in this case). 3. In our model, the formula ∀x ∃y (y ≤ x) says that every word has a prefix. This is clearly the case and there are in general multiple choices for y, which are dependent
on x. 4. In our model, the formula ∀x ∀y ∀z ((x ≤ y) → (x · z ≤ y · z)) says that whenever a word s1 is a prefix of s2 , then s1 s has to be a prefix of s2 s for every word s. This is clearly not the
case. For example, take s1 as 01, s2 as 011 and s to be 0. 5. In our model, the formula ¬∃x ∀y ((x ≤ y) → (y ≤ x)) says that there is no word s such that whenever s is a prefix of some other word s1 ,
it is the case that s1 is a prefix of s as well. This is true since there cannot be such an s. Assume, for the sake of argument, that there were such a word s. Then s is clearly a prefix of s0, but s0
cannot be a prefix of s since s0 contains one more bit than s.
It is crucial to realise that the notion of a model is extremely liberal and open-ended. All it takes is to choose a non-empty set A, whose elements
2.4 Semantics of predicate logic
model real-world objects, and a set of concrete functions and relations, one for each function, respectively predicate, symbol. The only mild requirement imposed on all of this is that the concrete
functions and relations on A have the same number of arguments as their syntactic counterparts. However, you, as a designer or implementor of such a model, have the responsibility of choosing your
model wisely. Your model should be a sufficiently accurate picture of whatever it is you want to model, but at the same time it should abstract away (= ignore) aspects of the world which are
irrelevant from the perspective of your task at hand. For example, if you build a database of family relationships, then it would be foolish to interpret father-of(x, y) by something like ‘x is the
daughter of y.’ By the same token, you probably would not want to have a predicate for ‘is taller than,’ since your focus in this model is merely on relationships defined by birth. Of course, there
are circumstances in which you may want to add additional features to your database. Given a model M for a pair (F, P) of function and predicate symbols, we are now almost in a position to formally
compute a truth value for all formulas in predicate logic which involve only function and predicate symbols from (F, P). There is still one thing, though, that we need to discuss. Given a formula ∀x
φ or ∃x φ, we intend to check whether φ holds for all, respectively some, value a in our model. While this is intuitive, we have no way of expressing this in our syntax: the formula φ usually has x
as a free variable; φ[a/x] is well-intended, but ill-formed since φ[a/x] is not a logical formula, for a is not a term but an element of our model. Therefore we are forced to interpret formulas
relative to an environment. You may think of environments in a variety of ways. Essentially, they are look-up tables for all variables; such a table l associates with every variable x a value l(x) of
the model. So you can also say that environments are functions l : var → A from the set of variables var to the universe of values A of the underlying model. Given such a look-up table, we can assign
truth values to all formulas. However, for some of these computations we need updated look-up tables. Definition 2.17 A look-up table or environment for a universe A of concrete values is a function
l : var → A from the set of variables var to A. For such an l, we denote by l[x → a] the look-up table which maps x to a and any other variable y to l(y). Finally, we are able to give a semantics to
formulas of predicate logic. For propositional logic, we did this by computing a truth value. Clearly, it suffices to know in which cases this value is T.
2 Predicate logic
Definition 2.18 Given a model M for a pair (F, P) and given an environment l, we define the satisfaction relation M l φ for each logical formula φ over the pair (F, P) and look-up table l by
structural induction on φ. If M l φ holds, we say that φ computes to T in the model M with respect to the environment l. P : If φ is of the form P (t1 , t2 , . . . , tn ), then we interpret the terms
t1 , t2 , . . . , tn in our set A by replacing all variables with their values according to l. In this way we compute concrete values a1 , a2 , . . . , an of A for each of these terms, where we
interpret any function symbol f ∈ F by f M . Now M l P (t1 , t2 , . . . , tn ) holds iff (a1 , a2 , . . . , an ) is in the set P M . ∀x: The relation M l ∀x ψ holds iff M l[x→a] ψ holds for all a ∈ A.
∃x: Dually, M l ∃x ψ holds iff M l[x→a] ψ holds for some a ∈ A. ¬: The relation M l ¬ψ holds iff it is not the case that M l ψ holds. ∨: The relation M l ψ1 ∨ ψ2 holds iff M l ψ1 or M l ψ2 holds. ∧: The
relation M l ψ1 ∧ ψ2 holds iff M l ψ1 and M l ψ2 hold. →: The relation M l ψ1 → ψ2 holds iff M l ψ2 holds whenever M l ψ1 holds.
We sometimes write M l φ to denote that M l φ does not hold. There is a straightforward inductive argument on the height of the parse tree of a formula which says that M l φ holds iff M l φ holds,
whenever l and l are two environments which are identical on the set of free variables of φ. In particular, if φ has no free variables at all, we then call φ a sentence; we conclude that M l φ holds,
or does not hold, regardless of the choice of l. Thus, for sentences φ we often elide l and write M φ since the choice of an environment l is then irrelevant. Example 2.19 Let us illustrate the
definitions above by means of andef def other simple example. Let F = {alma} and P = {loves} where alma is a constant and loves a predicate with two arguments. The model M we def choose here consists
of the privacy-respecting set A = {a, b, c}, the constant def def function almaM = a and the predicate lovesM = {(a, a), (b, a), (c, a)}, which has two arguments as required. We want to check whether
the model M satisfies None of Alma’s lovers’ lovers love her.
First, we need to express the, morally worrying, sentence in predicate logic. Here is such an encoding (as we already discussed, different but logically equivalent encodings are possible): ∀x ∀y
(loves(x, alma) ∧ loves(y, x) → ¬loves(y, alma)) .
2.4 Semantics of predicate logic
Does the model M satisfy this formula? Well, it does not; for we may choose a for x and b for y. Since (a, a) is in the set lovesM and (b, a) is in the set lovesM , we would need that the latter does
not hold since it is the interpretation of loves(y, alma); this cannot be. And what changes if we modify M to M , where we keep A and almaM , def but redefine the interpretation of loves as lovesM =
{(b, a), (c, b)}? Well, now there is exactly one lover of Alma’s lovers, namely c; but c is not one of Alma’s lovers. Thus, the formula in (2.8) holds in the model M .
2.4.2 Semantic entailment In propositional logic, the semantic entailment φ1 , φ2 , . . . , φn ψ holds iff: whenever all φ1 , φ2 , . . . , φn evaluate to T, the formula ψ evaluates to T as well. How
can we define such a notion for formulas in predicate logic, considering that M l φ is indexed with an environment? Definition 2.20 Let Γ be a (possibly infinite) set of formulas in predicate logic and
ψ a formula of predicate logic. 1. Semantic entailment Γ ψ holds iff for all models M and look-up tables l, whenever M l φ holds for all φ ∈ Γ, then M l ψ holds as well. 2. Formula ψ is satisfiable iff
there is some model M and some environment l such that M l ψ holds. 3. Formula ψ is valid iff M l ψ holds for all models M and environments l in which we can check ψ. 4. The set Γ is consistent or
satisfiable iff there is a model M and a look-up table l such that M l φ holds for all φ ∈ Γ.
In predicate logic, the symbol is overloaded: it denotes model checks ‘M φ’ and semantic entailment ‘φ1 , φ2 , . . . , φn ψ.’ Computationally, each of these notions means trouble. First, establishing
M φ will cause problems, if done on a machine, as soon as the universe of values A of M is infinite. In that case, checking the sentence ∀x ψ, where x is free in ψ, amounts to verifying M [x→a] ψ for
infinitely many elements a. Second, and much more seriously, in trying to verify that φ1 , φ2 , . . . , φn ψ holds, we have to check things out for all possible models, all models which are equipped
with the right structure (i.e. they have functions and predicates with the matching number of arguments). This task is impossible to perform mechanically. This should be contrasted to the situation
in propositional logic, where the computation of the truth tables for the propositions involved was the basis for computing this relationship successfully.
2 Predicate logic
However, we can sometimes reason that certain semantic entailments are valid. We do this by providing an argument that does not depend on the actual model at hand. Of course, this works only for a
very limited number of cases. The most prominent ones are the quantifier equivalences which we already encountered in the section on natural deduction. Let us look at a couple of examples of semantic
entailment. Example 2.21 The justification of the semantic entailment ∀x (P (x) → Q(x)) ∀x P (x) → ∀x Q(x) is as follows. Let M be a model satisfying ∀x (P (x) → Q(x)). We need to show that M satisfies
∀x P (x) → ∀x Q(x) as well. On inspecting the definition of M ψ1 → ψ2 , we see that we are done if not every element of our model satisfies P . Otherwise, every element does satisfy P . But since M
satisfies ∀x (P (x) → Q(x)), the latter fact forces every element of our model to satisfy Q as well. By combining these two cases (i.e. either all elements of M satisfy P , or not) we have shown that
M satisfies ∀x P (x) → ∀x Q(x). What about the converse of the above? Is ∀x P (x) → ∀x Q(x) ∀x (P (x) → Q(x)) valid as well? Hardly! Suppose that M is a model satisfying ∀x P (x) → ∀x Q(x). If A is
its underlying set and P M and QM are the corresponding interpretations of P and Q, then M ∀x P (x) → ∀x Q(x) simply says that, if P M equals A , then QM must equal A as well. However, if P M does
not equal A , then this implication is vacuously true (remember that F → · = T no matter what · actually is). In this case we do not get any additional constraints on our model M . After these
observations, it is now easy to def def def construct a counter-example model. Let A = {a, b}, P M = {a} and QM = {b}. Then M ∀x P (x) → ∀x Q(x) holds, but M ∀x (P (x) → Q(x)) does not.
2.4.3 The semantics of equality We have already pointed out the open-ended nature of the semantics of predicate logic. Given a predicate logic over a set of function symbols F and a set of predicate
symbols P, we need only a non-empty set A equipped with concrete functions or elements f M (for f ∈ F) and concrete predicates P M (for P ∈ P) in A which have the right arities agreed upon in our
specification. Of course, we also stressed that most models have natural interpretations of
2.5 Undecidability of predicate logic
functions and predicates, but central notions like that of semantic entailment (φ1 , φ2 , . . . , φn ψ) really depend on all possible models, even the ones that don’t seem to make any sense.
Apparently there is no way out of this peculiarity. For example, where would you draw the line between a model that makes sense and one that doesn’t? And would any such choice, or set of criteria,
not be subjective? Such constraints could also forbid a modification of your model if this alteration were caused by a slight adjustment of the problem domain you intended to model. You see that there
are a lot of good reasons for maintaining such a liberal stance towards the notion of models in predicate logic. However, there is one famous exception. Often one presents predicate logic such that
there is always a special predicate = available to denote equality (recall Section 2.3.1); it has two arguments and t1 = t2 has the intended meaning that the terms t1 and t2 compute the same thing.
We discussed its proof rule in natural deduction already in Section 2.3.1. Semantically, one recognises the special role of equality by imposing on an interpretation function =M to be actual equality
on the set A of M. Thus, (a, b) is in the set =M iff a and b are the same elements in the set A. def For example, given A = {a, b, c}, the interpretation =M of equality is forced to be {(a, a), (b,
b), (c, c)}. Hence the semantics of equality is easy, for it is always modelled extensionally.
2.5 Undecidability of predicate logic We continue our introduction to predicate logic with some negative results. Given a formula φ in propositional logic we can, at least in principle, determine
whether φ holds: if φ has n propositional atoms, then the truth table of φ contains 2n lines; and φ holds if, and only if, the column for φ (of length 2n ) contains only T entries. The bad news is
that such a mechanical procedure, working for all formulas φ, cannot be provided in predicate logic. We will give a formal proof of this negative result, though we rely on an informal (yet intuitive)
notion of computability. The problem of determining whether a predicate logic formula is valid is known as a decision problem. A solution to a decision problem is a program (written in Java, C, or
any other common language) that takes problem instances as input and always terminates, producing a correct ‘yes’ or ‘no’ output. In the case of the decision problem for predicate logic, the input to
the program is an arbitrary formula φ of predicate logic and the program
2 Predicate logic
is correct if it produces ‘yes’ whenever the input formula is valid and ‘no’ whenever it is not. Note that the program which solves a decision problem must terminate for all well-formed input: a
program which goes on thinking about it for ever is not allowed. The decision problem at hand is this: Validity in predicate logic. φ hold, yes or no?
Given a logical formula φ in predicate logic, does
We now show that this problem is not solvable; we cannot write a correct C or Java program that works for all φ. It is important to be clear about exactly what we are stating. Naturally, there are
some φ which can easily be seen to be valid; and others which can easily be seen to be invalid. However, there are also some φ for which it is not easy. Every φ can, in principle, be discovered to be
valid or not, if you are prepared to work arbitrarily hard at it; but there is no uniform mechanical procedure for determining whether φ is valid which will work for all φ. We prove this by a
well-known technique called problem reduction. That is, we take some other problem, of which we already know that it is not solvable, and we then show that the solvability of our problem entails the
solvability of the other one. This is a beautiful application of the proof rules ¬i and ¬e, since we can then infer that our own problem cannot be solvable as well. The problem that is known not to
be solvable, the Post correspondence problem, is interesting in its own right and, upon first reflection, does not seem to have a lot to do with predicate logic. The Post correspondence problem. Given
a finite sequence of pairs (s1 , t1 ), (s2 , t2 ), . . . , (sk , tk ) such that all si and ti are binary strings of positive length, is there a sequence of indices i1 , i2 , . . . , in with n ≥ 1
such that the concatenation of strings si1 si2 . . . sin equals ti1 ti2 . . . tin ? Here is an instance of the problem which we can solve successfully: the concrete correspondence problem instance C
is given by a sequence of three def pairs C = ((1, 101), (10, 00), (011, 11)) so def
s1 = 1 def
t1 = 101
s3 = 011
t3 = 11.
s2 = 10 t2 = 00
A solution to the problem is the sequence of indices (1, 3, 2, 3) since s1 s3 s2 s3 and t1 t3 t2 t3 both equal 101110011. Maybe you think that this problem must surely be solvable; but remember that
a computational solution would have
2.5 Undecidability of predicate logic
to be a program that solves all such problem instances. Things get a bit tougher already if we look at this (solvable) problem: def
s1 = 001 def
t1 = 0
s2 = 01 def
t2 = 011
s3 = 01 def
t3 = 101
s4 = 10 def
t4 = 001
which you are invited to solve by hand, or by writing a program for this specific instance. Note that the same number can occur in the sequence of indices, as happened in the first example in which 3
occurs twice. This means that the search space we are dealing with is infinite, which should give us some indication that the problem is unsolvable. However, we do not formally prove it in this book.
The proof of the following theorem is due to the mathematician A. Church. Theorem 2.22 The decision problem of validity in predicate logic is undecidable: no program exists which, given any φ,
decides whether φ. PROOF: As said before, we pretend that validity is decidable for predicate logic and thereby solve the (insoluble) Post correspondence problem. Given a correspondence problem
instance C: s1 s2 . . . sk t 1 t 2 . . . tk we need to be able to construct, within finite space and time and uniformly so for all instances, some formula φ of predicate logic such that φ holds iff the
correspondence problem instance C above has a solution. As function symbols, we choose a constant e and two function symbols f0 and f1 each of which requires one argument. We think of e as the empty
string, or word, and f0 and f1 symbolically stand for concatenation with 0, respectively 1. So if b1 b2 . . . bl is a binary string of bits, we can code that up as the term fbl (fbl−1 . . . (fb2 (fb1
(e))) . . . ). Note that this coding spells that word backwards. To facilitate reading those formulas, we abbreviate terms like fbl (fbl−1 . . . (fb2 (fb1 (t))) . . . ) by fb1 b2 ...bl (t). We also
require a predicate symbol P which expects two arguments. The intended meaning of P (s, t) is that there is some sequence of indices (i1 , i2 , . . . , im ) such that s is the term representing si1
si2 . . . sim and t represents ti1 ti2 . . . tim . Thus, s constructs a string using the same sequence of indices as does t; only s uses the si whereas t uses the ti .
2 Predicate logic
Our sentence φ has the coarse structure φ1 ∧ φ2 → φ3 where we set def
φ1 =
P (fsi (e), fti (e))
i=1 def
φ2 = ∀v ∀w
P (v, w) →
P (fsi (v), fti (w))
i=1 def
φ3 = ∃z P (z, z) . Our claim is φ holds iff the Post correspondence problem C has a solution. First, let us assume that φ holds. Our strategy is to find a model for φ which tells us there is a solution
to the correspondence problem C simply by inspecting what it means for φ to satisfy that particular model. The universe of concrete values A of that model is the set of all finite, binary strings
(including the empty string denoted by ). The interpretation eM of the constant e is just that empty string . The interpretation of f0 is the unary function f0M which appends a 0 to a given def def
string, f0M (s) = s0; similarly, f1M (s) = s1 appends a 1 to a given string. The interpretation of P on M is just what we expect it to be: P M = {(s, t) | there is a sequence of indices (i1 , i2 , .
. . , im ) such that s equals si1 si2 . . . sim and t equals ti1 ti2 . . . tim } def
where s and t are binary strings and the si and ti are the data of the correspondence problem C. A pair of strings (s, t) lies in P M iff, using the same sequence of indices (i1 , i2 , . . . , im ), s
is built using the corresponding si and t is built using the respective ti . Since φ holds we infer that M φ holds, too. We claim that M φ2 holds as well, which says that whenever the pair (s, t) is
in P M , then the pair (s si , t ti ) is also in P M for i = 1, 2, . . . , k (you can verify that is says this by inspecting the definition of P M ). Now (s, t) ∈ P M implies that there is some
sequence (i1 , i2 , . . . , im ) such that s equals si1 si2 . . . sim and t equals ti1 ti2 . . . tim . We simply choose the new sequence (i1 , i2 , . . . , im , i) and observe that s si equals si1
si2 . . . sim si and t ti equals ti1 ti2 . . . tim ti and so M φ2 holds as claimed. (Why does M φ1 hold?) Since M φ1 ∧ φ2 → φ3 and M φ1 ∧ φ2 hold, it follows that M φ3 holds as well. By definition of
φ3 and P M , this tells us there is a solution to C. Conversely, let us assume that the Post correspondence problem C has some solution, namely the sequence of indices (i1 , i2 , . . . , in ). Now we
have to show that, if M is any model having a constant eM , two unary functions,
2.5 Undecidability of predicate logic
f0M and f1M , and a binary predicate P M , then that model has to satisfy φ. Notice that the root of the parse tree of φ is an implication, so this is the crucial clause for the definition of M φ. By
that very definition, we are already done if M φ1 , or if M φ2 . The harder part is therefore the one where M φ1 ∧ φ2 , for in that case we need to verify M φ3 as well. The way we proceed here is by
interpreting finite, binary strings in the domain of values A of the model M . This is not unlike the coding of an interpreter for one programming language in another. The interpretation is done by a
function interpret which is defined inductively on the data structure of finite, binary strings:
interpret() = eM def
interpret(s0) = f0M (interpret(s)) def
interpret(s1) = f1M (interpret(s)) . def
Note that interpret(s) is defined inductively on the length of s. This interpretation is, like the coding above, backwards; for example, the string 0100110 gets interpreted as f0M (f1M (f1M (f0M (f0M
(f1M (f0M (eM ))))))). Note that (fbM (. . . (fb1 (eM ) . . . ))) is just the meaning of interpret(b1 b2 . . . bl ) = fbM l l−1 fs (e) in A , where s equals b1 b2 . . . bl . Using that and the fact
that M φ1 , we conclude that (interpret(si ), interpret(ti )) ∈ P M for i = 1, 2, . . . , k. Sim ilarly, since M φ2 , we know that for all (s, t) ∈ P M we have that (interpret(ssi ), interpret(tti ))
∈ P M for i = 1, 2, . . . , k. Using these two facts, starting with (s, t) = (si1 , ti1 ), we repeatedly use the latter observation to obtain
(interpret(si1 si2 . . . sin ), interpret(ti1 ti2 . . . tin )) ∈ P M .
Since si1 si2 . . . sin and ti1 ti2 . . . tin together form a solution of C, they are equal; and therefore interpret(si1 si2 . . . sin ) and interpret(ti1 ti2 . . . tin ) are the same elements in A ,
for interpreting the same thing gets you the same result. Hence (2.9) verifies ∃z P (z, z) in M and thus M φ3 . 2 There are two more negative results which we now get quite easily. Recall that a
formula φ is satisfiable if there is some model M and some environment l such that M l φ holds. This property is not to be taken for granted; the formula ∃x (P (x) ∧ ¬P (x)) is clearly unsatisfiable.
More interesting is the observation that φ is unsatisfiable if, and only if, ¬φ is valid, i.e. holds in all models. This is an immediate consequence of the definitional clause M l ¬φ for negation.
Since we can’t compute validity, it follows that we cannot compute satisfiability either.
2 Predicate logic
The other undecidability result comes from the soundness and completeness of predicate logic which, in special form for sentences, reads as φ iff φ
which we do not prove in this text. Since we can’t decide validity, we cannot decide provability either, on the basis of (2.10). One might reflect on that last negative result a bit. It means bad news
if one wants to implement perfect theorem provers which can mechanically produce a proof of a given formula, or refute it. It means good news, though, if we like the thought that machines still need
a little bit of human help. Creativity seems to have limits if we leave it to machines alone.
2.6 Expressiveness of predicate logic Predicate logic is much more expressive than propositional logic, having predicate and function symbols, as well as quantifiers. This expressivess comes at the
cost of making validity, satisfiability and provability undecidable. The good news, though, is that checking formulas on models is practical; SQL queries over relational databases or XQueries over XML
documents are examples of this in practice. Software models, design standards, and execution models of hardware or programs often are described in terms of directed graphs. Such models M are
interpretations of a two-argument predicate symbol R over a concrete set A of ‘states.’ Example 2.23 Given a set of states A = {s0 , s1 , s2 , s3 }, let RM be the set {(s0 , s1 ), (s1 , s0 ), (s1 ,
s1 ), (s1 , s2 ), (s2 , s0 ), (s3 , s0 ), (s3 , s2 )}. We may depict this model as a directed graph in Figure 2.5, where an edge (a transition) leads from a node s to a node s iff (s, s ) ∈ RM . In
that case, we often denote this as s → s . The validation of many applications requires to show that a ‘bad’ state cannot be reached from a ‘good’ state. What ‘good’ and ‘bad’ mean will depend on the
context. For example, a good state may be one in which an integer expression, say x ∗ (y − 1), evaluates to a value that serves as a safe index into an array a of length 10. A bad state would then be
one in which this integer expression evaluates to an unsafe value, say 11, causing an ‘outof-bounds exception.’ In its essence, deciding whether from a good state one can reach a bad state is the
reachability problem in directed graphs.
2.6 Expressiveness of predicate logic s0
Figure 2.5. A directed graph, which is a model M for a predicate symbol R with two arguments. A pair of nodes (n, n ) is in the interpretation R M of R iff there is a transition (an edge) from node n
to node n in that graph.
Reachability: Given nodes n and n in a directed graph, is there a finite path of transitions from n to n ? In Figure 2.5, state s2 is reachable from state s0 , e.g. through the path s0 → s1 → s2 . By
convention, every state reaches itself by a path of length 0. State s3 , however, is not reachable from s0 ; only states s0 , s1 , and s2 are reachable from s0 . Given the evident importance of this
concept, can we express reachability in predicate logic – which is, after all, so expressive that it is undecidable? To put this question more precisely: can we find a predicate-logic formula φ with u
and v as its only free variables and R as its only predicate symbol (of arity 2) such that φ holds in directed graphs iff there is a path in that graph from the node associated to u to the node
associated to v? For example, we might try to write: u = v ∨ ∃x(R(u, x) ∧ R(x, v)) ∨ ∃x1 ∃x2 (R(u, x1 ) ∧ R(x1 , x2 ) ∧ R(x2 , v)) ∨ . . . This is infinite, so it’s not a well-formed formula. The
question is: can we find a well-formed formula with the same meaning? Surprisingly, this is not the case. To show this we need to record an important consequence of the completeness of natural
deduction for predicate logic. Theorem 2.24 (Compactness Theorem) Let Γ be a set of sentences of predicate logic. If all finite subsets of Γ are satisfiable, then so is Γ. PROOF: We use proof by
contradiction: Assume that Γ is not satisfiable. Then the semantic entailment Γ ⊥ holds as there is no model in which all φ ∈ Γ are true. By completeness, this means that the sequent Γ ⊥ is valid.
(Note that this uses a slightly more general notion of sequent in which we may have infinitely many premises at our disposal. Soundness and
2 Predicate logic
completeness remain true for that reading.) Thus, this sequent has a proof in natural deduction; this proof – being a finite piece of text – can use only finitely many premises ∆ from Γ. But then ∆ ⊥
is valid, too, and so ∆ ⊥ follows by soundness. But the latter contradicts the fact that all finite subsets of Γ are consistent. 2 From this theorem one may derive a number of useful techniques. We
mention a technique for ensuring the existence of models of infinite size. Theorem 2.25 (L¨ owenheim-Skolem Theorem) Let ψ be a sentence of predicate logic such for any natural number n ≥ 1 there is a
model of ψ with at least n elements. Then ψ has a model with infinitely many elements. def PROOF: The formula φn = ∃x1 ∃x2 . . . ∃xn 1≤i 1, φn = ∃x1 . . . ∃xn−1 (R(c, x1 ) ∧ R(x1 , x2 ) ∧ · · · ∧ R
(xn−1 , c )). def
Let ∆ = {¬φi | i ≥ 0} ∪ {φ[c/u][c /v]}. All formulas in ∆ are sentences and ∆ is unsatisfiable, since the ‘conjunction’ of all sentences in ∆ says that there is no path of length 0, no path of length
1, etc. from the node denoted by c to the node denoted by c , but there is a finite path from c to c as φ[c/u][c /v] is true.
2.6 Expressiveness of predicate logic
However, every finite subset of ∆ is satisfiable since there are paths of any finite length. Therefore, by the Compactness Theorem, ∆ itself is satisfiable. This is a contradiction. Therefore, there
cannot be such a formula φ. 2
2.6.1 Existential second-order logic If predicate logic cannot express reachability in graphs, then what can, and at what cost? We seek an extension of predicate logic that can specify such important
properties, rather than inventing an entirely new syntax, semantics and proof theory from scratch. This can be realized by applying quantifiers not only to variables, but also to predicate symbols.
For a predicate symbol P with n ≥ 1 arguments, consider formulas of the form ∃P φ
where φ is a formula of predicate logic in which P occurs. Formulas of that form are the ones of existential second-order logic. An example of arity 2 is ∃P ∀x∀y∀z (C1 ∧ C2 ∧ C3 ∧ C4 )
where each Ci is a Horn clause4 C1 C2 C3 C4
= P (x, x) = P (x, y) ∧ P (y, z) → P (x, z) def = P (u, v) → ⊥ def = R(x, y) → P (x, y). def
If we think of R and P as two transition relations on a set of states, then C4 says that any R-edge is also a P -edge, C1 states that P is reflexive, C2 specifies that P is transitive, and C3 ensures
that there is no P -path from the node associated to u to the node associated to v. Given a model M with interpretations for all function and predicate symbols of φ in (2.11), except P , let MT be
that same model augmented with an interpretation T ⊆ A × A of P , i.e. P MT = T . For any look-up table l, the semantics of ∃P φ is then M l ∃P φ 4
for some T ⊆ A × A, MT l φ.
Meaning, a Horn clause after all atomic subformulas are replaced with propositional atoms.
2 Predicate logic
Example 2.27 Let ∃P φ be the formula in (2.12) and consider the model M of Example 2.23 and Figure 2.5. Let l be a look-up table with l(u) = s0 and l(v) = s3 . Does M l ∃P φ hold? For that, we need
an interpretation T ⊆ A × A of P such that MT l ∀x∀y∀x (C1 ∧ C2 ∧ C3 ∧ C4 ) holds. That is, we need to find a reflexive and transitive relation T ⊆ A × A that condef tains RM but not (s0 , s3 ). Please
verify that T = {(s, s ) ∈ A × A | s = s3 } ∪ {(s3 , s3 )} is such a T . Therefore, M l ∃P φ holds. In the exercises you are asked to show that the formula in (2.12) holds in a directed graph iff
there isn’t a finite path from node l(u) to node l(v) in that graph. Therefore, this formula specifies unreachability.
2.6.2 Universal second-order logic Of course, we can negate (2.12) and obtain ∀P ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 )
by relying on the familiar de Morgan laws. This is a formula of universal second-order logic. This formula expresses reachability. Theorem 2.28 Let M = (A, RM ) be any model. Then the formula in
(2.14) holds under look-up table l in M iff l(v) is R-reachable from l(u) in M. PROOF: 1. First, assume that MT l ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) holds for all interpretations T of P . Then it also
holds for the interpretation which is the reflexive, transitive closure of RM . But for that T , MT l ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) can hold only if MT l ¬C3 holds, as all other clauses Ci (i = 3)
are false. But this means that MT l P (u, v) has to hold. So (l(u), l(v)) ∈ T follows, meaning that there is a finite path from l(u) to l(v). 2. Conversely, let l(v) be R-reachable from l(u) in M. –
For any interpretation T of P which is not reflexive, not transitive or does not contain RM the relation MT l ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) holds, since T makes one of the clauses ¬C1 , ¬C2 or ¬C4
true. – The other possibility is that T be a reflexive, transitive relation containing RM . Then T contains the reflexive, transitive closure of RM . But (l(u), l(v)) is in that closure by assumption.
Therefore, ¬C3 is made true in the interpretation T under look-up table l, and so MT l ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) holds.
2.7 Micromodels of software
In summary, MT l ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) holds for all interpretations T ⊆ A × A. Therefore, M l ∀P ∃x∃y∃z (¬C1 ∨ ¬C2 ∨ ¬C3 ∨ ¬C4 ) holds.
It is beyond the scope of this text to show that reachability can also be expressed in existential second-order logic, but this is indeed the case. It is an important open problem to determine
whether existential second-order logic is closed under negation, i.e. whether for all such formulas ∃P φ there is a formula ∃Q ψ of existential second-order logic such that the latter is semantically
equivalent to the negation of the former. If we allow existential and universal quantifiers to apply to predicate symbols in the same formula, we arrive at fully-fledged second-order logic, e.g. ∃P ∀Q
(∀x∀y (Q(x, y) → Q(y, x)) → ∀u∀v (Q(u, v) → P (u, v))).
We have ∃P ∀Q (∀x∀y (Q(x, y) → Q(y, x)) → ∀u∀v (Q(u, v) → P (u, v))) iff there is some T such that for all U we have (MT )U ∀x∀y (Q(x, y) → Q(y, x)) → ∀u∀v (Q(u, v) → P (u, v)), the latter being a
model check in firstorder logic. If one wants to quantify over relations of relations, one gets third-order logic etc. Higher-order logics require great care in their design. Typical results such as
completeness and compactness may quickly fail to hold. Even worse, a naive higher-order logic may be inconsistent at the meta-level. Related problems were discovered in naive set theory, e.g. in the
attempt to define the ‘set’ A that contains as elements those sets X that do not contain themselves as an element: def
A = {X | X ∈ X}.
We won’t study higher-order logics in this text, but remark that many theorem provers or deductive frameworks rely on higher-order logical frameworks.
2.7 Micromodels of software Two of the central concepts developed so far are r model checking: given a formula φ of predicate logic and a matching model M determine whether M φ holds; and r semantic
entailment: given a set of formulas Γ of predicate logic, is Γ φ valid?
2 Predicate logic
How can we put these concepts to use in the modelling and reasoning about software? In the case of semantic entailment, Γ should contain all the requirements we impose on a software design and φ may
be a property we think should hold in any implementation that meets the requirements Γ. Semantic entailment therefore matches well with software specification and validation; alas, it is undecidable
in general. Since model checking is decidable, why not put all the requirements into a model M and then check M φ? The difficulty with this approach is that, by comitting to a particular model M, we
are comitting to a lot of detail which doesn’t form part of the requirements. Typically, the model instantiates a number of parameters which were left free in the requirements. From this point of
view, semantic entailment is better, because it allows a variety of models with a variety of different values for those parameters. We seek to combine semantic entailment and model checking in a way
which attempts to give us the advantages of both. We will extract from the requirements a relatively small number of small models, and check that they satisfy the property φ to be proved. This
satisfaction checking has the tractability of model checking, while the fact that we range over a set of models (albeit a small one) allows us to consider different values of parameters which are not
set in the requirements. This approach is implemented in a tool called Alloy, due to D. Jackson. The models we consider are what he calls ‘micromodels’ of software.
2.7.1 State machines We illlustrate this approach by revisiting Example 2.15 from page 125. Its models are state machines with F = {i} and P = {R, F }, where i is a constant, F a predicate symbol
with one argument and R a predicate symbol with two arguments. A (concrete) model M contains a set of concrete elements A – which may be a set of states of a computer program. The interpretations iM
∈ A, RM ∈ A × A, and F M ⊆ A are understood to be a designated initial state, a state transition relation, and a set of final (accepting) states, respectively. Model M is concrete since there is
nothing left un-specified and all checks M φ have definite answers: they either hold or they don’t. In practice not all functional or other requirements of a software system are known in advance, and
they are likely to change during its lifecycle. For example, we may not know how many states there will be; and some transitions may be mandatory whereas others may be optional in an implementation.
Conceptually, we seek a description M of all compliant
2.7 Micromodels of software
implementations Mi (i ∈ I) of some software system. Given some matching property ψ, we then want to know r (assertion checking) whether ψ holds in all implementations Mi ∈ M; or r (consistency
checking) whether ψ holds in some implementation Mi ∈ M.
For example, let M be the set of all concrete models of state machines, as above. A possible assertion check ψ is ‘Final states are never initial states.’ An example of a consistency check ψ is
‘There are state machines that contain a non-final but deadlocked state.’ As remarked earlier, if M were the set of all state machines, then checking properties would risk being undecidable, and would
at least be intractable. If M consists of a single model, then checking properties would be decidable; but a single model is not general enough. It would comit us to instantiating several parameters
which are not part of the requirements of a state machine, such as its size and detailed construction. A better idea is to fix a finite bound on the size of models, and check whether all models of that
size that satisfy the requirements also satisfy the property under consideration. r If we get a positive answer, we are somewhat confident that the property holds in all models. In this case, the
answer is not conclusive, because there could be a larger model which fails the property, but nevertheless a positive answer gives us some confidence. r If we get a negative answer, then we have found
a model in M which violates the property. In that case, we have a conclusive answer, and can inspect the model in question.
D. Jackson’s small scope hypothesis states that negative answers tend to occur in small models already, boosting the confidence we may have in a positive answer. Here is how one could write the
requirements for M for state machines in Alloy: sig State {} sig A i F R }
StateMachine { : set State, : A, : set A, : A -> A
The model specifies two signatures. Signature State is simple in that it has no internal structure, denoted by {}. Although the states of real systems may
2 Predicate logic
well have internal structure, our Alloy declaration abstracts it away. The second signature StateMachine has internal, composite structure, saying that every state machine has a set of states A, an
initial state i from A, a set of final states F from A, and a transition relation R of type A -> A. If we read -> as the cartesian product ×, we see that this internal structure is simply the
structural information needed for models of Example 2.15 (page 125). Concrete models of state machines are instances of signature StateMachine. It is useful to think of signatures as sets whose
elements are the instances of that signature. Elements possess all the structure declared in their signature. Given these signatures, we can code and check an assertion: assert FinalNotInitial { all
M : StateMachine | no M.i & M.F } check FinalNotIntial for 3 but 1 StateMachine declares an assertion named FinalNotInitial whose body specifies that for all models M of type StateMachine the property
no M.i & M.F is true. Read & for set intersection and no S (‘there is no S’) for ‘set S is empty.’ Alloy identifies elements a with singleton sets {a}, so this set intersection is well typed. The
relational dot operator . enables access to the internal components of a state machine: M.i is the initial state of M and M.F is its set of final states etc. Therefore, the expression no M.i & M.F
states ‘No initial state of M is also a final state of M.’ Finally, the check directive informs the analyzer of Alloy that it should try to find a counterexample of the assertion FinalNotInitial with
at most three elements for every signature, except for StateMachine which should have at most one. The results of Alloy’s assertion check are shown in Figure 2.7. This visualization has been
customized to decorate initial and final states with respective labels i and F. The transition relation is shown as a labeled graph and there is only one transition (from State 0 back to State 0) in
this example. Please verify that this is a counterexample to the claim of the assertion FinalNotInitial within the specified scopes. Alloy’s GUI lets you search for additional witnesses (here:
counterexamples), if they exist. Similarly, we can check a property of state machines for consistency with our model. Alloy uses the keyword fun for consistency checks. e.g. fun AGuidedSimulation(M :
StateMachine, s : M.A) { no s.(M.R) not s in M.F # M.A = 3 } run AGiudedSimulation for 3 but 1 StateMachine
2.7 Micromodels of software
module AboutStateMachines sig State {} sig A i F R }
-- simple states
StateMachine { -- composite state machines : set State, -- set of states of a state : A, -- initial state of a state : set A, -- set of final states of a : A -> A -- transition relation of a
machine machine state machine state machine
-- Claim that final states are never initial: false. assert FinalNotInitial { all M : StateMachine | no M.i & M.F } check FinalNotInitial for 3 but 1 StateMachine -- Is there a three-state machine
with a non-final deadlock? True. fun AGuidedSimulation(M : StateMachine, s : M.A) { no s.(M.R) not s in M.F # M.A = 3 } run AGuidedSimulation for 3 but 1 StateMachine Figure 2.6. The complete Alloy
module for models of state machines, with one assertion and one consistency check. The lexeme -- enables comments on the same line.
State_1 (F)
State_2 (i, F)
Figure 2.7. Alloy’s analyzer finds a state machine model (with one transition only) within the specified scope such that the assertion FinalNotInitial is false: the initial state State 2 is also
This consistency check is named AGuidedSimulation and followed by an ordered finite list of parameter/type pairs; the first parameter is M of type StateMachine, the second one is s of type M.A – i.e. s
is a state of M. The body of a consistency check is a finite list of constraints (here three), which are conjoined implicitly. In this case, we want to find a model with instances of the parameters M
and s such that s is a non-final state of M, the second constraint not s in M.F plus the type information s : M.A; and there is no transition out of s, the first constraint no s.(M.R). The latter
requires further explanation. The keyword no denotes ‘there is no;’ here it is applied to the set s.(M.R), expressing that there are no
2 Predicate logic State_0
R State_2 (i)
Figure 2.8. Alloy’s analyzer finds a state machine model within the specified scope such that the consistency check AGuidedSimulation is true: there is a non-final deadlocked state, here State 2.
elements in s.(M.R). Since M.R is the transition relation of M, we need to understand how s.(M.R) constructs a set. Well, s is an element of M.A and M.R has type M.A -> M.A. Therefore, we may form
the set of all elements s’ such that there is a M.R-transition from s to s’; this is the set s.(M.R). The third constraint states that M has exactly three states: in Alloy, # S = k declares that the
set S has exactly k elements. The run directive instructs to check the consistency of AGuidedSimulation for at most one state machine and at most three states; the constraint analyzer of Alloy
returns the witness (here: an example) of Figure 2.8. Please check that this witness satisfies all constraints of the consistency check and that it is within the specified scopes. The complete model of
state machines with these two checks is depicted in Figure 2.6. The keyword plus name module AboutStateMachines identify this under-specified model M, rightly suggesting that Alloy is a modular
specification and analysis platform.
2.7.2 Alma – re-visited Recall Example 2.19 from page 128. Its model had three elements and did not satisfy the formula in (2.8). We can now write a module in Alloy which checks whether all smaller
models have to satisfy (2.8). The code is given in Figure 2.9. It names the module AboutAlma and defines a simple signature of type Person. Then it declares a signature SoapOpera which has a cast – a
set of type Person – a designated cast member alma, and a relation loves of type cast -> cast. We check the assertion OfLovers in a scope of at most two persons and at most one soap opera. The body
of that assertion is the typed version of (2.8) and deserves a closer look: 1. Expressions of the form all x : T | F state that formula F is true for all instances x of type T. So the assertion
states that with S {...} is true for all soap operas S.
2.7 Micromodels of software
module AboutAlma sig Person {} sig SoapOpera { cast : set Person, alma : cast, loves : cast -> cast } assert OfLovers { all S : SoapOpera | with S { all x, y : cast | alma in x.loves && x in y.loves
=> not alma in y.loves } } check OfLovers for 2 but 1 SoapOpera Figure 2.9. In this module, the analysis of OfLovers checks whether there is a model of ≤ 2 persons and ≤ 1 soap operas for which the
query in (2.8), page 128, is false.
Person_1 loves Person_0 (cast, alma) Figure 2.10. Alloy’s analyzer finds a counterexample to the formula in (2.8): Alma is the only cast member and loves herself. 2. The expression with S {...} is a
convenient notation that allows us to write loves and cast instead of the needed S.loves and S.cast (respectively) within its curly brackets. 3. Its body ... states that for all x, and y in the cast
of S, if alma is loved by x and x is loved by y, then – the symbol => expresses implication – alma is not loved by y.
Alloy’s analysis finds a counterexample to this assertion, shown in Figure 2.10. It is a counterexample since alma is her own lover, and therefore also one of her lover’s lovers’. Apparently, we have
underspecified our model: we implicitly made the domain-specific assumption that self-love makes for
2 Predicate logic Person_1 (cast) loves
Person_2 (cast) loves
Person_0 (cast, alma)
Figure 2.11. Alloy’s analyzer finds a counterexample to the formula in (2.8) that meets the constraint of NoSelfLove with three cast members. The bidirectional arrow indicates that Person 1 loves
Person 2 and vice versa.
a poor script of jealousy and intrigue, but did not rule out self-love in our Alloy module. To remedy this, we can add a fact to the module; facts may have names and restrict the set of possible
models: assertions and consistency checks are conducted only over concrete models that satisfy all facts of the module. Adding the declaration fact NoSelfLove { all S : SoapOpera, p : S.cast | not p
in p.(S.loves) } to the module AboutAlma enforces that no member of any soap-opera cast loves him or herself. We re-check the assertion and the analyzer informs us that no solution was found. This
suggests that our model from Example 2.19 is indeed a minimal one in the presence of that domain assumption. If we retain that fact, but change the occurrence of 2 in the check directive to 3, we get
a counterexample, depicted in Figure 2.11. Can you see why it is a counterexample?
2.7.3 A software micromodel So far we used Alloy to generate instances of models of first-order logic that satisfy certain constraints expressed as formulas of first-order logic. Now we apply Alloy and
its constraint analyzer to a more serious task: we model a software system. The intended benefits provided by a system model are 1. it captures formally static and dynamic system structure and
behaviour; 2. it can verify consistency of the constrained design space;
2.7 Micromodels of software
3. it is executable, so it allows guided simulations through a potentially very complex design space; and 4. it can boost our confidence into the correctness of claims about static and dynamic aspects
of all its compliant implementations.
Moreover, formal models attached to software products can be seen as a reliability contract; a promise that the software implements the structure and behaviour of the model and is expected to meet
all of the assertions certified therein. (However, this may not be very useful for extremely under-specified models.) We will model a software package dependency system. This system is used when
software packages are installed or upgraded. The system checks to see if prerequisites in the form of libraries or other packages are present. The requirements on a software package dependency system
are not straightforward. As most computer users know, the upgrading process can go wrong in various ways. For example, upgrading a package can involve replacing shared libraries with newer versions.
But other packages which rely on the older versions of the shared libraries may then cease to work. Software package dependency systems are used in several computer systems, such as Red Hat Linux,
.NET’s Global Assembly Cache and others. Users often have to guess how technical questions get resolved within the dependency system. To the best of our knowledge, there is no publicly available
formal and executable model of any particular dependency system to which application programmers could turn if they had such non-trivial technical questions about its inner workings. In our model,
applications are built out of components. Components offer services to other components. A service can be a number of things. Typically, a service is a method (a modular piece of program code), a field
entry, or a type – e.g. the type of a class in an object-oriented programming language. Components typically require the import of services from other components. Technically speaking, such import
services resolve all un-resolved references within that component, making the component linkable. A component also has a name and may have a special service, called ‘main.’ We model components as a
signature in Alloy: sig Component { name: Name, main: option Service, export: set Service, import: set Service, version: Number }{ no import & export }
name of the component component may have a ‘main’ service services the component exports services the component imports version number of the component
2 Predicate logic
The signatures Service and Name won’t require any composite structure for our modelling purposes. The signature Number will get an ordering later on. A component is an instance of Component and
therefore has a name, a set of services export it offers to other components, and a set import of services it needs to import from other components. Last but not least, a component has a version
number. Observe the role of the modifiers set and option above. A declaration i : set S means that i is a subset of set S; but a declaration i : option S means that i is a subset of S with at most one
element. Thus, option enables us to model an element that may (non-empty, singleton set) or may not (empty set) be present; a very useful ability indeed. Finally, a declaration i : S states that i is
a subset of S containing exactly one element; this really specifies a scalar/element of type S since Alloy identifies elements a with sets {a}. We can constrain all instances of a signature with C by
adding { C } to its signature declaration. We did this for the signature Component, where C is the constraint no import & export, stating that, in all components, the intersection (&) of import and
export is empty (no). A Package Dependency System (PDS) consists of a set of components: sig PDS { components : set Component ... }{ components.import in components.export } and other structure that
we specify later on. The primary concern in a PDS is that its set of components be coherent: at all times, all imports of all of its components can be serviced within that PDS. This requirement is
enforced for all instances of PDS by adding the constraint components.import in components.export to its signature. Here components is a set of components and Alloy defines the meaning of
components.import as the union of all sets c.import, where c is an element of components. Therefore the requirement states that, for all c in components, all of c’s needed services can be provided by
some component in components as well. This is exactly the integrity constraint we need for the set of components of a PDS. Observe that this requirement does not specify which component provides
which service, which would be an unacceptable imposition on implementation freedom. Given this integrity constraint we can already model the installation (adding) or removal of a component in a PDS,
without having specified the remaining structure of a PDS. This is possible since, in the context of these operations, we may abstract a PDS into its set of components. We model
2.7 Micromodels of software
the addition of a component to a PDS as a parametrized fun-statement with name AddComponent and three parameters fun AddComponent(P, P’: PDS, c: Component) { not c in P.components P’.components =
P.components + c } run AddComponent for 3 where P is intended to be the PDS prior to the execution of that operation, P’ models the PDS after that execution, and c models the component that is to be
added. This intent interprets the parametric constraint AddComponent as an operation leading from one ‘state’ to another (obtained by removing c from the PDS P). The body of AddComponent states two
constraints, conjoined implicitly. Thus, this operation applies only if the component c is not already in the set of components of the PDS (not c in P.components; an example of a precondition) and if
the PDS adds only c and does not lose any other components (P’.components = P.components + c; an example of a postcondition). To get a feel for the complexities and vexations of designing software
systems, consider our conscious or implicit decision to enforce that all instances of PDS have a coherent set of components. This sounds like a very good idea, but what if a ‘real’ and faulty PDS
ever gets to a state in which it is incoherent? We would then be prevented from adding components that may restore its coherence! Therefore, the aspects of our model do not include issues such as
repair – which may indeed by an important software management aspect. The specification for the removal of a component is very similar to the one for AddComponent: fun RemoveComponent(P, P’: PDS, c:
Component) { c in P.components P’.components = P.components - c } run RemoveComponent for 3 except that the precondition now insists that c be in the set of components of the PDS prior to the
removal; and the postcondition specifies that the PDS lost component c but did not add or lose any other components. The expression S - T denotes exactly those ‘elements’ of S that are not in T. It
remains to complete the signature for PDS. Three additions are made. 1. A relation schedule assigns to each PDS component and any of its import services a component in that PDS that provides that
2 Predicate logic
fact SoundPDSs { all P : PDS | with P { all c : components, s : Service | --1 let c’ = c.schedule[s] { (some c’ iff s in c.import) && (some c’ => s in c’.export) } all c : components | c.requires =
c.schedule[Service] --2 } } Figure 2.12. A fact that constrains the state and schedulers of all PDSs.
2. Derived from schedule we obtain a relation requires between components of the PDS that expresses the dependencies between these components based on the schedule. 3. Finally, we add constraints
that ensure the integrity and correct handling of schedule and requires for all instances of PDS.
The complete signature of PDS is sig PDS { components : set Component, schedule : components -> Service ->? components, requires : components -> components } For any P : PDS, the expression
P.schedule denotes a relation of type P.components -> Service ->? P.components. The ? is a multiplicity constraint, saying that each component of the PDS and each service get related to at most one
component. This will ensure that the scheduler is deterministic and that it may not schedule anything – e.g. when the service is not needed by the component in the first argument. In Alloy there are
also multiplicity markings ! for ‘exactly one’ and + for ‘one or more.’ The absence of such markings means ‘zero or more.’ For example, the declaration of requires uses that default reading. We use a
fact-statement to constrain even further the structure and behaviour of all PDSs, depicted in Figure 2.12. The fact named SoundPDSs quantifies the constraints over all instances of PDSs (all P : PDS |
...) and uses with P {...} to avoid the use of navigation expressions of the form P.e. The body of that fact lists two constraints --1 and --2:
2.7 Micromodels of software
--1 states two constraints within a let-expression of the form let x = E {...}. Such a let-expression declares all free occurrences of x in {...} to be equal to E. Note that [] is a version of the
dot operator . with lower binding priority, so c.schedule[s] is syntactic sugar for s.(c.schedule). r In the first constraint, component c and a service s have another component c’ scheduled (some c’
is true iff set c’ is non-empty) if and only if s is actually in the import set of c. Only needed services are scheduled! r In the second constraint, if c’ is scheduled to provide service s for c,
then s is in the export set of c’ – we can only schedule components that can provide the scheduled services!
--2 defines requires in terms of schedule: a component c requires all those components that are scheduled to provide some service for c. Our complete Alloy model for PDSs is shown in Figure 2.13.
Using Alloy’s constraint analyzer we validate that all our fun-statements, notably the operations of removing and adding components to a PDS, are logically consistent for this design. The assertion
AddingIsFunctionalForPDSs claims that the execution of the operation which adds a component to a PDS renders a unique result PDS. Alloy’s analyzer finds a counterexample to this claim, where P has no
components, so nothing is scheduled or required; and P’ and P’’ have Component 2 as only component, added to P, so this component is required and scheduled in those PDSs. Since P’ and P’’ seem to be
equal, how can this be a counterexample? Well, we ran the analysis in scope 3, so PDS = {PDS 0, PDS 1, PDS 2} and Alloy chose PDS 0 as P, PDS 1 as P’, and PDS 2 as P’’. Since the set PDS contains
three elements, Alloy ‘thinks’ that they are all different from each other. This is the interpretation of equality enforced by predicate logic. Obviously, what is needed here is a structural equality
of types: we want to ensure that the addition of a component results into a PDS with unique structure. A fun-statement can be used to specify structural equality: fun StructurallyEqual(P, P’ : PDS) {
P.components = P’.components P.schedule = P’.schedule P.requires = P’.requires } run StructurallyEqual for 2 We then simply replace the expression P’ = P’’ in AdditionIsFunctional with the expression
StructurallyEqual(P’,P’’), increase the scope for
2 Predicate logic
154 module PDS open std/ord
-- opens specification template for linear order
sig Component { name: Name, main: option Service, export: set Service, import: set Service, version: Number }{ no import & export } sig PDS { components: set Component, schedule: components ->
Service ->? components, requires: components -> components }{ components.import in components.export } fact SoundPDSs { all P : PDS | with P { all c : components, s : Service | --1 let c’ =
c.schedule[s] { (some c’ iff s in c.import) && (some c’ => s in c’.export) } all c : components | c.requires = c.schedule[Service] } --2 } sig Name, Number, Service {} fun AddComponent(P, P’: PDS, c:
Component) { not c in P.components P’.components = P.components + c } run AddComponent for 3 but 2 PDS fun RemoveComponent(P, P’: PDS, c : Component) { c in P.components P’.components = P.components
- c } run RemoveComponent for 3 but 2 PDS fun HighestVersionPolicy(P: PDS) { with P { all s : Service, c : components, c’ : c.schedule[s], c’’ : components - c’ { s in c’’.export && c’’.name =
c’.name => c’’.version in c’.version.^(Ord[Number].prev) } } } run HighestVersionPolicy for 3 but 1 PDS fun AGuidedSimulation(P,P’,P’’ : PDS, c1, c2 : Component) { AddComponent(P,P’,c1)
RemoveComponent(P,P’’,c2) HighestVersionPolicy(P) HighestVersionPolicy(P’) HighestVersionPolicy(P’’) } run AGuidedSimulation for 3 assert AddingIsFunctionalForPDSs { all P, P’, P’’: PDS, c: Component
{ AddComponent(P,P’,c) && AddComponent(P,P’’,c) => P’ = P’’ } } check AddingIsFunctionalForPDSs for 3
Figure 2.13. Our Alloy model of the PDS.
2.7 Micromodels of software
that assertion to 7, re-built the model, and re-analyze that assertion. Perhaps surprisingly, we find as counterexample a PDS 0 with two components Component 0 and Component 1 such that Component
0.import = { Service 2 } and Component 1.import = { Service 1 }. Since Service 2 is contained in Component 2.export, we have two structurally different legitimate post states which are obtained by
adding Component 2 but which differ in their scheduler. In P’ we have the same scheduling instances as in PDS 0. Yet P’’ schedules Component 2 to provide service Service 2 for Component 0; and
Component 0 still provides Service 1 to Component 1. This analysis reveals that the addition of components creates opportunities to reschedule services, for better (e.g. optimizations) or for worse
(e.g. security breaches). The utility of a micromodel of software resides perhaps more in the ability to explore it through guided simulations, as opposed to verifying some of its properties with
absolute certainty. We demonstrate this by generating a simulation that shows the removal and the addition of a component to a PDS such that the scheduler always schedules components with the highest
version number possible in all PDSs. Therefore we know that such a scheduling policy is consistent for these two operations; it is by no means the only such policy and is not guaranteed to ensure
that applications won’t break when using scheduled services. The fun-statement fun HighestVersionPolicy(P: PDS) { with P { all s : Service, c : components, c’ : c.schedule[s], c’’ : components - c’ {
s in c’’.export && c’’.name = c’.name => c’’.version in c’.version.^(Ord[Number].prev) } } } run HighestVersionPolicy for 3 but 1 PDS specifies that, among those suppliers with identical name, the
scheduler chooses one with the highest available version number. The expression c’.version.^(Ord[Number].prev) needs explaining: c’.version is the version number of c’, an element of type Number. The
symbol ^ can be applied to a binary relation r : T -> T such that ^r has again type T -> T and denotes the transitive closure of r. In this case, T equals Number and r equals Ord[Number].prev.
2 Predicate logic
But what shall me make of the latter expression? It assumes that the module contains a statement open std/ord which opens the signature specifications from another module in file ord.als of the library
std. That module contains a signature named Ord which has a type variable as a parameter; it is polymorphic. The expression Ord[Number] instantiates that type variable with the type Number, and then
invokes the prev relation of that signature with that type, where prev is constrained in std/ord to be a linear order. The net effect is that we create a linear order on Number such that n.prev is the
previous element of n with respect to that order. Therefore, n.^prev lists all elements that are smaller than n in that order. Please reread the body of that fun-statement to convince yourself that
it states what is intended. Since fun-statements can be invoked with instances of their parameters, we can write the desired simulation based on HighestVersionPolicy: fun AGuidedSimulation(P,P’,P’’ :
PDS, c1, c2 : Component) { AddComponent(P,P’,c1) RemoveComponent(P,P’’,c2) HighestVersionPolicy(P) HighestVersionPolicy(P’) HighestVersionPolicy(P’’) } run AGuidedSimulation for 3 Alloy’s analyzer
generates a scenario for this simulation, which amounts to two different operation snapshots originating in P such that all three participating PDSs schedule according to HighestVersionPolicy. Can you
spot why we had to work with two components c1 and c2? We conclude this case study by pointing out limitations of Alloy and its analyzer. In order to be able to use a SAT solver for propositional
logic as an analysis engine, we can only check or run formulas of existential or universal second-order logic in the bodies of assertions or in the bodies of fun-statements (if they are wrapped in
existential quantifiers for all parameters). For example, we cannot even check whether there is an instance of AddComponent such that for the resulting PDS a certain scheduling policy is impossible.
For less explicit reasons it also seems unlikely that we can check in Alloy that every coherent set of components is realizable as P.components for some PDS P. This deficiency is due to the inherent
complexity of such problems and theorem provers may have to be used if such properties need to be guaranteed. On the other hand, the expressiveness of Alloy allows for the rapid prototyping of models
and the exploration of simulations and possible counterexamples which should enhance once understanding of a design and so improve that design’s reliability.
2.8 Exercises
2.8 Exercises Exercises 2.1 * 1. Use the predicates A(x, y) : B(x, y) : P (x) : S(x) : L(x) :
x x x x x
admires y attended y is a professor is a student is a lecture
and the nullary function symbol (constant) m : Mary to translate the following into predicate logic: (a) Mary admires every professor. (The answer is not ∀x A(m, P (x)).) (b) Some professor admires
Mary. (c) Mary admires herself. (d) No student attended every lecture. (e) No lecture was attended by every student. (f) No lecture was attended by any student. 2. Use the predicate specifications B
(x, y) : x beats y F (x) : x is an (American) football team Q(x, y) : x is quarterback of y L(x, y) : x loses to y and the constant symbols c: j:
Wildcats Jayhawks
to translate the following into predicate logic. (a) Every football team has a quarterback. (b) If the Jayhawks beat the Wildcats, then the Jayhawks do not lose to every football team. (c) The
Wildcats beat some team, which beat the Jayhawks. * 3. Find appropriate predicates and their specification to translate the following into predicate logic: (a) All red things are in the box. (b) Only
red things are in the box. (c) No animal is both a cat and a dog. (d) Every prize was won by a boy. (e) A boy won every prize.
2 Predicate logic
4. Let F (x, y) mean that x is the father of y; M (x, y) denotes x is the mother of y. Similarly, H(x, y), S(x, y), and B(x, y) say that x is the husband/sister/brother of y, respectively. You may
also use constants to denote individuals, like ‘Ed’ and ‘Patsy.’ However, you are not allowed to use any predicate symbols other than the above to translate the following sentences into predicate
logic: (a) Everybody has a mother. (b) Everybody has a father and a mother. (c) Whoever has a mother has a father. (d) Ed is a grandfather. (e) All fathers are parents. (f) All husbands are spouses.
(g) No uncle is an aunt. (h) All brothers are siblings. (i) Nobody’s grandmother is anybody’s father. (j) Ed and Patsy are husband and wife. (k) Carl is Monique’s brother-in-law. 5. The following
sentences are taken from the RFC3157 Internet Taskforce Document ‘Securely Available Credentials – Requirements.’ Specify each sentence in predicate logic, defining predicate symbols as appropriate:
(a) An attacker can persuade a server that a successful login has occurred, even if it hasn’t. (b) An attacker can overwrite someone else’s credentials on the server. (c) All users enter passwords
instead of names. (d) Credential transfer both to and from a device MUST be supported. (e) Credentials MUST NOT be forced by the protocol to be present in cleartext at any device other than the end
user’s. (f) The protocol MUST support a range of cryptographic algorithms, including syymetric and asymmetric algorithms, hash algorithms, and MAC algorithms. (g) Credentials MUST only be
downloadable following user authentication or else only downloadable in a format that requires completion of user authentication for deciphering. (h) Different end user devices MAY be used to
download, upload, or manage the same set of credentials.
Exercises 2.2 1. Let F be {d, f, g}, where d is a constant, f a function symbol with two arguments and g a function symbol with three arguments. (a) Which of the following strings are terms over F?
Draw the parse tree of those strings which are indeed terms: i. g(d, d) * ii. f (x, g(y, z), d)
2.8 Exercises
Figure 2.14. A parse tree representing an arithmetic term. * iii. g(x, f (y, z), d) iv. g(x, h(y, z), d) v. f (f (g(d, x), f (g(d, x), y, g(y, d)), g(d, d)), g(f (d, d, x), d), z) (b) The length of a
term over F is the length of its string representation, where we count all commas and parentheses. For example, the length of f (x, g(y, z), z) is 13. List all variable-free terms over F of length
less than 10. * (c) The height of a term over F is defined as 1 plus the length of the longest path in its parse tree, as in Definition 1.32. List all variable-free terms over F of height less than 4.
2. Draw the parse tree of the term (2 − s(x)) + (y ∗ x), considering that −, +, and ∗ are used in infix in this term. Compare your solution with the parse tree in Figure 2.14. 3. Which of the
following strings are formulas in predicate logic? Specify a reason for failure for strings which aren’t, draw parse trees of all strings which are. * (a) Let m be a constant, f a function symbol
with one argument and S and B two predicate symbols, each with two arguments: i. S(m, x) ii. B(m, f (m)) iii. f (m) iv. B(B(m, x), y) v. S(B(m), z) vi. (B(x, y) → (∃z S(z, y))) vii. (S(x, y) → S(y, f
(f (x)))) viii. (B(x) → B(B(x))). (b) Let c and d be constants, f a function symbol with one argument, g a function symbol with two arguments and h a function symbol with three arguments. Further, P
and Q are predicate symbols with three arguments:
2 Predicate logic
i. ∀x P (f (d), h(g(c, x), d, y)) ii. ∀x P (f (d), h(P (x, y), d, y)) iii. ∀x Q(g(h(x, f (d), x), g(x, x)), h(x, x, x), c) iv. ∃z (Q(z, z, z) → P (z)) v. ∀x ∀y (g(x, y) → P (x, y, x)) vi. Q(c, d, c).
4. Let φ be ∃x (P (y, z) ∧ (∀y (¬Q(y, x) ∨ P (y, z)))), where P and Q are predicate symbols with two arguments. * (a) Draw the parse tree of φ. * (b) Identify all bound and free variable leaves in φ.
(c) Is there a variable in φ which has free and bound occurrences? * (d) Consider the terms w (w is a variable), f (x) and g(y, z), where f and g are function symbols with arity 1 and 2,
respectively. i. Compute φ[w/x], φ[w/y], φ[f (x)/y] and φ[g(y, z)/z]. ii. Which of w, f (x) and g(y, z) are free for x in φ? iii. Which of w, f (x) and g(y, z) are free for y in φ? (e) What is the
scope of ∃x in φ? * (f) Suppose that we change φ to ∃x (P (y, z) ∧ (∀x (¬Q(x, x) ∨ P (x, z)))). What is the scope of ∃x now? def 5. (a) Let P be a predicate symbol with arity 3. Draw the parse tree
of ψ = ¬(∀x ((∃y P (x, y, z)) ∧ (∀z P (x, y, z)))). (b) Indicate the free and bound variables in that parse tree. (c) List all variables which occur free and bound therein. def (d) Compute ψ[t/x], ψ
[t/y] and ψ[t/z], where t = g(f (g(y, y)), y). Is t free for x in ψ; free for y in ψ; free for z in ψ? 6. Rename the variables for φ in Example 2.9 (page 106) such that the resulting formula ψ has
the same meaning as φ, but f (y, y) is free for x in ψ.
Exercises 2.3 1. Prove the validity of the following sequents using, among others, the rules =i and =e. Make sure that you indicate for each application of =e what the rule instances φ, t1 and t2
are. (a) (y = 0) ∧ (y = x) 0 = x (b) t1 = t2 (t + t2 ) = (t + t1 ) (c) (x = 0) ∨ ((x + x) > 0) (y = (x + x)) → ((y > 0) ∨ (y = (0 + x))). 2. Recall that we use = to express the equality of elements
in our models. Consider the formula ∃x ∃y (¬(x = y) ∧ (∀z ((z = x) ∨ (z = y)))). Can you say, in plain English, what this formula specifies? 3. Try to write down a sentence of predicate logic which
intuitively holds in a model iff the model has (respectively) * (a) exactly three distinct elements (b) at most three distinct elements * (c) only finitely many distinct elements.
2.8 Exercises
What ‘limitation’ of predicate logic causes problems in finding such a sentence for the last item? 4. (a) Find a (propositional) proof for φ → (q1 ∧ q2 ) |− (φ → q1 ) ∧ (φ → q2 ). (b) Find a
(predicate) proof for φ → ∀x Q(x) |− ∀x (φ → Q(x)), provided that x is not free in φ. (Hint: whenever you used ∧ rules in the (propositional) proof of the previous item, use ∀ rules in the
(predicate) proof.) (c) Find a proof for ∀x (P (x) → Q(x)) |− ∀x P (x) → ∀x Q(x). (Hint: try (p1 → q1 ) ∧ (p2 → q2 ) |− p1 ∧ p2 → q1 ∧ q2 first.) 5. Find a propositional logic sequent that corresponds
to ∃x ¬φ ¬∀x φ. Prove it. 6. Provide proofs for the following sequents: (a) ∀x P (x) ∀y P (y); using ∀x P (x) as a premise, your proof needs to end with an application of ∀i which requires the
formula P (y0 ). (b) ∀x (P (x) → Q(x)) (∀x ¬Q(x)) → (∀x ¬P (x)) (c) ∀x (P (x) → ¬Q(x)) ¬(∃x (P (x) ∧ Q(x))). 7. The sequents below look a bit tedious, but in proving their validity you make sure that
you really understand how to nest the proof rules: * (a) ∀x ∀y P (x, y) |− ∀u ∀v P (u, v) (b) ∃x ∃y F (x, y) |− ∃u ∃v F (u, v) * (c) ∃x ∀y P (x, y) |− ∀y ∃x P (x, y). 8. In this exercise, whenever
you use a proof rule for quantifiers, you should mention how its side condition (if applicable) is satisfied. (a) Prove 2(b-h) of Theorem 2.13 from page 117. (b) Prove one direction of 1(b) of Theorem
2.13: ¬∃x φ ∀x ¬φ. (c) Prove 3(a) of Theorem 2.13: (∀x φ) ∧ (∀x ψ) ∀x (φ ∧ ψ); recall that you have to do two separate proofs. (d) Prove both directions of 4(a) of Theorem 2.13: ∀x ∀y φ ∀y ∀x φ. 9.
Prove the validity of the following sequents in predicate logic, where F , G, P , and Q have arity 1, and S has arity 0 (a ‘propositional atom’): * (a) ∃x (S → Q(x)) |− S → ∃x Q(x) (b) S → ∃x Q(x) |−
∃x (S → Q(x)) (c) ∃x P (x) → S |− ∀x (P (x) → S) * (d) ∀x P (x) → S |− ∃x (P (x) → S) (e) ∀x (P (x) ∨ Q(x)) |− ∀x P (x) ∨ ∃x Q(x) (f) ∀x ∃y (P (x) ∨ Q(y)) |− ∃y ∀x (P (x) ∨ Q(y)) (g) ∀x (¬P (x) ∧ Q
(x)) ∀x (P (x) → Q(x)) (h) ∀x (P (x) ∧ Q(x)) ∀x (P (x) → Q(x)) (i) ∃x (¬P (x) ∧ ¬Q(x)) ∃x (¬(P (x) ∧ Q(x))) (j) ∃x (¬P (x) ∨ Q(x)) ∃x (¬(P (x) ∧ ¬Q(x))) * (k) ∀x (P (x) ∧ Q(x)) |− ∀x P (x) ∧ ∀x Q(x).
* (l) ∀x P (x) ∨ ∀x Q(x) |− ∀x (P (x) ∨ Q(x)). *(m) ∃x (P (x) ∧ Q(x)) |− ∃x P (x) ∧ ∃x Q(x). * (n) ∃x F (x) ∨ ∃x G(x) |− ∃x (F (x) ∨ G(x)). (o) ∀x ∀y (S(y) → F (x)) |− ∃yS(y) → ∀x F (x).
2 Predicate logic
* (p) ¬∀x ¬P (x) |− ∃x P (x). * (q) ∀x ¬P (x) |− ¬∃x P (x). * (r) ¬∃x P (x) |− ∀x ¬P (x). 10. Just like natural deduction proofs for propositional logic, certain things that look easy can be hard to
prove for predicate logic. Typically, these involve the ¬¬e rule. The patterns are the same as in propositional logic: (a) Proving that p ∨ q |− ¬(¬p ∧ ¬q) is valid is quite easy. Try it. (b) Show
that ∃x P (x) |− ¬∀x ¬P (x) is valid. (c) Proving that ¬(¬p ∧ ¬q) |− p ∨ q is valid is hard; you have to try to prove ¬¬(p ∨ q) first and then use the ¬¬e rule. Do it. (d) Re-express the sequent from
the previous item such that p and q are unary predicates and both formulas are universally quantified. Prove its validity. 11. The proofs of the sequents below combine the proof rules for equality and
quantifiers. We write φ ↔ ψ as an abbreviation for (φ → ψ) ∧ (ψ → φ). Find proofs for * (a) P (b) |− ∀x (x = b → P (x)) (b) P (b), ∀x∀y (P (x) ∧ P (y) → x = y) |− ∀x (P (x) ↔ x = b) * (c) ∃x ∃y (H(x,
y) ∨ H(y, x)), ¬∃x H(x, x) |− ∃x∃y ¬(x = y) (d) ∀x (P (x) ↔ x = b) |− P (b) ∧ ∀x∀y (P (x) ∧ P (y) → x = y). * 12. Prove the validity of S → ∀x Q(x) |− ∀x (S → Q(x)), where S has arity 0 (a
‘propositional atom’). 13. By natural deduction, show the validity of * (a) ∀x P (a, x, x), ∀x ∀y ∀z (P (x, y, z) → P (f (x), y, f (z))) |− P (f (a), a, f (a)) * (b) ∀x P (a, x, x), ∀x ∀y ∀z (P (x,
y, z) → P (f (x), y, f (z))) |− ∃z P (f (a), z, f (f (a))) * (c) ∀y Q(b, y), ∀x ∀y (Q(x, y) → Q(s(x), s(y))) |− ∃z (Q(b, z) ∧ Q(z, s(s(b)))) (d) ∀x ∀y ∀z (S(x, y) ∧ S(y, z) → S(x, z)), ∀x ¬S(x, x) ∀x
∀y (S(x, y) → ¬S(y, x)) (e) ∀x (P (x) ∨ Q(x)), ∃x ¬Q(x), ∀x (R(x) → ¬P (x)) ∃x ¬R(x) (f) ∀x (P (x) → (Q(x) ∨ R(x))), ¬∃x (P (x) ∧ R(x)) ∀x (P (x) → Q(x)) (g) ∃x ∃y (S(x, y) ∨ S(y, x)) ∃x ∃y S(x, y)
(h) ∃x (P (x) ∧ Q(x)), ∀y (P (x) → R(x)) ∃x (R(x) ∧ Q(x)). 14. Translate the following argument into a sequent in predicate logic using a suitable set of predicate symbols: If there are any tax
payers, then all politicians are tax payers. If there are any philanthropists, then all tax payers are philanthropists. So, if there are any tax-paying philanthropists, then all politicians are
Now come up with a proof of that sequent in predicate logic.
2.8 Exercises
15. Discuss in what sense the equivalences of Theorem 2.13 (page 117) form the basis of an algorithm which, given φ, pushes quantifiers to the top of the formula’s parse tree. If the result is ψ, what
can you say about commonalities and differences between φ and ψ?
Exercises 2.4 * 1. Consider the formula φ = ∀x ∀y Q(g(x, y), g(y, y), z), where Q and g have arity 3 and 2, respectively. Find two models M and M with respective environments l and l such that M l φ
but M l φ. def 2. Consider the sentence φ = ∀x ∃y ∃z (P (x, y) ∧ P (z, y) ∧ (P (x, z) → P (z, x))). Which of the following models satisfies φ? def (a) The model M consists of the set of natural
numbers with P M = {(m, n) | m < n}. def (b) The model M consists of the set of natural numbers with P M = {(m, 2 ∗ m) | m natural number}. def (c) The model M consists of the set of natural numbers
with P M = {(m, n) | m < n + 1}. 3. Let P be a predicate with two arguments. Find a model which satisfies the sentence ∀x ¬P (x, x); also find one which doesn’t. 4. Consider the sentence ∀x(∃yP (x, y)
∧ (∃zP (z, x) → ∀yP (x, y))). Please simulate the evaluation of this sentence in a model and look-up table of your choice, focusing on how the initial look-up table l grows and shrinks like a stack
when you evaluate its subformulas according to the definition of the satisfaction relation. 5. Let φ be the sentence ∀x ∀y ∃z (R(x, y) → R(y, z)), where R is a predicate symbol of two arguments. def
def * (a) Let A = {a, b, c, d} and RM = {(b, c), (b, b), (b, a)}. Do we have M φ? Justify your answer, whatever it is. def def * (b) Let A = {a, b, c} and RM = {(b, c), (a, b), (c, b)}. Do we have M
φ? Justify your answer, whatever it is. * 6. Consider the three sentences def
φ1 = ∀x P (x, x) def
φ2 = ∀x ∀y (P (x, y) → P (y, x)) def
φ3 = ∀x ∀y ∀z ((P (x, y) ∧ P (y, z) → P (x, z))) def
which express that the binary predicate P is reflexive, symmetric and transitive, respectively. Show that none of these sentences is semantically entailed by the other ones by choosing for each pair
of sentences above a model which satisfies these two, but not the third sentence – essentially, you are asked to find three binary relations, each satisfying just two of these properties.
2 Predicate logic
7. Show the semantic entailment ∀x ¬φ ¬∃x φ; for that you have to take any model which satisfies ∀x ¬φ and you have to reason why this model must also satisfy ¬∃x φ. You should do this in a similar
way to the examples in Section 2.4.2. * 8. Show the semantic entailment ∀x P (x) ∨ ∀x Q(x) ∀x (P (x) ∨ Q(x)). 9. Let φ and ψ and η be sentences of predicate logic. (a) If ψ is semantically entailed
by φ, is it necessarily the case that ψ is not semantically entailed by ¬φ? * (b) If ψ is semantically entailed by φ ∧ η, is it necessarily the case that ψ is semantically entailed by φ and
semantically entailed by η? (c) If ψ is semantically entailed by φ or by η, is it necessarily the case that ψ is semantically entailed by φ ∨ η? (d) Explain why ψ is semantically entailed by φ iff φ →
ψ is valid. 10. Is ∀x (P (x) ∨ Q(x)) ∀x P (x) ∨ ∀x Q(x) a semantic entailment? Justify your answer. 11. For each set of formulas below show that they are consistent: (a) ∀x ¬S(x, x), ∃x P (x), ∀x ∃y
S(x, y), ∀x (P (x) → ∃y S(y, x)) * (b) ∀x ¬S(x, x), ∀x ∃y S(x, y), ∀x ∀y ∀z ((S(x, y) ∧ S(y, z)) → S(x, z)) (c) (∀x (P (x) ∨ Q(x))) → ∃y R(y), ∀x (R(x) → Q(x)), ∃y (¬Q(y) ∧ P (y)) * (d) ∃x S(x, x),
∀x ∀y (S(x, y) → (x = y)). 12. For each of the formulas of predicate logic below, either find a model which does not satisfy it, or prove it is valid: (a) (∀x ∀y (S(x, y) → S(y, x))) → (∀x ¬S(x, x)) *
(b) ∃y ((∀x P (x)) → P (y)) (c) (∀x (P (x) → ∃y Q(y))) → (∀x ∃y (P (x) → Q(y))) (d) (∀x ∃y (P (x) → Q(y))) → (∀x (P (x) → ∃y Q(y))) (e) ∀x ∀y (S(x, y) → (∃z (S(x, z) ∧ S(z, y)))) (f) (∀x ∀y (S(x, y)
→ (x = y))) → (∀z ¬S(z, z)) * (g) (∀x ∃y (S(x, y) ∧ ((S(x, y) ∧ S(y, x)) → (x = y)))) → (¬∃z ∀w (S(z, w))). (h) ∀x ∀y ((P (x) → P (y)) ∧ (P (y) → P (x))) (i) (∀x ((P (x) → Q(x)) ∧ (Q(x) → P (x)))) →
((∀x P (x)) → (∀x Q(x))) (j) ((∀x P (x)) → (∀x Q(x))) → (∀x ((P (x) → Q(x)) ∧ (Q(x) → P (x)))) (k) Difficult: (∀x ∃y (P (x) → Q(y))) → (∃y ∀x (P (x) → Q(y))).
Exercises 2.5 1. Assuming that our proof calculus for predicate logic is sound (see exercise 3 below), show that the validity of the following sequents cannot be proved by finding for each sequent a
model such that all formulas to the left of evaluate to T and the sole formula to the right of evaluates to F (explain why this guarantees the non-existence of a proof):
2.8 Exercises
(a) ∀x (P (x) ∨ Q(x)) ∀x P (x) ∨ ∀x Q(x) * (b) ∀x (P (x) → R(x)), ∀x (Q(x) → R(x)) ∃x (P (x) ∧ Q(x)) (c) (∀x P (x)) → L ∀x (P (x) → L), where L has arity 0 * (d) ∀x ∃y S(x, y) ∃y ∀x S(x, y) (e) ∃x P
(x), ∃y Q(y) ∃z (P (z) ∧ Q(z)). * (f) ∃x (¬P (x) ∧ Q(x)) ∀x (P (x) → Q(x)) * (g) ∃x (¬P (x) ∨ ¬Q(x)) ∀x (P (x) ∨ Q(x)). 2. Assuming that is sound and complete for in first-order logic, explain in
detail why the undecidability of implies that satisfiability, validity, and provability are all undecidable for that logic. 3. To show the soundness of our natural deduction rules for predicate logic,
it intuitively suffices to show that the conclusion of a proof rule is true provided that all its premises are true. What additional complication arises due to the presence of variables and quantifiers?
Can you precisely formalise the necessary induction hypothesis for proving soundness?
Exercises 2.6 1. In Example 2.23, page 136, does M l ∃P φ hold if l satisfies * (a) l(u) = s3 and l(v) = s1 ; (b) l(u) = s1 and l(v) = s3 ? Justify your answers. 2. Prove that M l ∃P ∀x∀y∀z (C1 ∧ C2 ∧
C3 ∧ C4 ) holds iff state l(v) is not reachable from state l(u) in the model M, where the Ci are the ones of (2.12) on page 139. 3. Does Theorem 2.26 from page 138 apply or remain valid if we allow φ
to contain function symbols of any finite arity? * 4. In the directed graph of Figure 2.5 from page 137, how many paths are there that witness the reachability of node s3 from s2 ? 5. Let P and R be
predicate symbols of arity 2. Write formulas of existential secondorder logic of the form ∃P ψ that hold in all models of the form M = (A, RM ) iff * (a) R contains a reflexive and symmetric relation;
(b) R contains an equivalence relation (c) there is an R-path that visits each node of the graph exactly once – such a path is called Hamiltonian (d) R can be extended to an equivalence relation:
there is some equivalence relation T with RM ⊆ T * (e) the relation ‘there is an R-path of length 2’ is transitive. * 6. Show informally that (2.16) on page 141 gives rise to Russell’s paradox: A has
to be, and cannot be, an element of A. 7. The second item in the proof of Theorem 2.28 (page 140) relies on the fact that if a binary relation R is contained in a reflexive, transitive relation T of
2 Predicate logic
the same type, then T also contains the reflexive, transitive closure of R. Prove this. 8. For the model of Example 2.23 and Figure 2.5 (page 137), determine which model checks hold and justify your
answer: * (a) ∃P (∀x∀y P (x, y) → ¬P (y, x)) ∧ (∀u∀v R(u, v) → P (v, u)); (b) ∀P (∃x∃y∃z P (x, y) ∧ P (y, z) ∧ ¬P (x, z)) → (∀u∀v R(u, v) → P (u, v)); and (c) ∀P (∀x ¬P (x, x)) ∨ (∀u∀v R(u, v) → P
(u, v)). 9. Express the following statements about a binary relation R in predicate logic, universal second-order logic, or existential second-order logic – if at all possible: (a) All symmetric,
transitive relations either don’t contain R or are equivalence relations. * (b) All nodes are on at least one R-cycle. (c) There is a smallest relation containing R which is symmetric. (d) There is a
smallest relation containing R which is reflexive. * (e) The relation R is a maximal equivalence relation: R is an equivalence relation; and there is no relation contained in R that is an equivalence
Exercises 2.7 1.* (a) Explain why the model of Figure 2.11 (page 148) is a counterexample to OfLovers in the presence of the fact NoSelfLove. (b) Can you identify the set {a, b, c} from Example 2.19
(page 128) with the model of Figure 2.11 such that these two models are structurally the same? Justify your answer. * (c) Explain informally why no model with less than three elements can satisfy
(2.8) from page 128 and the fact NoSelfLove. 2. Use the following fragment of an Alloy module module AboutGraphs sig Element {} sig Graph { nodes : set Element, edges : nodes -> nodes } for these
modelling tasks: (a) Recall Exercise 6 from page 163 and its three sentences, where P (x, y) specifies that there is an edge from x to y. For each sentence, write a consistency check that attempts to
generate a model of a graph in which that sentence is false, but the other two are true. Analyze it within Alloy. What it the smallest scope, if any, in which the analyzer finds a model for this?
2.8 Exercises
* (b) (Recall that the expression # S = n specifies that set S has n elements.) Use Alloy to generate a graph with seven nodes such that each node can reach exactly five nodes on finite paths (not
necessarily the same five nodes). (c) A cycle of length n is a set of n nodes and a path through each of them, beginning and ending with the same node. Generate a cycle of length 4. 3. An undirected
graph has a set of nodes and a set of edges, except that every edge connects two nodes without any sense of direction. (a) Adjust the Alloy module from the previous item – e.g. by adding an
appropriate fact – to ‘simulate’ undirected graphs. (b) Write some consistency and assertion checks and analyze them to boost the confidence you may have in your Alloy module of undirected graphs. *
4. A colorable graph consists of a set of nodes, a binary symmetric relation (the edges) between nodes and a function that assigns to each node a color. This function is subject to the constraint
that no nodes have the same color if they are related by an edge. (a) Write a signature AboutColoredGraphs for this structure and these constraints. (b) Write a fun-statement that generates a graph
whose nodes are colored by two colors only. Such a graph is 2-colorable. (c) For eack k = 3, 4 write a fun-statement that generates a graph whose nodes are colored by k colors such that all k colors
are being used. Such a graph is k-colorable. (d) Test these three functions in a module. (e) Try to write a fun-statement that generates a graph that is 3-colorable but definitely not 2-colorable.
What does Alloy’s model builder report? Consider the formula obtained from that fun-statement’s body by existentially quantifying that body with all its parameters. Determine whether is belongs to
predicate logic, existential or universal second-order logic. * 5. A Kripke model is a state machine with a non-empty set of initial states init, a mapping prop from states to atomic properties
(specifying which properties are true at which states), a state transition relation next, and a set of final states final (states that don’t have a next state). With a module KripkeModel: (a) Write a
signature StateMachine and some basic facts that reflect this structure and these constraints. (b) Write a fun-statement Reaches which takes a state machine as first parameter and a set of states as a
second parameter such that the second parameter denotes the first parameter’s set of states reachable from any initial state. Note: Given the type declaration r : T -> T, the expression *r has type T
-> T as well and denotes the reflexive, transitive closure of r. (c) Write these fun-statements and check their consistency: i. DeadlockFree(m: StateMachine), among the reachable states of m only the
final ones can deadlock;
2 Predicate logic State_0 prop: Prop_1
next State_2 prop: Prop_0
Figure 2.15. A snapshot of a non-deterministic state machine in which no non-final state deadlocks and where states that satisfy the same properties are identical. ii. Deterministic(m: StateMachine),
at all reachable states of m the state transition relation is deterministic: each state has at most one outgoing transition; iii. Reachability(m: StateMachine, p: Prop), some state which has property
p can be reached in m; and iv. Liveness(m: StateMachine, p: Prop), no matter which state m reaches, it can – from that state – reach a state in which p holds. (d) i. Write an assertion Implies which
says that whenever a state machine satisfies Liveness for a property then it also satisfies Reachability for that property. ii. Analyze that assertion in a scope of your choice. What conclusions can
you draw from the analysis’ findings? (e) Write an assertion Converse which states that Reachability of a property implies its Liveness. Analyze it in a scope of 3. What do you conclude, based on the
analysis’ result? (f) Write a fun-statement that, when analyzed, generates a statemachine with two propositions and three states such that it satisfies the statement of the sentence in the caption of
Figure 2.15. * 6. Groups are the bread and butter of cryptography and group operations are applied in the silent background when you use PUTTY, Secure Socket Layers etc. A group is a tuple (G, , 1),
where : G × G → G is a function and 1 ∈ G such that G1 for every x ∈ G there is some y ∈ G such that x y = y x = 1 (any such y is called an inverse of x); G2 for all x, y, z ∈ G, we have x (y z) = (x
y) z; and G3 for all x ∈ G, we have x 1 = 1 x = x. (a) Specify a signature for groups that realizes this functionality and its constraints. (b) Write a fun-statement AGroup that generates a group
with three elements. (c) Write an assertion Inverse saying that inverse elements are unique. Check it in the scope of 5. Report your findings. What would the small scope hypothesis suggest?
2.8 Exercises
(d) i. Write an assertion Commutative saying that all groups are commutative. A group is commutative iff x y = y x for all its elements x and y. ii. Check the assertion Commutative in scope 5 and
report your findings. What would the small scope hypothesis suggest? iii. Re-check assertion Commutative in scope 6 and record how long the tool takes to find a solution. What lesson(s) do you learn
from this? (e) For the functions and assertions above, is it safe to restrict the scope for groups to 1? And how does one do this in Alloy? 7. In Alloy, one can extend a signature. For example, we
may declare sig Program extends PDS { m : components -- initial main of PDS } This declares instances of Program to be of type PDS, but to also possess a designated component named m. Observe how the
occurrence of components in m : components refers to the set of components of a program, viewed as a PDS5 . In this exercise, you are asked to modify the Alloy module of Figure 2.13 on page 154. (a)
Include a signature Program as above. Add a fact stating that all programs’ designated component has a main method; and for all programs, their set of components is the reflexive, transitive closure
of their relation requires applied to the designated component m. Alloy uses *r to denote the reflexive, transitive closure of relation r. (b) Write a guided simulation that, if consistent, produces a
model with three PDSs, exactly one of them being a program. The program has four components – including the designated m – all of which schedule services from the remaining three components. Use
Alloy’s analyzer to detemine whether your simulation is consistent and compliant with the specification given in this item. (c) Let’s say that a component of a program is garbage for that program if
no service reachable from the main service of m via requires schedules that component. Explain whether, and if so how, the constraints of AddComponent and RemoveComponent already enforce the presence
of ‘garbage collection’ if the instances of P and P’ are constrained to be programs. 8. Recall our discussion of existential and universal second-order logic from Section 2.6. Then study the
structure of the fun-statements and assertions in Figure 2.13 on page 154. As you may know, Alloy analyzes such statements by deriving from them a formula for which it tries to find a model within the
specified scope: the negation of the body of an assertion; or the body of a fun-statement, existentially quantified with all its parameters. For each of these derived formulas, 5
In most object-oriented languages, e.g. Java, extends creates a new type. In Alloy 2.0 and 2.1, it creates a subset of a type and not a new type as such, where the subset has additional structure and
may need to satisfy additional constraints.
2 Predicate logic
determine whether they can be expressed in first-order logic, existential secondorder logic or universal second-order logic. 9. Recalling the comment on page 142 that Alloy combines model checking M φ
and validity checking Γ φ, can you discuss to what extent this is so?
2.9 Bibliographic notes Many design decisions have been taken in the development of predicate logic in the form known today. The Greeks and the medievals had systems in which many of the examples and
exercises in this book could be represented, but nothing that we would recognise as predicate logic emerged until the work of Gottlob Frege in 1879, printed in [Fre03]. An account of the
contributions of the many other people involved in the development of logic can be found in the first few pages of W. Hodges’ chapter in [Hod83]. There are many books covering classical logic and its
use in computer science; we give a few incomplete pointers to the literature. The books [SA91], [vD89] and [Gal87] cover more theoretical applications than those in this book, including type theory,
logic programming, algebraic specification and term-rewriting systems. An approach focusing on automatic theorem proving is taken by [Fit96]. Books which study the mathematical aspects of predicate
logic in greater detail, such as completeness of the proof systems and incompleteness of first-order arithmetic, include [Ham78] and [Hod83]. Most of these books present other proof systems besides
natural deduction such as axiomatic systems and tableau systems. Although natural deduction has the advantages of elegance and simplicity over axiomatic methods, there are few expositions of it in
logic books aimed at a computer science audience. One exception to this is the book [BEKV94], which is the first one to present the rules for quantifiers in the form we used here. A natural deduction
theorem prover called Jape has been developed, in which one can vary the set of available rules and specify new ones6 . A standard reference for computability theory is [BJ80]. A proof for the
undecidability of the Post correspondence problem can be found in the text book [Tay98]. The second instance of a Post correspondence problem is taken from [Sch92]. A text on the fundamentals of
databases systems is [EN94]. The discussion of Section 2.6 is largely based on the text [Pap94] which we highly recommend if you mean to find out more about the intimate connections between logic and
computational complexity. 6
2.9 Bibliographic notes
The source code of all complete Alloy modules from this chapter (working under Alloy 2.0 and 2.1) as well as source code compliant with Alloy 3.0 are available under ‘ancillary material’ at the
book’s website. The PDS model grew out of a coursework set in the Fall 2002 for C475 Software Engineering Environments, co-taught by Susan Eisenbach and the first author; a published model customized
for the .NET global assembly cache will appeared in [EJC03]. The modelling language Alloy and its constraint analyzer [JSS01] have been developed by D. Jackson and his Software Design Group at the
Laboratory for Computer Science at the Massachusetts Institute of Technology. The tool has a dedicated repository website at alloy.mit.edu. More information on typed higher-order logics and their use
in the modelling and verifying of programming frameworks can be found on F. Pfenning’s course homepage7 on Computation and Deduction. 7
3 Verification by model checking
3.1 Motivation for verification There is a great advantage in being able to verify the correctness of computer systems, whether they are hardware, software, or a combination. This is most obvious in
the case of safety-critical systems, but also applies to those that are commercially critical, such as mass-produced chips, mission critical, etc. Formal verification methods have quite recently
become usable by industry and there is a growing demand for professionals able to apply them. In this chapter, and the next one, we examine two applications of logics to the question of verifying the
correctness of computer systems, or programs. Formal verification techniques can be thought of as comprising three parts: r a framework for modelling systems, typically a description language of some
sort; r a specification language for describing the properties to be verified; r a verification method to establish whether the description of a system satisfies the specification.
Approaches to verification can be classified according to the following criteria: Proof-based vs. model-based. In a proof-based approach, the system description is a set of formulas Γ (in a suitable
logic) and the specification is another formula φ. The verification method consists of trying to find a proof that Γ |− φ. This typically requires guidance and expertise from the user. In a model-based
approach, the system is represented by a model M for an appropriate logic. The specification is again represented by a formula φ and the verification method consists of computing whether a model M
satisfies φ (written M φ). This computation is usually automatic for finite models. 172
3.1 Motivation for verification
In Chapters 1 and 2, we could see that logical proof systems are often sound and complete, meaning that Γ |− φ (provability) holds if, and only if, Γ φ (semantic entailment) holds, where the latter
is defined as follows: for all models M, if for all ψ ∈ Γ we have M ψ, then M φ. Thus, we see that the model-based approach is potentially simpler than the proof-based approach, for it is based on a
single model M rather than a possibly infinite class of them. Degree of automation. Approaches differ on how automatic the method is; the extremes are fully automatic and fully manual. Many of the
computer-assisted techniques are somewhere in the middle. Full- vs. property-verification. The specification may describe a single property of the system, or it may describe its full behaviour. The
latter is typically expensive to verify. Intended domain of application, which may be hardware or software; sequential or concurrent; reactive or terminating; etc. A reactive system is one which
reacts to its environment and is not meant to terminate (e.g., operating systems, embedded systems and computer hardware). Pre- vs. post-development. Verification is of greater advantage if introduced
early in the course of system development, because errors caught earlier in the production cycle are less costly to rectify. (It is alleged that Intel lost millions of dollars by releasing their
Pentium chip with the FDIV error.) This chapter concerns a verification method called model checking. In terms of the above classification, model checking is an automatic, modelbased,
property-verification approach. It is intended to be used for concurrent, reactive systems and originated as a post-development methodology. Concurrency bugs are among the most difficult to find by
testing (the activity of running several simulations of important scenarios), since they tend to be non-reproducible or not covered by test cases, so it is well worth having a verification technique
that can help one to find them. The Alloy system described in Chapter 2 is also an automatic, modelbased, property-verification approach. The way models are used is slightly different, however. Alloy
finds models which form counterexamples to assertions made by the user. Model checking starts with a model described by the user, and discovers whether hypotheses asserted by the user are valid on the
model. If they are not, it can produce counterexamples, consisting of execution traces. Another difference between Alloy and model checking is that model checking (unlike Alloy) focuses explicitly on
temporal properties and the temporal evolution of systems.
3 Verification by model checking
By contrast, Chapter 4 describes a very different verification technique which in terms of the above classification is a proof-based, computer-assisted, property-verification approach. It is intended to
be used for programs which we expect to terminate and produce a result. Model checking is based on temporal logic. The idea of temporal logic is that a formula is not statically true or false in a
model, as it is in propositional and predicate logic. Instead, the models of temporal logic contain several states and a formula can be true in some states and false in others. Thus, the static
notion of truth is replaced by a dynamic one, in which the formulas may change their truth values as the system evolves from state to state. In model checking, the models M are transition systems and
the properties φ are formulas in temporal logic. To verify that a system satisfies a property, we must do three things: r model the system using the description language of a model checker, arriving
at a model M; r code the property using the specification language of the model checker, resulting in a temporal logic formula φ; r Run the model checker with inputs M and φ.
The model checker outputs the answer ‘yes’ if M φ and ‘no’ otherwise; in the latter case, most model checkers also produce a trace of system behaviour which causes this failure. This automatic
generation of such ‘counter traces’ is an important tool in the design and debugging of systems. Since model checking is a model-based approach, in terms of the classification given earlier, it
follows that in this chapter, unlike in the previous two, we will not be concerned with semantic entailment (Γ φ), or with proof theory (Γ φ), such as the development of a natural deduction calculus
for temporal logic. We will work solely with the notion of satisfaction, i.e. the satisfaction relation between a model and a formula (M φ). There is a whole zoo of temporal logics that people have
proposed and used for various things. The abundance of such formalisms may be organised by classifying them according to their particular view of ‘time.’ Lineartime logics think of time as a set of
paths, where a path is a sequence of time instances. Branching-time logics represent time as a tree, rooted at the present moment and branching out into the future. Branching time appears to make the
non-deterministic nature of the future more explicit. Another quality of time is whether we think of it as being continuous or discrete. The former would be suggested if we study an analogue
computer, the latter might be preferred for a synchronous network.
3.2 Linear-time temporal logic
Temporal logics have a dynamic aspect to them, since the truth of a formula is not fixed in a model, as it is in predicate or propositional logic, but depends on the time-point inside the model. In
this chapter, we study a logic where time is linear, called Linear-time Temporal Logic (LTL), and another where time is branching, namely Computation Tree Logic (CTL). These logics have proven to be
extremely fruitful in verifying hardware and communication protocols; and people are beginning to apply them to the verification of software. Model checking is the process of computing an answer to
the question of whether M, s φ holds, where φ is a formula of one of these logics, M is an appropriate model of the system under consideration, s is a state of that model and is the underlying
satisfaction relation. Models like M should not be confused with an actual physical system. Models are abstractions that omit lots of real features of a physical system, which are irrelevant to the
checking of φ. This is similar to the abstractions that one does in calculus or mechanics. There we talk about straight lines, perfect circles, or an experiment without friction. These abstractions
are very powerful, for they allow us to focus on the essentials of our particular concern.
3.2 Linear-time temporal logic Linear-time temporal logic, or LTL for short, is a temporal logic, with connectives that allow us to refer to the future. It models time as a sequence of states,
extending infinitely into the future. This sequence of states is sometimes called a computation path, or simply a path. In general, the future is not determined, so we consider several paths,
representing different possible futures, any one of which might be the ‘actual’ path that is realised. We work with a fixed set Atoms of atomic formulas (such as p, q, r, . . . , or p1 , p2 , . . . ).
These atoms stand for atomic facts which may hold of a system, like ‘Printer Q5 is busy,’ or ‘Process 3259 is suspended,’ or ‘The content of register R1 is the integer value 6.’ The choice of atomic
descriptions obviously depends on our particular interest in a system at hand.
3.2.1 Syntax of LTL Definition 3.1 Linear-time temporal logic (LTL) has the following syntax given in Backus Naur form: φ ::= | ⊥ | p | (¬φ) | (φ ∧ φ) | (φ ∨ φ) | (φ → φ) | (X φ) | (F φ) | (G φ) | (φ
U φ) | (φ W φ) | (φ R φ) where p is any propositional atom from some set Atoms.
3 Verification by model checking
r Figure 3.1. The parse tree of (F (p → G r) ∨ ((¬q) U p)).
Thus, the symbols and ⊥ are LTL formulas, as are all atoms from Atoms; and ¬φ is an LTL formula if φ is one, etc. The connectives X, F, G, U, R, and W are called temporal connectives. X means ‘neXt
state,’ F means ‘some Future state,’ and G means ‘all future states (Globally).’ The next three, U, R and W are called ‘Until,’ ‘Release’ and ‘Weak-until’ respectively. We will look at the precise
meaning of all these connectives in the next section; for now, we concentrate on their syntax. Here are some examples of LTL formulas: r (((F p) ∧ (G q)) → (p W r)) r (F (p → (G r)) ∨ ((¬q) U p)),
the parse tree of this formula is illustrated in Figure 3.1. r (p W (q W r)) r ((G (F p)) → (F (q ∨ s))).
It’s boring to write all those brackets, and makes the formulas hard to read. Many of them can be omitted without introducing ambiguities; for example, (p → (F q)) could be written p → F q without
ambiguity. Others, however, are required to resolve ambiguities. In order to omit some of those, we assume similar binding priorities for the LTL connectives to those we assumed for propositional and
predicate logic.
3.2 Linear-time temporal logic
q Figure 3.2. The parse tree of F p → G r ∨ ¬q U p, assuming binding priorities of Convention 3.2.
Convention 3.2 The unary connectives (consisting of ¬ and the temporal connectives X, F and G) bind most tightly. Next in the order come U, R and W; then come ∧ and ∨; and after that comes →. These
binding priorities allow us to drop some brackets without introducing ambiguity. The examples above can be written: r r r r
Fp ∧ Gq → p W r F (p → G r) ∨ ¬q U p p W (q W r) G F p → F (q ∨ s).
The brackets we retained were in order to override the priorities of Convention 3.2, or to disambiguate cases which the convention does not resolve. For example, with no brackets at all, the second
formula would become F p → G r ∨ ¬q U p, corresponding to the parse tree of Figure 3.2, which is quite different. The following are not well-formed formulas: r U r – since U is binary, not unary r p G
q – since G is unary, not binary.
3 Verification by model checking
Definition 3.3 A subformula of an LTL formula φ is any formula ψ whose parse tree is a subtree of φ’s parse tree. The subformulas of p W (q U r), e.g., are p, q, r, q U r and p W (q U r).
3.2.2 Semantics of LTL The kinds of systems we are interested in verifying using LTL may be modelled as transition systems. A transition system models a system by means of states (static structure)
and transitions (dynamic structure). More formally: Definition 3.4 A transition system M = (S, →, L) is a set of states S endowed with a transition relation → (a binary relation on S), such that
every s ∈ S has some s ∈ S with s → s , and a labelling function L : S → P(Atoms). Transition systems are also simply called models in this chapter. So a model has a collection of states S, a
relation →, saying how the system can move from state to state, and, associated with each state s, one has the set of atomic propositions L(s) which are true at that particular state. We write P
(Atoms) for the power set of Atoms, a collection of atomic descriptions. For example, the power set of {p, q} is {∅, {p}, {q}, {p, q}}. A good way of thinking about L is that it is just an assignment
of truth values to all the propositional atoms, as it was the case for propositional logic (we called that a valuation). The difference now is that we have more than one state, so this assignment
depends on which state s the system is in: L(s) contains all atoms which are true in state s. We may conveniently express all the information about a (finite) transition system M using directed graphs
whose nodes (which we call states) contain all propositional atoms that are true in that state. For example, if our system has only three states s0 , s1 and s2 ; if the only possible transitions
between states are s0 → s1 , s0 → s2 , s1 → s0 , s1 → s2 and s2 → s2 ; and if L(s0 ) = {p, q}, L(s1 ) = {q, r} and L(s2 ) = {r}, then we can condense all this information into Figure 3.3. We prefer
to present models by means of such pictures whenever that is feasible. The requirement in Definition 3.4 that for every s ∈ S there is at least one s ∈ S such that s → s means that no state of the
system can ‘deadlock.’ This is a technical convenience, and in fact it does not represent any real restriction on the systems we can model. If a system did deadlock, we could always add an extra
state sd representing deadlock, together with new
3.2 Linear-time temporal logic
p, q
q, r
Figure 3.3. A concise representation of a transition system M = (S, → ,L) as a directed graph. We label state s with l iff l ∈ L(s).
sd Figure 3.4. On the left, we have a system with a state s4 that does not have any further transitions. On the right, we expand that system with a ‘deadlock’ state sd such that no state can
deadlock; of course, it is then our understanding that reaching the ‘deadlock’ state sd corresponds to deadlock in the original system.
transitions s → sd for each s which was a deadlock in the old system, as well as sd → sd . See Figure 3.4 for such an example. Definition 3.5 A path in a model M = (S, →, L) is an infinite sequence of
states s1 , s2 , s3 , . . . in S such that, for each i ≥ 1, si → si+1 . We write the path as s1 → s2 → . . . . Consider the path π = s1 → s2 → . . . . It represents a possible future of our system:
first it is in state s1 , then it is in state s2 , and so on. We write π i for the suffix starting at si , e.g., π 3 is s3 → s4 → . . . .
3 Verification by model checking
p, q
q, r
p, q
q, r
Figure 3.5. Unwinding the system of Figure 3.3 as an infinite tree of all computation paths beginning in a particular state.
It is useful to visualise all possible computation paths from a given state s by unwinding the transition system to obtain an infinite computation tree. For example, if we unwind the state graph of
Figure 3.3 for the designated starting state s0 , then we get the infinite tree in Figure 3.5. The execution paths of a model M are explicitly represented in the tree obtained by unwinding the model.
Definition 3.6 Let M = (S, →, L) be a model and π = s1 → . . . be a path in M. Whether π satisfies an LTL formula is defined by the satisfaction relation as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.
π π π π π π π π π
⊥ p iff p ∈ L(s1 ) ¬φ iff π φ φ1 ∧ φ2 iff π φ1 and π φ2 φ1 ∨ φ2 iff π φ1 or π φ2 φ1 → φ2 iff π φ2 whenever π φ1 X φ iff π 2 φ G φ iff, for all i ≥ 1, π i φ
3.2 Linear-time temporal logic
s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10
Figure 3.6. An illustration of the meaning of Until in the semantics of LTL. Suppose p is satisfied at (and only at) s3 , s4 , s5 , s6 , s7 , s8 and q is satisfied at (and only at) s9 . Only the
states s3 to s9 each satisfy p U q along the path shown. 10. π F φ iff there is some i ≥ 1 such that π i φ 11. π φ U ψ iff there is some i ≥ 1 such that π i ψ and for all j = 1, . . . , i − 1 we have π
j φ 12. π φ W ψ iff either there is some i ≥ 1 such that π i ψ and for all j = 1, . . . , i − 1 we have π j φ; or for all k ≥ 1 we have π k φ 13. π φ R ψ iff either there is some i ≥ 1 such that π i φ
and for all j = 1, . . . , i we have π j ψ, or for all k ≥ 1 we have π k ψ.
Clauses 1 and 2 reflect the facts that is always true, and ⊥ is always false. Clauses 3–7 are similar to the corresponding clauses we saw in propositional logic. Clause 8 removes the first state from
the path, in order to create a path starting at the ‘next’ (second) state. Notice that clause 3 means that atoms are evaluated in the first state along the path in consideration. However, that doesn’t
mean that all the atoms occuring in an LTL formula refer to the first state of the path; if they are in the scope of a temporal connective, e.g., in G (p → X q), then the calculation of satisfaction
involves taking suffices of the path in consideration, and the atoms refer to the first state of those suffices. Let’s now look at clauses 11–13, which deal with the binary temporal connectives. U, which
stands for ‘Until,’ is the most commonly encountered one of these. The formula φ1 U φ2 holds on a path if it is the case that φ1 holds continuously until φ2 holds. Moreover, φ1 U φ2 actually demands
that φ2 does hold in some future state. See Figure 3.6 for illustration: each of the states s3 to s9 satisfies p U q along the path shown, but s0 to s2 don’t. The other binary connectives are W,
standing for ‘Weak-until,’ and R, standing for ‘Release.’ Weak-until is just like U, except that φ W ψ does not require that ψ is eventually satisfied along the path in question, which is required by
φ U ψ. Release R is the dual of U; that is, φ R ψ is equivalent to ¬(¬φ U ¬ψ). It is called ‘Release’ because clause 11 determines that ψ must remain true up to and including the moment when φ
becomes true (if there is one); φ ‘releases’ ψ. R and W are actually quite similar; the differences are that they swap the roles of φ and ψ, and the clause for W has an i − 1
3 Verification by model checking
where R has i. Since they are similar, why do we need both? We don’t; they are interdefinable, as we will see later. However, it’s useful to have both. R is useful because it is the dual of U, while W
is useful because it is a weak form of U. Note that neither the strong version (U) or the weak version (W) of until says anything about what happens after the until has been realised. This is in
contrast with some of the readings of ‘until’ in natural language. For example, in the sentence ‘I smoked until I was 22’ it is not only expressed that the person referred to continually smoked up
until he or she was 22 years old, but we also would interpret such a sentence as saying that this person gave up smoking from that point onwards. This is different from the semantics of until in
temporal logic. We could express the sentence about smoking by combining U with other connectives; for example, by asserting that it was once true that s U (t ∧ G ¬s), where s represents ‘I smoke’
and t represents ‘I am 22.’ Remark 3.7 Notice that, in clauses 9–13 above, the future includes the present. This means that, when we say ‘in all future states,’ we are including the present state as
a future state. It is a matter of convention whether we do this, or not. As an exercise, you may consider developing a version of LTL in which the future excludes the present. A consequence of
adopting the convention that the future shall include the present is that the formulas G p → p, p → q U p and p → F p are true in every state of every model. So far we have defined a satisfaction
relation between paths and LTL formulas. However, to verify systems, we would like to say that a model as a whole satisfies an LTL formula. This is defined to hold whenever every possible execution
path of the model satisfies the formula. Definition 3.8 Suppose M = (S, →, L) is a model, s ∈ S, and φ an LTL formula. We write M, s φ if, for every execution path π of M starting at s, we have π φ.
If M is clear from the context, we may abbreviate M, s φ by s φ. It should be clear that we have outlined the formal foundations of a procedure that, given φ, M and s, can check whether M, s φ holds.
Later in this chapter, we will examine algorithms which implement this calculation. Let us now look at some example checks for the system in Figures 3.3 and 3.5. 1. M, s0 p ∧ q holds since the atomic
symbols p and q are contained in the node of s0 : π p ∧ q for every path π beginning in s0 .
3.2 Linear-time temporal logic
2. M, s0 ¬r holds since the atomic symbol r is not contained in node s0 . 3. M, s0 holds by definition. 4. M, s0 X r holds since all paths from s0 have either s1 or s2 as their next state, and each of
those states satisfies r. 5. M, s0 X (q ∧ r) does not hold since we have the rightmost computation path s0 → s2 → s2 → s2 → . . . in Figure 3.5, whose second node s2 contains r, but not q. 6. M, s0 G
¬(p ∧ r) holds since all computation paths beginning in s0 satisfy G ¬(p ∧ r), i.e. they satisfy ¬(p ∧ r) in each state along the path. Notice that G φ holds in a state if, and only if, φ holds in
all states reachable from the given state. 7. For similar reasons, M, s2 G r holds (note the s2 instead of s0 ). 8. For any state s of M, we have M, s F (¬q ∧ r) → F G r. This says that if any path π
beginning in s gets to a state satisfying (¬q ∧ r), then the path π satisfies F G r. Indeed this is true, since if the path has a state satisfying (¬q ∧ r) then (since that state must be s2 ) the path
does satisfy F G r. Notice what F G r says about a path: eventually, you have continuously r. 9. The formula G F p expresses that p occurs along the path in question infinitely often. Intuitively,
it’s saying: no matter how far along the path you go (that’s the G part) you will find you still have a p in front of you (that’s the F part). For example, the path s0 → s1 → s0 → s1 → . . . satisfies
G F p. But the path s0 → s2 → s2 → s2 → . . . doesn’t. 10. In our model, if a path from s0 has infinitely many ps on it then it must be the path s0 → s1 → s0 → s1 → . . . , and in that case it also
has infinitely many rs on it. So, M, s0 G F p → G F r. But it is not the case the other way around! It is not the case that M, s0 G F r → G F p, because we can find a path from s0 which has infinitely
many rs but only one p.
3.2.3 Practical patterns of specifications What kind of practically relevant properties can we check with formulas of LTL? We list a few of the common patterns. Suppose atomic descriptions include
some words such as busy and requested. We may require some of the following properties of real systems: r It is impossible to get to a state where started holds, but ready does not hold: G¬(started ∧
¬ready) The negation of this formula expresses that it is possible to get to such a state, but this is only so if interpreted on paths (π φ). We cannot assert such a possibility if interpreted on
states (s φ) since we cannot express the existence of paths; for that interpretation, the negation of the formula above asserts that all paths will eventually get to such a state.
3 Verification by model checking
r For any state, if a request (of some resource) occurs, then it will eventually be acknowledged: G (requested → F acknowledged). r A certain process is enabled infinitely often on every computation
path: G F enabled. r Whatever happens, a certain process will eventually be permanently deadlocked: F G deadlock. r If the process is enabled infinitely often, then it runs infinitely often. G F
enabled → G F running. r An upwards travelling lift at the second floor does not change its direction when it has passengers wishing to go to the fifth floor: G (floor2 ∧ directionup ∧ ButtonPressed5 →
(directionup U floor5)) Here, our atomic descriptions are boolean expressions built from system variables, e.g., floor2.
There are some things which are not possible to say in LTL, however. One big class of such things are statements which assert the existence of a path, such as these ones: r From any state it is
possible to get to a restart state (i.e., there is a path from all states to a state satisfying restart). r The lift can remain idle on the third floor with its doors closed (i.e., from the state in
which it is on the third floor, there is a path along which it stays there).
LTL can’t express these because it cannot directly assert the existence of paths. In Section 3.4, we look at Computation Tree Logic (CTL) which has operators for quantifying over paths, and can
express these properties.
3.2.4 Important equivalences between LTL formulas Definition 3.9 We say that two LTL formulas φ and ψ are semantically equivalent, or simply equivalent, writing φ ≡ ψ, if for all models M and all
paths π in M: π φ iff π ψ. The equivalence of φ and ψ means that φ and ψ are semantically interchangeable. If φ is a subformula of some bigger formula χ, and ψ ≡ φ, then we can make the substitution
of ψ for φ in χ without changing the meaning of χ. In propositional logic, we saw that ∧ and ∨ are duals of each other, meaning that if you push a ¬ past a ∧, it becomes a ∨, and vice versa: ¬(φ ∧ ψ)
≡ ¬φ ∨ ¬ψ
¬(φ ∨ ψ) ≡ ¬φ ∧ ¬ψ.
(Because ∧ and ∨ are binary, pushing a negation downwards in the parse tree past one of them also has the effect of duplicating that negation.)
3.2 Linear-time temporal logic
Similarly, F and G are duals of each other, and X is dual with itself: ¬G φ ≡ F ¬φ
¬F φ ≡ G ¬φ
¬X φ ≡ X ¬φ.
Also U and R are duals of each other: ¬(φ U ψ) ≡ ¬φ R ¬ψ
¬(φ R ψ) ≡ ¬φ U ¬ψ.
We should give formal proofs of these equivalences. But they are easy, so we leave them as an exercise to the reader. ‘Morally’ there ought to be a dual for W, and you can invent one if you like.
Work out what it might mean, and then pick a symbol based on the first letter of the meaning. However, it might not be very useful. It’s also the case that F distributes over ∨ and G over ∧, i.e., F
(φ ∨ ψ) ≡ F φ ∨ F ψ G (φ ∧ ψ) ≡ G φ ∧ G ψ. Compare this with the quantifier equivalences in Section 2.3.2. But F does not distribute over ∧. What this means is that there is a model with a path which
distinguishes F (φ ∧ ψ) and F φ ∧ F ψ, for some φ, ψ. Take the path s0 → s1 → s0 → s1 → . . . from the system of Figure 3.3, for example; it satisfies F p ∧ F r but it doesn’t satisfy F (p ∧ r). Here
are two more equivalences in LTL: Fφ ≡ U φ
G φ ≡ ⊥ R φ.
The first one exploits the fact that the clause for Until states two things: the second formula φ must become true; and until then, the first formula must hold. So, if we put ‘no constraint’ for the
first formula, it boils down to asking that the second formula holds, which is what F asks. (The formula represent ‘no constraint.’ If you ask me to bring it about that holds, I need do nothing, it
enforces no constraint. In the same sense, ⊥ is ‘every constraint.’ If you ask me to bring it about that ⊥ holds, I’ll have to meet every constraint there is, which is impossible.) The second
formula, that G φ ≡ ⊥ R φ, can be obtained from the first by putting a ¬ in front of each side, and applying the duality rules. Another more intuitive way of seeing this is to recall the meaning of
‘release:’ ⊥ releases φ, but ⊥ will never be true, so φ doesn’t get released. Another pair of equivalences relates the strong and weak versions of Until, U and W. Strong until may be seen as weak
until plus the constraint that the eventuality must actually occur: φ U ψ ≡ φ W ψ ∧ Fψ.
3 Verification by model checking
To prove equivalence (3.2), suppose first that a path satisfies φ U ψ. Then, from clause 11, we have i ≥ 1 such that π i ψ and for all j = 1, . . . , i − 1 we have π j φ. From clause 12, this proves φ
W ψ, and from clause 10 it proves F ψ. Thus for all paths π, if π φ U ψ then π φ W ψ ∧ F ψ. As an exercise, the reader can prove it the other way around. Writing W in terms of U is also possible: W
is like U but also allows the possibility of the eventuality never occurring: φ W ψ ≡ φ U ψ ∨ G φ.
Inspection of clauses 12 and 13 reveals that R and W are rather similar. The differences are that they swap the roles of their arguments φ and ψ; and the clause for W has an i − 1 where R has i.
Therefore, it is not surprising that they are expressible in terms of each other, as follows: φ W ψ ≡ ψ R (φ ∨ ψ) φ R ψ ≡ ψ W (φ ∧ ψ).
(3.4) (3.5)
3.2.5 Adequate sets of connectives for LTL Recall that φ ≡ ψ holds iff any path in any transition system which satisfies φ also satisfies ψ, and vice versa. As in propositional logic, there is some
redundancy among the connectives. For example, in Chapter 1 we saw that the set {⊥, ∧, ¬} forms an adequate set of connectives, since the other connectives ∨, →, , etc., can be written in terms of
those three. Small adequate sets of connectives also exist in LTL. Here is a summary of the situation. r X is completely orthogonal to the other connectives. That is to say, its presence doesn’t help
in defining any of the other ones in terms of each other. Moreover, X cannot be derived from any combination of the others. r Each of the sets {U, X}, {R, X}, {W, X} is adequate. To see this, we note
that – R and W may be defined from U, by the duality φ R ψ ≡ ¬(¬φ U ¬ψ) and equivalence (3.4) followed by the duality, respectively. – U and W may be defined from R, by the duality φ U ψ ≡ ¬(¬φ R ¬ψ)
and equivalence (3.4), respectively. – R and U may be defined from W, by equivalence (3.5) and the duality φ U ψ ≡ ¬(¬φ R ¬ψ) followed by equivalence (3.5).
Sometimes it is useful to look at adequate sets of connectives which do not rely on the availability of negation. That’s because it is often convenient to assume formulas are written in
negation-normal form, where all the negation symbols are applied to propositional atoms (i.e., they are near the leaves
3.3 Model checking: systems, tools, properties
of the parse tree). In this case, these sets are adequate for the fragment without X, and no strict subset is: {U, R}, {U, W}, {U, G}, {R, F}, {W, F}. But {R, G} and {W, G} are not adequate. Note
that one cannot define G with {U,F}, and one cannot define F with {R,G} or {W,G}. We finally state and prove a useful equivalence about U. Theorem 3.10 The equivalence φ U ψ ≡ ¬(¬ψ U (¬φ ∧ ¬ψ)) ∧ F ψ
holds for all LTL formulas φ and ψ. PROOF: Take any path s0 → s1 → s2 → . . . in any model. First, suppose s0 φ U ψ holds. Let n be the smallest number such that sn ψ; such a number has to exist
since s0 φ U ψ; then, for each k < n, sk φ. We immediately have s0 F ψ, so it remains to show s0 ¬(¬ψ U (¬φ ∧ ¬ψ)), which, if we expand, means: (∗) for each i > 0, if si ¬φ ∧ ¬ψ, then there is some j
< i with sj ψ. def Take any i > 0 with si ¬φ ∧ ¬ψ; i > n, so we can take j = n and have sj ψ. Conversely, suppose s0 ¬(¬ψ U (¬φ ∧ ¬ψ)) ∧ F ψ holds; we prove s0 φ U ψ. Since s0 F ψ, we have a minimal
n as before. We show that, for any i < n, si φ. Suppose si ¬φ; since n is minimal, we know si ¬ψ, so by (∗) there is some j < i < n with sj ψ, contradicting the minimality of n. 2
3.3 Model checking: systems, tools, properties 3.3.1 Example: mutual exclusion Let us now look at a larger example of verification using LTL, having to do with mutual exclusion. When concurrent
processes share a resource (such as a file on a disk or a database entry), it may be necessary to ensure that they do not have access to it at the same time. Several processes simultaneously editing
the same file would not be desirable. We therefore identify certain critical sections of each process’ code and arrange that only one process can be in its critical section at a time. The critical
section should include all the access to the shared resource (though it should be as small as possible so that no unnecessary exclusion takes place). The problem we are faced with is to find a
protocol for determining which process is allowed to enter its critical section at which time. Once we have found one which we think works, we verify our solution by checking that it has some
expected properties, such as the following ones: Safety: Only one process is in its critical section at any time.
3 Verification by model checking
n1 n2 s5
n1 t2
t1 n2
c 1 n2 s4
c1 t2
t1 t2
n1 c 2 s7
t1 c2
Figure 3.7. A first-attempt model for mutual exclusion.
This safety property is not enough, since a protocol which permanently excluded every process from its critical section would be safe, but not very useful. Therefore, we should also require:
Liveness: Whenever any process requests to enter its critical section, it will eventually be permitted to do so. Non-blocking: A process can always request to enter its critical section. Some rather
crude protocols might work on the basis that they cycle through the processes, making each one in turn enter its critical section. Since it might be naturally the case that some of them request
access to the shared resource more often than others, we should make sure our protocol has the property: No strict sequencing: Processes need not enter their critical section in strict sequence. The
first modelling attempt We will model two processes, each of which is in its non-critical state (n), or trying to enter its critical state (t), or in its critical state (c). Each individual process
undergoes transitions in the cycle n → t → c → n → . . . , but the two processes interleave with each other. Consider the protocol given by the transition system M in Figure 3.7. (As usual, we write
p1 p2 . . . pm in a node s to denote that p1 , p2 , . . . , pm are the only propositional atoms true at s.) The two processes start off in their non-critical sections (global state s0 ). State s0 is
the only initial state, indicated by the incoming edge with no source. Either of them may now
3.3 Model checking: systems, tools, properties
move to its trying state, but only one of them can ever make a transition at a time (asynchronous interleaving). At each step, an (unspecified) scheduler determines which process may run. So there is
a transition arrow from s0 to s1 and s5 . From s1 (i.e., process 1 trying, process 2 non-critical) again two things can happen: either process 1 moves again (we go to s2 ), or process 2 moves (we go
to s3 ). Notice that not every process can move in every state. For example, process 1 cannot move in state s7 , since it cannot go into its critical section until process 2 comes out of its critical
section. We would like to check the four properties by first describing them as temporal logic formulas. Unfortunately, they are not all expressible as LTL formulas. Let us look at them case-by-case.
Safety: This is expressible in LTL, as G ¬(c1 ∧ c2 ). Clearly, G ¬(c1 ∧ c2 ) is satisfied in the initial state (indeed, in every state). Liveness: This is also expressible: G (t1 → F c1 ). However, it
is not satisfied by the initial state, for we can find a path starting at the initial state along which there is a state, namely s1 , in which t1 is true but from there along the path c1 is false. The
path in question is s0 → s1 → s3 → s7 → s1 → s3 → s7 . . . on which c1 is always false. Non-blocking: Let’s just consider process 1. We would like to express the property as: for every state
satisfying n1 , there is a successor satisfying t1 . Unfortunately, this existence quantifier on paths (‘there is a successor satisfying. . . ’) cannot be expressed in LTL. It can be expressed in the
logic CTL, which we will turn to in the next section (for the impatient, see page 215). No strict sequencing: We might consider expressing this as saying: there is a path with two distinct states
satisfying c1 such that no state in between them has that property. However, we cannot express ‘there exists a path,’ so let us consider the complement formula instead. The complement says that all
paths having a c1 period which ends cannot have a further c1 state until a c2 state occurs. We write this as: G (c1 → c1 W (¬c1 ∧ ¬c1 W c2 )). This says that anytime we get into a c1 state, either
that condition persists indefinitely, or it ends with a nonc1 state and in that case there is no further c1 state unless and until we obtain a c2 state. This formula is false, as exemplified by the
path s0 → s5 → s3 → s4 → s5 → s3 → s4 . . . . Therefore the original condition expressing that strict sequencing need not occur, is true. Before further considering the mutual exclusion example, some
comments about expressing properties in LTL are appropriate. Notice that in the
3 Verification by model checking
no-strict-sequencing property, we overcame the problem of not being able to express the existence of paths by instead expressing the complement property, which of course talks about all paths. Then
we can perform our check, and simply reverse the answer; if the complement property is false, we declare our property to be true, and vice versa. Why was that tactic not available to us to express
the non-blocking property? The reason is that it says: every path to a n1 state may be continued by a one-step path to a t1 state. The presence of both universal and existential quantifiers is the
problem. In the no-strict-sequencing property, we had only an existential quantifier; thus, taking the complement property turned it into a universal path quantifier, which can be expressed in LTL. But
where we have alternating quantifiers, taking the complement property doesn’t help in general. Let’s go back to the mutual exclusion example. The reason liveness failed in our first attempt at
modelling mutual exclusion is that non-determinism means it might continually favour one process over another. The problem is that the state s3 does not distinguish between which of the processes
first went into its trying state. We can solve this by splitting s3 into two states. The second modelling attempt The two states s3 and s9 in Figure 3.8 both correspond to the state s3 in our first
modelling attempt. They both record that the two processes are in their trying states, but in s3 it is implicitly recorded that it is process 1’s turn, whereas in s9 it is process 2’s turn. Note that
states s3 and s9 both have the labelling t1 t2 ; the definition of transition systems does not preclude this. We can think of there being some other, hidden, variables which are not part of the
initial labelling, which distinguish s3 and s9 . Remark 3.11 The four properties of safety, liveness, non-blocking and nostrict-sequencing are satisfied by the model in Figure 3.8. (Since the
nonblocking property has not yet been written in temporal logic, we can only check it informally.) In this second modelling attempt, our transition system is still slightly over-simplified, because we
are assuming that it will move to a different state on every tick of the clock (there are no transitions to the same state). We may wish to model that a process can stay in its critical state for
several ticks, but if we include an arrow from s4 , or s7 , to itself, we will again violate liveness. This problem will be solved later in this chapter when we consider ‘fairness constraints’
(Section 3.6.2).
3.3 Model checking: systems, tools, properties
n1 n2 s5
n1 t2
t1 n2 s3
c 1 n2
t1 t2 s4
c1 t2
n1 c 2
t1 t2 s7
t1 c2
Figure 3.8. A second-attempt model for mutual exclusion. There are now two states representing t1 t2 , namely s3 and s9 .
3.3.2 The NuSMV model checker So far, this chapter has been quite theoretical; and the sections after this one continue in this vein. However, one of the exciting things about model checking is that
it is also a practical subject, for there are several efficient implementations which can check large systems in realistic time. In this section, we look at the NuSMV model-checking system. NuSMV
stands for ‘New Symbolic Model Verifier.’ NuSMV is an Open Source product, is actively supported and has a substantial user community. For details on how to obtain it, see the bibliographic notes at
the end of the chapter. NuSMV (sometimes called simply SMV) provides a language for describing the models we have been drawing as diagrams and it directly checks the validity of LTL (and also CTL)
formulas on those models. SMV takes as input a text consisting of a program describing a model and some specifications (temporal logic formulas). It produces as output either the word ‘true’ if the
specifications hold, or a trace showing why the specification is false for the model represented by our program. SMV programs consist of one or more modules. As in the programming language C, or Java,
one of the modules must be called main. Modules can declare variables and assign to them. Assignments usually give the initial value of a variable and its next value as an expression in terms of the
current values of variables. This expression can be non-deterministic (denoted by several expressions in braces, or no assignment at all). Non-determinism is used to model the environment and for
3 Verification by model checking
The following input to SMV: MODULE main VAR request : boolean; status : {ready,busy}; ASSIGN init(status) := ready; next(status) := case request : busy; 1 : {ready,busy}; esac; LTLSPEC G(request -> F
status=busy) consists of a program and a specification. The program has two variables, request of type boolean and status of enumeration type {ready, busy}: 0 denotes ‘false’ and 1 represents ‘true.’
The initial and subsequent values of variable request are not determined within this program; this conservatively models that these values are determined by an external environment. This
under-specification of request implies that the value of variable status is partially determined: initially, it is ready; and it becomes busy whenever request is true. If request is false, the next
value of status is not determined. Note that the case 1: signifies the default case, and that case statements are evaluated from the top down: if several expressions to the left of a ‘:’ are true,
then the command corresponding to the first, top-most true expression will be executed. The program therefore denotes the transition system shown in Figure 3.9; there are four states, each one
corresponding to a possible value of the two binary variables. Note that we wrote ‘busy’ as a shorthand for ‘status=busy’ and ‘req’ for ‘request is true.’ It takes a while to get used to the syntax
of SMV and its meaning. Since variable request functions as a genuine environment in this model, the program and the transition system are non-deterministic: i.e., the ‘next state’ is not uniquely
defined. Any state transition based on the behaviour of status comes in a pair: to a successor state where request is false, or true, respectively. For example, the state ‘¬req, busy’ has four states
it can move to (itself and three others). LTL specifications are introduced by the keyword LTLSPEC and are simply LTL formulas. Notice that SMV uses &, |, -> and ! for ∧, ∨, → and ¬, respectively,
since they are available on standard keyboards. We may
3.3 Model checking: systems, tools, properties
req ready
req busy
¬req ready
¬req busy
Figure 3.9. The model corresponding to the SMV program in the text.
easily verify that the specification of our module main holds of the model in Figure 3.9. Modules in SMV SMV supports breaking a system description into several modules, to aid readability and to
verify interaction properties. A module is instantiated when a variable having that module name as its type is declared. This defines a set of variables, one for each one declared in the module
description. In the example below, which is one of the ones distributed with SMV, a counter which repeatedly counts from 000 through to 111 is described by three single-bit counters. The module
counter cell is instantiated three times, with the names bit0, bit1 and bit2. The counter module has one formal parameter, carry in, which is given the actual value 1 in bit0, and bit0.carry out in
the instance bit1. Hence, the carry in of module bit1 is the carry out of module bit0. Note that we use the period ‘.’ in m.v to access the variable v in module m. This notation is also used by Alloy
(see Chapter 2) and a host of programming languages to access fields in record structures, or methods in objects. The keyword DEFINE is used to assign the expression value & carry in to the symbol
carry out (such definitions are just a means for referring to the current value of a certain expression). MODULE main VAR bit0 : counter_cell(1); bit1 : counter_cell(bit0.carry_out); bit2 :
counter_cell(bit1.carry_out); LTLSPEC G F bit2.carry_out
3 Verification by model checking
MODULE counter_cell(carry_in) VAR value : boolean; ASSIGN init(value) := 0; next(value) := (value + carry_in) mod 2; DEFINE carry_out := value & carry_in; The effect of the DEFINE statement could have
been obtained by declaring a new variable and assigning its value thus: VAR carry_out : boolean; ASSIGN carry_out := value & carry_in; Notice that, in this assignment, the current value of the
variable is assigned. Defined symbols are usually preferable to variables, since they don’t increase the state space by declaring new variables. However, they cannot be assigned non-deterministically
since they refer only to another expression. Synchronous and asynchronous composition By default, modules in SMV are composed synchronously: this means that there is a global clock and, each time it
ticks, each of the modules executes in parallel. By use of the process keyword, it is possible to compose the modules asynchronously. In that case, they run at different ‘speeds,’ interleaving
arbitrarily. At each tick of the clock, one of them is non-deterministically chosen and executed for one cycle. Asynchronous interleaving composition is useful for describing communication protocols,
asynchronous circuits and other systems whose actions are not synchronised to a global clock. The bit counter above is synchronous, whereas the examples below of mutual exclusion and the alternating
bit protocol are asynchronous.
3.3.3 Running NuSMV The normal use of NuSMV is to run it in batch mode, from a Unix shell or command prompt in Windows. The command line NuSMV counter3.smv
3.3 Model checking: systems, tools, properties
will analyse the code in the file counter3.smv and report on the specifications it contains. One can also run NuSMV interactively. In that case, the command line NuSMV -int counter3.smv enters NuSMV’s
command-line interpreter. From there, there is a variety of commands you can use which allow you to compile the description and run the specification checks, as well as inspect partial results and set
various parameters. See the NuSMV user manual for more details. NuSMV also supports bounded model checking, invoked by the commandline option -bmc. Bounded model checking looks for counterexamples in
order of size, starting with counterexamples of length 1, then 2, etc., up to a given threshold (10 by default). Note that bounded model checking is incomplete: failure to find a counterexample does
not mean that there is none, but only that there is none of length up to the threshold. For related reasons, this incompleteness features also in Alloy and its constraint analyzer. Thus, while a
negative answer can be relied on (if NuSMV finds a counterexample, it is valid), a positive one cannot. References on bounded model checking can be found in the bibliographic notes on page 254. Later
on, we use bounded model checking to prove the optimality of a scheduler.
3.3.4 Mutual exclusion revisited Figure 3.10 gives the SMV code for a mutual exclusion protocol. This code consists of two modules, main and prc. The module main has the variable turn, which
determines whose turn it is to enter the critical section if both are trying to enter (recall the discussion about the states s3 and s9 in Section 3.3.1). The module main also has two instantiations
of prc. In each of these instantiations, st is the status of a process (saying whether it is in its critical section, or not, or trying) and other-st is the status of the other process (notice how
this is passed as a parameter in the third and fourth lines of main). The value of st evolves in the way described in a previous section: when it is n, it may stay as n or move to t. When it is t, if
the other one is n, it will go straight to c, but if the other one is t, it will check whose turn it is before going to c. Then, when it is c, it may move back to n. Each instantiation of prc gives
the turn to the other one when it gets to its critical section. An important feature of SMV is that we can restrict its search tree to execution paths along which an arbitrary boolean formula about
the state
3 Verification by model checking
196 MODULE main VAR
pr1: process prc(pr2.st, turn, 0); pr2: process prc(pr1.st, turn, 1); turn: boolean; ASSIGN init(turn) := 0; -- safety LTLSPEC
G!((pr1.st = c) & (pr2.st = c))
-- liveness LTLSPEC
G((pr1.st = t) -> F (pr1.st = c))
G((pr2.st = t) -> F (pr2.st = c))
-- ‘negation’ of strict sequencing (desired to be false) LTLSPEC G(pr1.st=c -> ( G pr1.st=c | (pr1.st=c U (!pr1.st=c & G !pr1.st=c | ((!pr1.st=c) U pr2.st=c))))) MODULE prc(other-st, turn, myturn)
VAR st: {n, t, c}; ASSIGN init(st) := n; next(st) := case (st = n)
: {t,n};
(st = t) & (other-st = n)
: c;
(st = t) & (other-st = t) & (turn = myturn): c; (st = c)
: {c,n};
: st;
esac; next(turn) := case turn = myturn & st = c : !turn; 1
: turn;
esac; FAIRNESS running FAIRNESS
!(st = c)
Figure 3.10. SMV code for mutual exclusion. Because W is not supported by SMV, we had to make use of equivalence (3.3) to write the no-strict-sequencing formula as an equivalent but longer formula
involving U.
3.3 Model checking: systems, tools, properties
φ is true infinitely often. Because this is often used to model fair access to resources, it is called a fairness constraint and introduced by the keyword FAIRNESS. Thus, the occurrence of FAIRNESS φ
means that SMV, when checking a specification ψ, will ignore any path along which φ is not satisfied infinitely often. In the module prc, we restrict model checks to computation paths along which st is
infinitely often not equal to c. This is because our code allows the process to stay in its critical section as long as it likes. Thus, there is another opportunity for liveness to fail: if process 2
stays in its critical section forever, process 1 will never be able to enter. Again, we ought not to take this kind of violation into account, since it is patently unfair if a process is allowed to
stay in its critical section for ever. We are looking for more subtle violations of the specifications, if there are any. To avoid the one above, we stipulate the fairness constraint !(st=c). If the
module in question has been declared with the process keyword, then at each time point SMV will non-deterministically decide whether or not to select it for execution, as explained earlier. We may
wish to ignore paths in which a module is starved of processor time. The reserved word running can be used instead of a formula in a fairness constraint: writing FAIRNESS running restricts attention
to execution paths along which the module in which it appears is selected for execution infinitely often. In prc, we restrict ourselves to such paths, since, without this restriction, it would be easy
to violate the liveness constraint if an instance of prc were never selected for execution. We assume the scheduler is fair; this assumption is codified by two FAIRNESS clauses. We return to the issue
of fairness, and the question of how our model-checking algorithm copes with it, in the next section. Please run this program in NuSMV to see which specifications hold for it. The transition system
corresponding to this program is shown in Figure 3.11. Each state shows the values of the variables; for example, ct1 is the state in which process 1 and 2 are critical and trying, respectively, and
turn=1. The labels on the transitions show which process was selected for execution. In general, each state has several transitions, some in which process 1 moves and others in which process 2 moves.
This model is a bit different from the previous model given for mutual exclusion in Figure 3.8, for these two reasons: r Because the boolean variable turn has been explicitly introduced to distinguish
between states s3 and s9 of Figure 3.8, we now distinguish between certain states
3 Verification by model checking
tc1 2
2 1,2
cn1 2
nn1 2
Figure 3.11. The transition system corresponding to the SMV code in Figure 3.10. The labels on the transitions denote the process which makes the move. The label 1, 2 means that either process could
make that move.
3.3 Model checking: systems, tools, properties
(for example, ct0 and ct1) which were identical before. However, these states are not distinguished if you look just at the transitions from them. Therefore, they satisfy the same LTL formulas which
don’t mention turn. Those states are distinguished only by the way they can arise. r We have eliminated an over-simplification made in the model of Figure 3.8. Recall that we assumed the system would
move to a different state on every tick of the clock (there were no transitions from a state to itself). In Figure 3.11, we allow transitions from each state to itself, representing that a process was
chosen for execution and did some private computation, but did not move in or out of its critical section. Of course, by doing this we have introduced paths in which one process gets stuck in its
critical section, whence the need to invoke a fairness constraint to eliminate such paths.
3.3.5 The ferryman You may recall the puzzle of a ferryman, goat, cabbage, and wolf all on one side of a river. The ferryman can cross the river with at most one passenger in his boat. There is a
behavioural conflict between: 1. the goat and the cabbage; and 2. the goat and the wolf;
if they are on the same river bank but the ferryman crosses the river or stays on the other bank. Can the ferryman transport all goods to the other side, without any conflicts occurring? This is a
planning problem, but it can be solved by model checking. We describe a transition system in which the states represent which goods are at which side of the river. Then we ask if the goal state is
reachable from the initial state: Is there a path from the initial state such that it has a state along it at which all the goods are on the other side, and during the transitions to that state the
goods are never left in an unsafe, conflicting situation? We model all possible behaviour (including that which results in conflicts) as a NuSMV program (Figure 3.12). The location of each agent is
modelled as a boolean variable: 0 denotes that the agent is on the initial bank, and 1 the destination bank. Thus, ferryman = 0 means that the ferryman is on the initial bank, ferryman = 1 that he is
on the destination bank, and similarly for the variables goat, cabbage and wolf. The variable carry takes a value indicating whether the goat, cabbage, wolf or nothing is carried by the ferryman. The
definition of next(carry) works as follows. It is non-deterministic, but the set from which a value is non-deterministically chosen is determined by the values of ferryman, goat,
MODULE main VAR ferryman : boolean; goat : boolean; cabbage : boolean; wolf : boolean; carry : {g,c,w,0}; ASSIGN init(ferryman) := 0; init(goat) init(cabbage) := 0; init(wolf) init(carry) := 0;
:= 0; := 0;
next(ferryman) := 0,1; next(carry) := case ferryman=goat : g; 1 : 0; esac union case ferryman=cabbage : c; 1 : 0; esac union case ferryman=wolf : w; 1 : 0; esac union 0; next(goat) := case ferryman=
goat & next(carry)=g : next(ferryman); 1 : goat; esac; next(cabbage) := case ferryman=cabbage & next(carry)=c : next(ferryman); 1 : cabbage; esac; next(wolf) := case ferryman=wolf & next(carry)=w :
next(ferryman); 1 : wolf; esac; LTLSPEC !((
(goat=cabbage | goat=wolf) -> goat=ferryman) U (cabbage & goat & wolf & ferryman))
Figure 3.12. NuSMV code for the ferryman planning problem.
3.3 Model checking: systems, tools, properties
etc., and always includes 0. If ferryman = goat (i.e., they are on the same side) then g is a member of the set from which next(carry) is chosen. The situation for cabbage and wolf is similar. Thus,
if ferryman = goat = wolf = cabbage then that set is {g, w, 0}. The next value assigned to ferryman is non-deterministic: he can choose to cross or not to cross the river. But the next values of
goat, cabbage and wolf are deterministic, since whether they are carried or not is determined by the ferryman’s choice, represented by the non-deterministic assignment to carry; these values follow
the same pattern. Note how the boolean guards refer to state bits at the next state. The SMV compiler does a dependency analysis and rejects circular dependencies on next values. (The dependency
analysis is rather pessimistic: sometimes NuSMV complains of circularity even in situations when it could be resolved. The original CMU-SMV is more liberal in this respect.)
Running NuSMV We seek a path satisfying φ U ψ, where ψ asserts the final goal state, and φ expresses the safety condition (if the goat is with the cabbage or the wolf, then the ferryman is there, too,
to prevent any untoward behaviour). Thus, we assert that all paths satisfy ¬(φ U ψ), i.e., no path satisfies φ U ψ. We hope this is not the case, and NuSMV will give us an example path which does
satisfy φ U ψ. Indeed, running NuSMV gives us the path of Figure 3.13, which represents a solution to the puzzle. The beginning of the generated path represents the usual solution to this puzzle: the
ferryman takes the goat first, then goes back for the cabbage. To avoid leaving the goat and the cabbage together, he takes the goat back, and picks up the wolf. Now the wolf and the cabbage are on
the destination side, and he goes back again to get the goat. This brings us to State 1.9, where the ferryman appears to take a well-earned break. But the path continues. States 1.10 to 1.15 show
that he takes his charges back to the original side of the bank; first the cabbage, then the wolf, then the goat. Unfortunately it appears that the ferryman’s clever plan up to state 1.9 is now
spoiled, because the goat meets an unhappy end in state 1.11. What went wrong? Nothing, actually. NuSMV has given us an infinite path, which loops around the 15 illustrated states. Along the infinite
path, the ferryman repeatedly takes his goods across (safely), and then back again (unsafely). This path does indeed satisfy the specification φ U ψ, which asserts the safety of the forward journey
but says nothing about what happens after that. In other words, the path is correct; it satisfies φ U ψ (with ψ occurring at state 8). What happens along the path after that has no bearing on φ U ψ.
3 Verification by model checking
acws-0116% nusmv ferryman.smv *** This is NuSMV 2.1.2 (compiled 2002-11-22 12:00:00) *** For more information of NuSMV see *** or email to . *** Please report bugs to . -- specification !(((goat =
cabbage | goat = wolf) -> goat = ferryman) U (((cabbage & goat) & wolf) & ferryman)) is false -- as demonstrated by the following execution sequence -- loop starts here --> State 1.1 State 1.8 State
1.9 State 1.2 State 1.10 State 1.3 State 1.11 State 1.4 State 1.12 State 1.5 State 1.13 State 1.14 State 1.6 State 1.15 State 1.7 msg_chan.output1=1) Figure 3.17. The main ABP module.
acknowledgement, so that sender does not think that its very first message is being acknowledged before anything has happened. For the same reason, the output of the channels is initialised to 1. The
specifications for ABP. ifications:
Our SMV program satisfies the following spec-
Safety: If the message bit 1 has been sent and the correct acknowledgement has been returned, then a 1 was indeed received by the receiver: G (S.st=sent & S.message1=1 -> msg chan.output1=1).
Liveness: Messages get through eventually. Thus, for any state there is inevitably a future state in which the current message has got through. In the module sender, we specified G F st=sent. (This
specification could equivalently have been written in the main module, as G F S.st=sent.) Similarly, acknowledgements get through eventually. In the module receiver, we write G F st=received.
3.4 Branching-time logic In our analysis of LTL (linear-time temporal logic) in the preceding sections, we noted that LTL formulas are evaluated on paths. We defined that a state of a system satisfies
an LTL formula if all paths from the given state satisfy it. Thus, LTL implicitly quantifies universally over paths. Therefore, properties which assert the existence of a path cannot be expressed in
LTL. This problem can partly be alleviated by considering the negation of the property in question, and interpreting the result accordingly. To check whether there
3 Verification by model checking
exists a path from s satisfying the LTL formula φ, we check whether all paths satisfy ¬φ; a positive answer to this is a negative answer to our original question, and vice versa. We used this
approach when analysing the ferryman puzzle in the previous section. However, as already noted, properties which mix universal and existential path quantifiers cannot in general be model checked using
this approach, because the complement formula still has a mix. Branching-time logics solve this problem by allowing us to quantify explicitly over paths. We will examine a logic known as Computation
Tree Logic, or CTL. In CTL, as well as the temporal operators U, F, G and X of LTL we also have quantifiers A and E which express ‘all paths’ and ‘exists a path’, respectively. For example, we can
write: r There is a reachable state satisfying q: this is written EF q. r From all reachable states satisfying p, it is possible to maintain p continuously until reaching a state satisfying q: this
is written AG (p → E[p U q]). r Whenever a state satisfying p is reached, the system can exhibit q continuously forevermore: AG (p → EG q). r There is a reachable state from which all reachable
states satisfy p: EF AG p.
3.4.1 Syntax of CTL Computation Tree Logic, or CTL for short, is a branching-time logic, meaning that its model of time is a tree-like structure in which the future is not determined; there are
different paths in the future, any one of which might be the ‘actual’ path that is realised. As before, we work with a fixed set of atomic formulas/descriptions (such as p, q, r, . . . , or p1 , p2 , .
. . ). Definition 3.12 We define CTL formulas inductively via a Backus Naur form as done for LTL: φ ::= ⊥ | | p | (¬φ) | (φ ∧ φ) | (φ ∨ φ) | (φ → φ) | AX φ | EX φ | AF φ | EF φ | AG φ | EG φ | A[φ U
φ] | E[φ U φ] where p ranges over a set of atomic formulas. Notice that each of the CTL temporal connectives is a pair of symbols. The first of the pair is one of A and E. A means ‘along All paths’
(inevitably) and E means ‘along at least (there Exists) one path’ (possibly). The second one of the pair is X, F, G, or U, meaning ‘neXt state,’ ‘some Future state,’ ‘all future states (Globally)’
and Until, respectively. The pair of symbols in E[φ1 U φ2 ], for example, is EU. In CTL, pairs of symbols like EU are
3.4 Branching-time logic
indivisible. Notice that AU and EU are binary. The symbols X, F, G and U cannot occur without being preceded by an A or an E; similarly, every A or E must have one of X, F, G and U to accompany it.
Usually weak-until (W) and release (R) are not included in CTL, but they are derivable (see Section 3.4.5). Convention 3.13 We assume similar binding priorities for the CTL connectives to what we did
for propositional and predicate logic. The unary connectives (consisting of ¬ and the temporal connectives AG, EG, AF, EF, AX and EX) bind most tightly. Next in the order come ∧ and ∨; and after that
come →, AU and EU . Naturally, we can use brackets in order to override these priorities. Let us see some examples of well-formed CTL formulas and some examples which are not well-formed, in order to
understand the syntax. Suppose that p, q and r are atomic formulas. The following are well-formed CTL formulas: r AG (q → EG r), note that this is not the same as AG q → EG r, for according to
Convention 3.13, the latter formula means (AG q) → (EG r) r EF E[r U q] r A[p U EF r] r EF EG p → AF r, again, note that this binds as (EF EG p) → AF r, not EF (EG p → AF r) or EF EG (p → AF r) r A
[p1 U A[p2 U p3 ]] r E[A[p1 U p2 ] U p3 ] r AG (p → A[p U (¬p ∧ A[¬p U q])]).
It is worth spending some time seeing how the syntax rules allow us to construct each of these. The following are not well-formed formulas: r r r r r r
EF G r A¬G ¬p F [r U q] EF (r U q) AEF r A[(r U q) ∧ (p U r)].
It is especially worth understanding why the syntax rules don’t allow us to construct these. For example, take EF (r U q). The problem with this string is that U can occur only when paired with an A
or an E. The E we have is paired with the F. To make this into a well-formed CTL formula, we would have to write EF E[r U q] or EF A[r U q].
3 Verification by model checking
Figure 3.18. The parse tree of a CTL formula without infix notation.
Notice that we use square brackets after the A or E, when the paired operator is a U. There is no strong reason for this; you could use ordinary round brackets instead. However, it often helps one to
read the formula (because we can more easily spot where the corresponding close bracket is). Another reason for using the square brackets is that SMV insists on it. The reason A[(r U q) ∧ (p U r)] is
not a well-formed formula is that the syntax does not allow us to put a boolean connective (like ∧) directly inside A[ ] or E[ ]. Occurrences of A or E must be followed by one of G, F, X or U; when
they are followed by U, it must be in the form A[φ U ψ]. Now, the φ and the ψ may contain ∧, since they are arbitrary formulas; so A[(p ∧ q) U (¬r → q)] is a well-formed formula. Observe that AU and
EU are binary connectives which mix infix and prefix notation. In pure infix, we would write φ1 AU φ2 , whereas in pure prefix we would write AU(φ1 , φ2 ). As with any formal language, and as we did in
the previous two chapters, it is useful to draw parse trees for well-formed formulas. The parse tree for A[AX ¬p U E[EX (p ∧ q) U ¬p]] is shown in Figure 3.18. Definition 3.14 A subformula of a CTL
formula φ is any formula ψ whose parse tree is a subtree of φ’s parse tree.
3.4 Branching-time logic
3.4.2 Semantics of computation tree logic CTL formulas are interpreted over transition systems (Definition 3.4). Let M = (S, →, L) be such a model, s ∈ S and φ a CTL formula. The definition of whether
M, s φ holds is recursive on the structure of φ, and can be roughly understood as follows: r If φ is atomic, satisfaction is determined by L. r If the top-level connective of φ (i.e., the connective
occurring top-most in the parse tree of φ) is a boolean connective (∧, ∨, ¬, etc.) then the satisfaction question is answered by the usual truth-table definition and further recursion down φ. r If the
top level connective is an operator beginning A, then satisfaction holds if all paths from s satisfy the ‘LTL formula’ resulting from removing the A symbol. r Similarly, if the top level connective
begins with E, then satisfaction holds if some path from s satisfy the ‘LTL formula’ resulting from removing the E.
In the last two cases, the result of removing A or E is not strictly an LTL formula, for it may contain further As or Es below. However, these will be dealt with by the recursion. The formal
definition of M, s φ is a bit more verbose: Definition 3.15 Let M = (S, →, L) be a model for CTL, s in S, φ a CTL formula. The relation M, s φ is defined by structural induction on φ: M, s and M, s ⊥
M, s p iff p ∈ L(s) M, s ¬φ iff M, s φ M, s φ1 ∧ φ2 iff M, s φ1 and M, s φ2 M, s φ1 ∨ φ2 iff M, s φ1 or M, s φ2 M, s φ1 → φ2 iff M, s φ1 or M, s φ2 . M, s AX φ iff for all s1 such that s → s1 we have M, s1
φ. Thus, AX says: ‘in every next state.’ 8. M, s EX φ iff for some s1 such that s → s1 we have M, s1 φ. Thus, EX says: ‘in some next state.’ E is dual to A – in exactly the same way that ∃ is dual to
∀ in predicate logic. 9. M, s AG φ holds iff for all paths s1 → s2 → s3 → . . ., where s1 equals s, and all si along the path, we have M, si φ. Mnemonically: for All computation paths beginning in s
the property φ holds Globally. Note that ‘along the path’ includes the path’s initial state s. 10. M, s EG φ holds iff there is a path s1 → s2 → s3 → . . ., where s1 equals s, and for all si along the
path, we have M, si φ. Mnemonically: there Exists a path beginning in s such that φ holds Globally along the path. 1. 2. 3. 4. 5. 6. 7.
3 Verification by model checking
Figure 3.19. A system whose starting state satisfies EF φ.
11. M, s AF φ holds iff for all paths s1 → s2 → . . ., where s1 equals s, there is some si such that M, si φ. Mnemonically: for All computation paths beginning in s there will be some Future state
where φ holds. 12. M, s EF φ holds iff there is a path s1 → s2 → s3 → . . ., where s1 equals s, and for some si along the path, we have M, si φ. Mnemonically: there Exists a computation path beginning
in s such that φ holds in some Future state; 13. M, s A[φ1 U φ2 ] holds iff for all paths s1 → s2 → s3 → . . ., where s1 equals s, that path satisfies φ1 U φ2 , i.e., there is some si along the path,
such that M, si φ2 , and, for each j < i, we have M, sj φ1 . Mnemonically: All computation paths beginning in s satisfy that φ1 Until φ2 holds on it. 14. M, s E[φ1 U φ2 ] holds iff there is a path s1
→ s2 → s3 → . . ., where s1 equals s, and that path satisfies φ1 U φ2 as specified in 13. Mnemonically: there Exists a computation path beginning in s such that φ1 Until φ2 holds on it.
Clauses 9–14 above refer to computation paths in models. It is therefore useful to visualise all possible computation paths from a given state s by unwinding the transition system to obtain an
infinite computation tree, whence ‘computation tree logic.’ The diagrams in Figures 3.19–3.22 show schematically systems whose starting states satisfy the formulas EF φ, EG φ, AG φ and AF φ,
respectively. Of course, we could add more φ to any of these diagrams and still preserve the satisfaction – although there is nothing to add for AG . The diagrams illustrate a ‘least’ way of
satisfying the formulas.
3.4 Branching-time logic
φ φ φ
Figure 3.20. A system whose starting state satisfies EG φ.
φ φ φ φ
φ φ
φ φ
Figure 3.21. A system whose starting state satisfies AG φ.
Recall the transition system of Figure 3.3 for the designated starting state s0 , and the infinite tree illustrated in Figure 3.5. Let us now look at some example checks for this system. 1. M, s0 p ∧
q holds since the atomic symbols p and q are contained in the node of s0 . 2. M, s0 ¬r holds since the atomic symbol r is not contained in node s0 .
3 Verification by model checking
φ φ
Figure 3.22. A system whose starting state satisfies AF φ. 3. M, s0 holds by definition. 4. M, s0 EX (q ∧ r) holds since we have the leftmost computation path s0 → s1 → s0 → s1 → . . . in Figure 3.5,
whose second node s1 contains q and r. 5. M, s0 ¬AX (q ∧ r) holds since we have the rightmost computation path s0 → s2 → s2 → s2 → . . . in Figure 3.5, whose second node s2 only contains r, but not
q. 6. M, s0 ¬EF (p ∧ r) holds since there is no computation path beginning in s0 such that we could reach a state where p ∧ r would hold. This is so because there is simply no state whatsoever in
this system where p and r hold at the same time. 7. M, s2 EG r holds since there is a computation path s2 → s2 → s2 → . . . beginning in s2 such that r holds in all future states of that path; this
is the only computation path beginning at s2 and so M, s2 AG r holds as well. 8. M, s0 AF r holds since, for all computation paths beginning in s0 , the system reaches a state (s1 or s2 ) such that r
holds. 9. M, s0 E[(p ∧ q) U r] holds since we have the rightmost computation path s0 → s2 → s2 → s2 → . . . in Figure 3.5, whose second node s2 (i = 1) satisfies r, but all previous nodes (only j = 0,
i.e., node s0 ) satisfy p ∧ q. 10. M, s0 A[p U r] holds since p holds at s0 and r holds in any possible successor state of s0 , so p U r is true for all computation paths beginning in s0 (so we may
choose i = 1 independently of the path). 11. M, s0 AG (p ∨ q ∨ r → EF EG r) holds since in all states reachable from s0 and satisfying p ∨ q ∨ r (all states in this case) the system can reach a state
satisfying EG r (in this case state s2 ).
3.4 Branching-time logic
3.4.3 Practical patterns of specifications It’s useful to look at some typical examples of formulas, and compare the situation with LTL (Section 3.2.3). Suppose atomic descriptions include some words
such as busy and requested. r It is possible to get to a state where started holds, but ready doesn’t: EF (started ∧ ¬ready). To express impossibility, we simply negate the formula. r For any state,
if a request (of some resource) occurs, then it will eventually be acknowledged: AG (requested → AF acknowledged). r The property that if the process is enabled infinitely often, then it runs
infinitely often, is not expressible in CTL. In particular, it is not expressed by AG AF enabled → AG AF running, or indeed any other insertion of A or E into the corresponding LTL formula. The CTL
formula just given expresses that if every path has infinitely often enabled, then every path is infinitely often taken; this is much weaker than asserting that every path which has infinitely often
enabled is infinitely often taken. r A certain process is enabled infinitely often on every computation path: AG (AF enabled). r Whatever happens, a certain process will eventually be permanently
deadlocked: AF (AG deadlock). r From any state it is possible to get to a restart state: AG (EF restart). r An upwards travelling lift at the second floor does not change its direction when it has
passengers wishing to go to the fifth floor: AG (floor2 ∧ directionup ∧ ButtonPressed5 → A[directionup U floor5]) Here, our atomic descriptions are boolean expressions built from system variables,
e.g., floor2. r The lift can remain idle on the third floor with its doors closed: AG (floor3 ∧ idle ∧ doorclosed → EG (floor3 ∧ idle ∧ doorclosed)). r A process can always request to enter its
critical section. Recall that this was not expressible in LTL. Using the propositions of Figure 3.8, this may be written AG (n1 → EX t1 ) in CTL. r Processes need not enter their critical section in
strict sequence. This was also not expressible in LTL, though we expressed its negation. CTL allows us to express it directly: EF (c1 ∧ E[c1 U (¬c1 ∧ E[¬c2 U c1 ])]).
3.4.4 Important equivalences between CTL formulas Definition 3.16 Two CTL formulas φ and ψ are said to be semantically equivalent if any state in any model which satisfies one of them also satisfies
the other; we denote this by φ ≡ ψ.
3 Verification by model checking
We have already noticed that A is a universal quantifier on paths and E is the corresponding existential quantifier. Moreover, G and F are also universal and existential quantifiers, ranging over the
states along a particular path. In view of these facts, it is not surprising to find that de Morgan rules exist: ¬AF φ ≡ EG ¬φ ¬EF φ ≡ AG ¬φ
¬AX φ ≡ EX ¬φ. We also have the equivalences AF φ ≡ A[ U φ]
EF φ ≡ E[ U φ]
which are similar to the corresponding equivalences in LTL.
3.4.5 Adequate sets of CTL connectives As in propositional logic and in LTL, there is some redundancy among the CTL connectives. For example, the connective AX can be written ¬EX ¬; and AG, AF, EG
and EF can be written in terms of AU and EU as follows: first, write AG φ as ¬EF ¬φ and EG φ as ¬AF ¬φ, using (3.6), and then use AF φ ≡ A[ U φ] and EF φ ≡ E[ U φ]. Therefore AU, EU and EX form an
adequate set of temporal connectives. Also EG, EU, and EX form an adequate set, for we have the equivalence A[φ U ψ] ≡ ¬(E[¬ψ U (¬φ ∧ ¬ψ)] ∨ EG ¬ψ)
which can be proved as follows: A[φ U ψ] ≡ A[¬(¬ψ U (¬φ ∧ ¬ψ)) ∧ F ψ] ≡ ¬E¬[¬(¬ψ U (¬φ ∧ ¬ψ)) ∧ F ψ] ≡ ¬E[(¬ψ U (¬φ ∧ ¬ψ)) ∨ G ¬ψ] ≡ ¬(E[¬ψ U (¬φ ∧ ¬ψ)] ∨ EG ¬ψ). The first line is by Theorem 3.10,
and the remainder by elementary manipulation. (This proof involves intermediate formulas which violate the syntactic formation rules of CTL; however, it is valid in the logic CTL* introduced in the
next section.) More generally, we have: Theorem 3.17 A set of temporal connectives in CTL is adequate if, and only if, it contains at least one of {AX , EX }, at least one of {EG , AF , AU } and EU .
3.5 CTL* and the expressive powers of LTL and CTL
This theorem is proved in a paper referenced in the bibliographic notes at the end of the chapter. The connective EU plays a special role in that theorem because neither weak-until W nor release R
are primitive in CTL (Definition 3.12). The temporal connectives AR, ER, AW and EW are all definable in CTL: r r r r
A[φ R ψ] = ¬E[¬φ U ¬ψ] E[φ R ψ] = ¬A[¬φ U ¬ψ] A[φ W ψ] = A[ψ R (φ ∨ ψ)], and then use the first equation above E[φ W ψ] = E[ψ R (φ ∨ ψ)], and then use the second one.
These definitions are justified by LTL equivalences in Sections 3.2.4 and 3.2.5. Some other noteworthy equivalences in CTL are the following: AG φ EG φ AF φ EF φ A[φ U ψ] E[φ U ψ]
≡ ≡ ≡ ≡ ≡ ≡
φ ∧ AX AG φ φ ∧ EX EG φ φ ∨ AX AF φ φ ∨ EX EF φ ψ ∨ (φ ∧ AX A[φ U ψ]) ψ ∨ (φ ∧ EX E[φ U ψ]).
For example, the intuition for the third one is the following: in order to have AF φ in a particular state, φ must be true at some point along each path from that state. To achieve this, we either
have φ true now, in the current state; or we postpone it, in which case we must have AF φ in each of the next states. Notice how this equivalence appears to define AF in terms of AX and AF itself, an
apparently circular definition. In fact, these equivalences can be used to define the six connectives on the left in terms of AX and EX , in a non-circular way. This is called the fixed-point
characterisation of CTL; it is the mathematical foundation for the model-checking algorithm developed in Section 3.6.1; and we return to it later (Section 3.7).
3.5 CTL* and the expressive powers of LTL and CTL CTL allows explicit quantification over paths, and in this respect it is more expressive than LTL, as we have seen. However, it does not allow one to
select a range of paths by describing them with a formula, as LTL does. In that respect, LTL is more expressive. For example, in LTL we can say ‘all paths which have a p along them also have a q
along them,’ by writing F p → F q. It is not possible to write this in CTL because of the constraint that every F has an associated A or E. The formula AF p → AF q means
3 Verification by model checking
something quite different: it says ‘if all paths have a p along them, then all paths have a q along them.’ One might write AG (p → AF q), which is closer, since it says that every way of extending
every path to a p eventually meets a q, but that is still not capturing the meaning of F p → F q. CTL* is a logic which combines the expressive powers of LTL and CTL, by dropping the CTL constraint
that every temporal operator (X, U, F, G) has to be associated with a unique path quantifier (A, E). It allows us to write formulas such as r A[(p U r) ∨ (q U r)]: along all paths, either p is true
until r, or q is true until r. r A[X p ∨ X X p]: along all paths, p is true in the next state, or the next but one. r E[G F p]: there is a path along which p is infinitely often true.
These formulas are not equivalent to, respectively, A[(p ∨ q) U r)], AX p ∨ AX AX p and EG EF p. It turns out that the first of them can be written as a (rather long) CTL formula. The second and third
do not have a CTL equivalent. The syntax of CTL* involves two classes of formulas: r state formulas, which are evaluated in states: φ ::= | p | (¬φ) | (φ ∧ φ) | A[α] | E[α] where p is any atomic
formula and α any path formula; and r path formulas, which are evaluated along paths: α ::= φ | (¬α) | (α ∧ α) | (α U α) | (G α) | (F α) | (X α)
where φ is any state formula. This is an example of an inductive definition which is mutually recursive: the definition of each class depends upon the definition of the other, with base cases p and .
LTL and CTL as subsets of CTL* Although the syntax of LTL does not include A and E, the semantic viewpoint of LTL is that we consider all paths. Therefore, the LTL formula α is equivalent to the CTL*
formula A[α]. Thus, LTL can be viewed as a subset of CTL*. CTL is also a subset of CTL*, since it is the fragment of CTL* in which we restrict the form of path formulas to α ::= (φ U φ) | (G φ) | (F
φ) | (X φ) Figure 3.23 shows the relationship among the expressive powers of CTL, LTL and CTL*. Here are some examples of formulas in each of the subsets
3.5 CTL* and the expressive powers of LTL and CTL
Figure 3.23. The expressive powers of CTL, LTL and CTL*.
shown: def
In CTL but not in LTL: ψ1 = AG EF p. This expresses: wherever we have got to, we can always get to a state in which p is true. This is also useful, e.g., in finding deadlocks in protocols. The proof
that AG EF p is not expressible in LTL is as follows. Let φ be an LTL formula such that A[φ] is allegedly equivalent to AG EF p. Since M, s AG EF p in the left-hand diagram below, we have M, s A[φ].
Now let M be as shown in the right-hand diagram. The paths from s in M are a subset of those from s in M, so we have M , s A[φ]. Yet, it is not the case that M , s AG EF p; a contradiction. s
In CTL*, but neither in CTL nor in LTL: ψ4 = E[G F p], saying that there is a path with infinitely many p. The proof that this is not expressible in CTL is quite complex and may be found in the papers
co-authored by E. A. Emerson with others, given in the references. (Why is it not expressible in LTL?) def In LTL but not in CTL: ψ3 = A[G F p → F q], saying that if there are infinitely many p along
the path, then there is an occurrence of q. This is an interesting thing to be able to say; for example, many fairness constraints are of the form ‘infinitely often requested implies eventually
acknowledged’. def In LTL and CTL: ψ2 = AG (p → AF q) in CTL, or G (p → F q) in LTL: any p is eventually followed by a q. Remark 3.18 We just saw that some (but not all) LTL formulas can be converted
into CTL formulas by adding an A to each temporal operator. For
3 Verification by model checking
a positive example, the LTL formula G (p → F q) is equivalent to the CTL formula AG (p → AF q). We discuss two more negative examples: r F G p and AF AG p are not equivalent, since F G p is satisfied,
whereas AF AG p is not satisfied, in the model
In fact, AF AG p is strictly stronger than F G p. r While the LTL formulas X F p and F X p are equivalent, and they are equivalent to the CTL formula AX AF p, they are not equivalent to AF AX p. The
latter is strictly stronger, and has quite a strange meaning (try working it out).
Remark 3.19 There is a considerable literature comparing linear-time and branching-time logics. The question of which one is ‘better’ has been debated for about 20 years. We have seen that they have
incomparable expressive powers. CTL* is more expressive than either of them, but is computationally much more expensive (as will be seen in Section 3.6). The choice between LTL and CTL depends on the
application at hand, and on personal preference. LTL lacks CTL’s ability to quantify over paths, and CTL lacks LTL’s finer-grained ability to describe individual paths. To many people, LTL appears to
be more straightforward to use; as noted above, CTL formulas like AF AX p seem hard to understand.
3.5.1 Boolean combinations of temporal formulas in CTL Compared with CTL*, the syntax of CTL is restricted in two ways: it does not allow boolean combinations of path formulas and it does not allow
nesting of the path modalities X, F and G. Indeed, we have already seen examples of the inexpressibility in CTL of nesting of path modalities, namely the formulas ψ3 and ψ4 above. In this section, we
see that the first of these restrictions is only apparent; we can find equivalents in CTL for formulas having boolean combinations of path formulas. The idea is to translate any CTL formula having
boolean combinations of path formulas into a CTL formula that doesn’t. For example, we may see that E[F p ∧ F q] ≡ EF [p ∧ EF q] ∨ EF [q ∧ EF p] since, if we have F p ∧ F q along any path, then
either the p must come before the q, or the other way around, corresponding to the two disjuncts on the right. (If the p and q occur simultaneously, then both disjuncts are true.)
3.6 Model-checking algorithms
Since U is like F (only with the extra complication of its first argument), we find the following equivalence: E[(p1 U q1 ) ∧ (p2 U q2 )] ≡ E[(p1 ∧ p2 ) U (q1 ∧ E[p2 U q2 ])] ∨ E[(p1 ∧ p2 ) U (q2 ∧ E
[p1 U q1 ])]. And from the CTL equivalence A[p U q] ≡ ¬(E[¬q U (¬p ∧ ¬q)] ∨ EG ¬q) (see Theorem 3.10) we can obtain E[¬(p U q)] ≡ E[¬q U (¬p ∧ ¬q)] ∨ EG ¬q. Other identities we need in this
translation include E[¬X p] ≡ EX ¬p.
3.5.2 Past operators in LTL The temporal operators X, U, F, etc. which we have seen so far refer to the future. Sometimes we want to encode properties that refer to the past, such as: ‘whenever q
occurs, then there was some p in the past.’ To do this, we may add the operators Y, S, O, H. They stand for yesterday, since, once, and historically, and are the past analogues of X, U, F, G,
respectively. Thus, the example formula may be written G (q → O p). NuSMV supports past operators in LTL. One could also add past operators to CTL (AY, ES, etc.) but NuSMV does not support them.
Somewhat counter-intuitively, past operators do not increase the expressive power of LTL. That is to say, every LTL formula with past operators can be written equivalently without them. The example
formula above can be written ¬p W q, or equivalently ¬(¬q U (p ∧ ¬q)) if one wants to avoid W. This result is surprising, because it seems that being able to talk about the past as well as the future
allows more expressivity than talking about the future alone. However, recall that LTL equivalence is quite crude: it says that the two formulas are satisfied by exactly the same set of paths. The
past operators allow us to travel backwards along the path, but only to reach points we could have reached by travelling forwards from its beginning. In contrast, adding past operators to CTL does
increase its expressive power, because they can allow us to examine states not forward-reachable from the present one.
3.6 Model-checking algorithms The semantic definitions for LTL and CTL presented in Sections 3.2 and 3.4 allow us to test whether the initial states of a given system satisfy an LTL or CTL formula.
This is the basic model-checking question. In general, interesting transition systems will have a huge number of states and the formula
3 Verification by model checking
we are interested in checking may be quite long. It is therefore well worth trying to find efficient algorithms. Although LTL is generally preferred by specifiers, as already noted, we start with CTL
model checking because its algorithm is simpler.
3.6.1 The CTL model-checking algorithm Humans may find it easier to do model checks on the unwindings of models into infinite trees, given a designated initial state, for then all possible paths are
plainly visible. However, if we think of implementing a model checker on a computer, we certainly cannot unwind transition systems into infinite trees. We need to do checks on finite data structures.
For this reason, we now have to develop new insights into the semantics of CTL. Such a deeper understanding will provide the basis for an efficient algorithm which, given M, s ∈ S and φ, computes
whether M, s φ holds. In the case that φ is not satisfied, such an algorithm can be augmented to produce an actual path (= run) of the system demonstrating that M cannot satisfy φ. That way, we may
debug a system by trying to fix what enables runs which refute φ. There are various ways in which one could consider ?
M, s0 φ as a computational problem. For example, one could have the model M, the formula φ and a state s0 as input; one would then expect a reply of the form ‘yes’ (M, s0 φ holds), or ‘no’ (M, s0 φ
does not hold). Alternatively, the inputs could be just M and φ, where the output would be all states s of the model M which satisfy φ. It turns out that it is easier to provide an algorithm for
solving the second of these two problems. This automatically gives us a solution to the first one, since we can simply check whether s0 is an element of the output set. The labelling algorithm We
present an algorithm which, given a model and a CTL formula, outputs the set of states of the model that satisfy the formula. The algorithm does not need to be able to handle every CTL connective
explicitly, since we have already seen that the connectives ⊥, ¬ and ∧ form an adequate set as far as the propositional connectives are concerned; and AF , EU and EX form an adequate set of temporal
connectives. Given an arbitrary CTL formula φ, we would simply pre-process φ in order to write it in an equivalent form in terms of the adequate set of connectives, and then
3.6 Model-checking algorithms
Repeat. . . AFψ1
. . . until no change. Figure 3.24. The iteration step of the procedure for labelling states with subformulas of the form AF ψ1 .
call the model-checking algorithm. Here is the algorithm: INPUT: a CTL model M = (S, →, L) and a CTL formula φ. OUTPUT: the set of states of M which satisfy φ. First, change φ to the output of
TRANSLATE (φ), i.e., we write φ in terms of the connectives AF, EU, EX, ∧, ¬ and ⊥ using the equivalences given earlier in the chapter. Next, label the states of M with the subformulas of φ that are
satisfied there, starting with the smallest subformulas and working outwards towards φ. Suppose ψ is a subformula of φ and states satisfying all the immediate subformulas of ψ have already been
labelled. We determine by a case analysis which states to label with ψ. If ψ is r r r r r
⊥: then no states are labelled with ⊥. p: then label s with p if p ∈ L(s). ψ1 ∧ ψ2 : label s with ψ1 ∧ ψ2 if s is already labelled both with ψ1 and with ψ2 . ¬ψ1 : label s with ¬ψ1 if s is not
already labelled with ψ1 . AF ψ1 : – If any state s is labelled with ψ1 , label it with AF ψ1 . – Repeat: label any state with AF ψ1 if all successor states are labelled with AF ψ1 , until there is
no change. This step is illustrated in Figure 3.24. r E[ψ1 U ψ2 ]: – If any state s is labelled with ψ2 , label it with E[ψ1 U ψ2 ]. – Repeat: label any state with E[ψ1 U ψ2 ] if it is labelled with
ψ1 and at least one of its successors is labelled with E[ψ1 U ψ2 ], until there is no change. This step is illustrated in Figure 3.25. r EX ψ : label any state with EX ψ if one of its successors is
labelled with ψ . 1 1 1
3 Verification by model checking
E[ψ1 U ψ2 ]
E[ψ1 U ψ2 ]
ψ1 E[ψ1 U ψ2 ]
Figure 3.25. The iteration step of the procedure for labelling states with subformulas of the form E[ψ1 U ψ2 ].
Having performed the labelling for all the subformulas of φ (including φ itself), we output the states which are labelled φ. The complexity of this algorithm is O(f · V · (V + E)), where f is the
number of connectives in the formula, V is the number of states and E is the number of transitions; the algorithm is linear in the size of the formula and quadratic in the size of the model. Handling
EG directly Instead of using a minimal adequate set of connectives, it would have been possible to write similar routines for the other connectives. Indeed, this would probably be more efficient. The
connectives AG and EG require a slightly different approach from that for the others, however. Here is the algorithm to deal with EG ψ1 directly: r EG ψ : 1 – Label all the states with EG ψ1 . – If
any state s is not labelled with ψ1 , delete the label EG ψ1 . – Repeat: delete the label EG ψ1 from any state if none of its successors is labelled with EG ψ1 ; until there is no change.
Here, we label all the states with the subformula EG ψ1 and then whittle down this labelled set, instead of building it up from nothing as we did in the case for EU. Actually, there is no real
difference between this procedure for EG ψ and what you would do if you translated it into ¬AF ¬ψ as far as the final result is concerned. A variant which is more efficient We can improve the efficiency
of our labelling algorithm by using a cleverer way of handling EG. Instead of using EX, EU and AF as the adequate set, we use EX, EU and EG instead. For EX and EU we do as before (but take care to
search the model by
3.6 Model-checking algorithms
states satisfying ψ EG ψ
Figure 3.26. A better way of handling EG.
backwards breadth-first search, for this ensures that we won’t have to pass over any node twice). For the EG ψ case: r Restrict the graph to states satisfying ψ, i.e., delete all other states and
their transitions; r Find the maximal strongly connected components (SCCs); these are maximal regions of the state space in which every state is linked with (= has a finite path to) every other one in
that region. r Use backwards breadth-first search on the restricted graph to find any state that can reach an SCC; see Figure 3.26.
The complexity of this algorithm is O(f · (V + E)), i.e., linear both in the size of the model and in the size of the formula. Example 3.20 We applied the basic algorithm to our second model of
mutual exclusion with the formula E[¬c2 U c1 ]; see Figure 3.27. The algorithm labels all states which satisfy c1 during phase 1 with E[¬c2 U c1 ]. This labels s2 and s4 . During phase 2, it labels
all states which do not satisfy c2 and have a successor state that is already labelled. This labels states s1 and s3 . During phase 3, we label s0 because it does not satisfy c2 and has a successor
state (s1 ) which is already labelled. Thereafter, the algorithm terminates because no additional states get labelled: all unlabelled states either satisfy c2 , or must pass through such a state to
reach a labelled state. The pseudo-code of the CTL model-checking algorithm We present the pseudo-code for the basic labelling algorithm. The main function SAT (for ‘satisfies’) takes as input a CTL
formula. The program SAT expects a parse tree of some CTL formula constructed by means of the grammar in Definition 3.12. This expectation reflects an important precondition on the correctness of the
algorithm SAT. For example, the program simply would not know what to do with an input of the form X ( ∧ EF p3 ), since this is not a CTL formula.
3 Verification by model checking
s0 0 : n1 n2 3 : E[¬c2 U c1 ]
0 : t1 n2 2 : E[¬c2 U c1 ]
0 : c1 n2 1 : E[¬c2 U c1 ]
0 : n1 t2
0 : t1 t2 2 : E[¬c2 U c1 ]
0 : t1 t2
0 : n1 c2
s4 0 : c1 t2 1 : E[¬c2 U c1 ]
s7 0 : t1 c2
Figure 3.27. An example run of the labelling algorithm in our second model of mutual exclusion applied to the formula E[¬c2 U c1 ].
The pseudo-code we write for SAT looks a bit like fragments of C or Java code; we use functions with a keyword return that indicates which result the function should return. We will also use natural
language to indicate the case analysis over the root node of the parse tree of φ. The declaration local var declares some fresh variables local to the current instance of the procedure in question,
whereas repeat until executes the command which follows it repeatedly, until the condition becomes true. Additionally, we employ suggestive notation for the operations on sets, like intersection, set
complement and so forth. In reality we would need an abstract data type, together with implementations of these operations, but for now we are interested only in the mechanism in principle of the
algorithm for SAT; any (correct and efficient) implementation of sets would do and we study such an implementation in Chapter 6. We assume that SAT has access to all the relevant parts of the model: S,
→ and L. In particular, we ignore the fact that SAT would require a description of M as input as well. We simply assume that SAT operates directly on any such given model. Note how SAT translates φ
into an equivalent formula of the adequate set chosen.
3.6 Model-checking algorithms
function SAT (φ) /* determines the set of states satisfying φ */ begin case φ is : return S φ is ⊥ : return ∅ φ is atomic: return {s ∈ S | φ ∈ L(s)} φ is ¬φ1 : return S − SAT (φ1 ) φ is φ1 ∧ φ2 :
return SAT (φ1 ) ∩ SAT (φ2 ) φ is φ1 ∨ φ2 : return SAT (φ1 ) ∪ SAT (φ2 ) φ is φ1 → φ2 : return SAT (¬φ1 ∨ φ2 ) φ is AX φ1 : return SAT (¬EX ¬φ1 ) φ is EX φ1 : return SATEX (φ1 ) φ is A[φ1 U φ2 ] :
return SAT(¬(E[¬φ2 U (¬φ1 ∧ ¬φ2 )] ∨ EG ¬φ2 )) φ is E[φ1 U φ2 ] : return SATEU (φ1 , φ2 ) φ is EF φ1 : return SAT (E( U φ1 )) φ is EG φ1 : return SAT(¬AF ¬φ1 ) φ is AF φ1 : return SATAF (φ1 ) φ is AG
φ1 : return SAT (¬EF ¬φ1 ) end case end function Figure 3.28. The function SAT. It takes a CTL formula as input and returns the set of states satisfying the formula. It calls the functions SATEX ,
SATEU and SATAF , respectively, if EX , EU or AF is the root of the input’s parse tree.
The algorithm is presented in Figure 3.28 and its subfunctions in Figures 3.29–3.31. They use program variables X, Y , V and W which are sets of states. The program for SAT handles the easy cases
directly and passes more complicated cases on to special procedures, which in turn might call SAT recursively on subexpressions. These special procedures rely on implementations of the functions pre∃
(Y ) = {s ∈ S | exists s , (s → s and s ∈ Y )} pre∀ (Y ) = {s ∈ S | for all s , (s → s implies s ∈ Y )}. ‘Pre’ denotes travelling backwards along the transition relation. Both functions compute a
pre-image of a set of states. The function pre∃ (instrumental in SATEX and SATEU ) takes a subset Y of states and returns the set of states which can make a transition into Y . The function pre∀ ,
used in SATAF , takes
3 Verification by model checking
function SATEX (φ) /* determines the set of states satisfying EX φ */ local var X, Y begin X := SAT (φ); Y := pre∃ (X); return Y end Figure 3.29. The function SATEX . It computes the states
satisfying φ by calling SAT. Then, it looks backwards along → to find the states satisfying EX φ.
function SATAF (φ) /* determines the set of states satisfying AF φ */ local var X, Y begin X := S; Y := SAT (φ); repeat until X = Y begin X := Y ; Y := Y ∪ pre∀ (Y ) end return Y end Figure 3.30. The
function SATAF . It computes the states satisfying φ by calling SAT. Then, it accumulates states satisfying AF φ in the manner described in the labelling algorithm.
a set Y and returns the set of states which make transitions only into Y . Observe that pre∀ can be expressed in terms of complementation and pre∃ , as follows: pre∀ (Y ) = S − pre∃ (S − Y )
where we write S − Y for the set of all s ∈ S which are not in Y . The correctness of this pseudocode and the model checking algorithm is discussed in Section 3.7.
3.6 Model-checking algorithms
function SATEU (φ, ψ) /* determines the set of states satisfying E[φ U ψ] */ local var W, X, Y begin W := SAT (φ); X := S; Y := SAT (ψ); repeat until X = Y begin X := Y ; Y := Y ∪ (W ∩ pre∃ (Y )) end
return Y end Figure 3.31. The function SATEU . It computes the states satisfying φ by calling SAT. Then, it accumulates states satisfying E[φ U ψ] in the manner described in the labelling algorithm.
The ‘state explosion’ problem Although the labelling algorithm (with the clever way of handling EG) is linear in the size of the model, unfortunately the size of the model is itself more often than
not exponential in the number of variables and the number of components of the system which execute in parallel. This means that, for example, adding a boolean variable to your program will double
the complexity of verifying a property of it. The tendency of state spaces to become very large is known as the state explosion problem. A lot of research has gone into finding ways of overcoming it,
including the use of: r Efficient data structures, called ordered binary decision diagrams (OBDDs), which represent sets of states instead of individual states. We study these in Chapter 6 in detail.
SMV is implemented using OBDDs. r Abstraction: one may interpret a model abstractly, uniformly or for a specific property. r Partial order reduction: for asynchronous systems, several interleavings of
component traces may be equivalent as far as satisfaction of the formula to be checked is concerned. This can often substantially reduce the size of the model-checking problem. r Induction:
model-checking systems with (e.g.) large numbers of identical, or similar, components can often be implemented by ‘induction’ on this number.
3 Verification by model checking
r Composition: break the verification problem down into several simpler verification problems.
The last four issues are beyond the scope of this book, but references may be found at the end of this chapter.
3.6.2 CTL model checking with fairness The verification of M, s0 φ might fail because the model M may contain behaviour which is unrealistic, or guaranteed not to occur in the actual system being
analysed. For example, in the mutual exclusion case, we expressed that the process prc can stay in its critical section (st=c) as long as it needs. We modelled this by the non-deterministic
assignment next(st) := case ... (st = c) ... esac;
: {c,n};
However, if we really allow process 2 to stay in its critical section as long as it likes, then we have a path which violates the liveness constraint AG (t1 → AF c1 ), since, if process 2 stays
forever in its critical section, t1 can be true without c1 ever becoming true. We would like to ignore this path, i.e., we would like to assume that the process can stay in its critical section as
long as it needs, but will eventually exit from its critical section after some finite time. In LTL, we could handle this by verifying a formula like FG¬c2 → φ, where φ is the formula we actually want
to verify. This whole formula asserts that all paths which satisfy infinitely often ¬c2 also satisfy φ. However, we cannot do this in CTL because we cannot write formulas of the form FG¬c2 → φ in CTL.
The logic CTL is not expressive enough to allow us to pick out the ‘fair’ paths, i.e., those in which process 2 always eventually leaves its critical section. It is for that reason that SMV allows us
to impose fairness constraints on top of the transition system it describes. These assumptions state that a given formula is true infinitely often along every computation path. We call such paths fair
computation paths. The presence of fairness constraints means that, when evaluating the truth of CTL formulas in specifications, the connectives A and E range only over fair paths.
3.6 Model-checking algorithms
We therefore impose the fairness constraint that !st=c be true infinitely often. This means that, whatever state the process is in, there will be a state in the future in which it is not in its
critical section. Similar fairness constraints were used for the Alternating Bit Protocol. Fairness constraints of the form (where φ is a state formula) Property φ is true infinitely often
are known as simple fairness constraints. Other types include those of the form If φ is true infinitely often, then ψ is also true infinitely often.
SMV can deal only with simple fairness constraints; but how does it do that? To answer that, we now explain how we may adapt our model-checking algorithm so that A and E are assumed to range only
over fair computation paths. def
Definition 3.21 Let C = {ψ1 , ψ2 , . . . , ψn } be a set of n fairness constraints. A computation path s0 → s1 → . . . is fair with respect to these fairness constraints iff for each i there are
infinitely many j such that sj ψi , that is, each ψi is true infinitely often along the path. Let us write AC and EC for the operators A and E restricted to fair paths. For example, M, s0 AC G φ iff φ
is true in every state along all fair paths; and similarly for AC F, AC U, etc. Notice that these operators explicitly depend on the chosen set C of fairness constraints. We already know that EC U,
EC G and EC X form an adequate set; this can be shown in the same manner as was done for the temporal connectives without fairness constraints (Section 3.4.4). We also have that EC [φ U ψ] ≡ E[φ U (ψ
∧ EC G )] EC X φ ≡ EX (φ ∧ EC G ). To see this, observe that a computation path is fair iff any suffix of it is fair. Therefore, we need only provide an algorithm for EC G φ. It is similar to Algorithm
2 for EG, given earlier in this chapter: r Restrict the graph to states satisfying φ; of the resulting graph, we want to know from which states there is a fair path. r Find the maximal strongly
connected components (SCCs) of the restricted graph; r Remove an SCC if, for some ψi , it does not contain a state satisfying ψi . The resulting SCCs are the fair SCCs. Any state of the restricted
graph that can reach one has a fair path from it.
3 Verification by model checking
states satisfying φ fair SCC
Ef Gφ ψ1 ψ2
fair SCC ψ3
Figure 3.32. Computing the states satisfying EC G φ. A state satisfies EC G φ iff, in the graph resulting from the restriction to states satisfying φ, the state has a fair path from it. A fair path
is one which leads to an SCC with a cycle passing through at least one state that satisfies each fairness constraint; in the example, C equals {ψ1 , ψ2 , ψ3 }. r Use backwards breadth-first search to
find the states on the restricted graph that can reach a fair SCC.
See Figure 3.32. The complexity of this algorithm is O(n · f · (V + E)), i.e., still linear in the size of the model and formula. It should be noted that writing fairness conditions using SMV’s
FAIRNESS keyword is necessary only for CTL model checking. In the case of LTL, we can assert the fairness condition as part of the formula to be checked. For example, if we wish to check the LTL
formula ψ under the assumption that φ is infinitely often true, we check G F φ → ψ. This means: all paths satisfying infinitely often φ also satisfy ψ. It is not possible to express this in CTL. In
particular, any way of adding As or Es to G F φ → ψ will result in a formula with a different meaning from the intended one. For example, AG AF φ → ψ means that if all paths are fair then ψ holds,
rather than what was intended: ψ holds along all paths which are fair.
3.6.3 The LTL model-checking algorithm The algorithm presented in the sections above for CTL model checking is quite intuitive: given a system and a CTL formula, it labels states of the system with
the subformulas of the formula which are satisfied there. The state-labelling approach is appropriate because subformulas of the formula may be evaluated in states of the system. This is not the case
for LTL: subformulas of the formula must be evaluated not in states but along paths of the system. Therefore, LTL model checking has to adopt a different strategy. There are several algorithms for LTL
model checking described in the literature. Although they differ in detail, nearly all of them adopt the same
3.6 Model-checking algorithms
basic strategy. We explain that strategy first; then, we describe some algorithms in more detail. The basic strategy Let M = (S, →, L) be a model, s ∈ S, and φ an LTL formula. We determine whether M,
s φ, i.e., whether φ is satisfied along all paths of M starting at s. Almost all LTL model checking algorithms proceed along the following three steps. 1. Construct an automaton, also known as a
tableau, for the formula ¬φ. The automaton for ψ is called Aψ . Thus, we construct A¬φ . The automaton has a notion of accepting a trace. A trace is a sequence of valuations of the propositional
atoms. From a path, we can abstract its trace. The construction has the property that for all paths π: π ψ iff the trace of π is accepted by Aψ . In other words, the automaton Aψ encodes precisely the
traces which satisfy ψ. Thus, the automaton A¬φ which we construct for ¬φ has the property that it encodes all the traces satisfying ¬φ; i.e., all the traces which do not satisfy φ. 2. Combine the
automaton A¬φ with the model M of the system. The combination operation results in a transition system whose paths are both paths of the automaton and paths of the system. 3. Discover whether there
is any path from a state derived from s in the combined transition system. Such a path, if there is one, can be interpreted as a path in M beginning at s which does not satisfy φ. If there was no
such path, then output: ‘Yes, M, s φ.’ Otherwise, if there is such a path, output ‘No, M, s φ.’ In the latter case, the counterexample can be extracted from the path found.
Let us consider an example. The system is described by the SMV program and its model M, shown in Figure 3.33. We consider the formula ¬(a U b). Since it is not the case that all paths of M satisfy
the formula (for example, the path q3 , q2 , q2 . . . does not satisfy it) we expect the model check to fail. In accordance with Step 1, we construct an automaton AaUb which characterises precisely
the traces which satisfy a U b. (We use the fact that ¬¬(a U b) is equivalent to a U b.) Such an automaton is shown in Figure 3.34. We will look at how to construct it later; for now, we just try to
understand how and why it works. A trace t is accepted by an automaton like the one of Figure 3.34 if there exists a path π through the automaton such that: r π starts in an initial state (i.e. one
containing φ); r it respects the transition relation of the automaton; r t is the trace of π; matches the corresponding state of π;
3 Verification by model checking
init(a) := 1; init(b) := 0; next(a) := case !a : 0; b : 1; 1 : {0,1}; esac; next(b) := case a & next(a) : !b; !a : 1; 1 : {0,1}; esac;
q1 ab
ab q4
Figure 3.33. An SMV program and its model M. q2
q1 abφ
q4 abφ q3
Figure 3.34. Automaton accepting precisely traces satisfying φ = a U b. The transitions with no arrows can be taken in either direction. The acceptance condition is that the path of the automaton
cannot loop indefinitely through q3 . def
r the path respects a certain ‘accepting condition.’ For the automaton of Figure 3.34, the accepting condition is that the path should not end q3 , q3 , q3 . . . , indefinitely.
For example, suppose t is a b, a b, a b, a b, a b, a b, a b, a b, . . . , eventually repeating forevermore the state a b. Then we choose the path q3 , q3 , q3 , q4 , q4 , q1 , q3 , q3 . . . . We
start in q3 because the first state is a b and it is an initial
3.6 Model-checking algorithms
state. The next states we choose just follow the valuation of the states of π. For example, at q1 the next valuation is a b and the transitions allow us to choose q3 or q3 . We choose q3 , and loop
there forevermore. This path meets the conditions, and therefore the trace t is accepted. Observe that the definition states ‘there exists a path.’ In the example above, there are also paths which
don’t meet the conditions: r Any path beginning q , q , . . . doesn’t meet the condition that we have to respect 3 3 the transition relation. r The path q , q , q , q , q , q , q , q . . . doesn’t
meet the condition that we must 3 3 3 4 4 1 3 3 not end on a loop of q3 .
These paths need not bother us, because it is sufficient to find one which does meet the conditions in order to declare that π is accepted. Why does the automaton of Figure 3.34 work as intended? To
understand it, observe that it has enough states to distinguish the values of the propositions – that is, a state for each of the valuations {a b, a b, a b, a b}, and in fact two states for the
valuation a b. One state for each of {a b, a b, a b} is intuitively enough, because those valuations determine whether a U b holds. But a U b could be false or true in a b, so we have to consider the
two cases. def The presence of φ = a U b in a state indicates that either we are still expecting φ to become true, or we have just obtained it. Whereas φ indicates we no longer expect φ, and have not
just obtained it. The transitions of the automaton are such that the only way out of q3 is to obtain b, i.e., to move to q2 or q4 . Apart from that, the transitions are liberal, allowing any path to
be followed; each of q1 , q2 , q3 can transition to any valuation, and so can q3 , q3 taken together, provided we are careful to choose the right one to enter. The acceptance condition, which allows
any path except one looping indefinitely on q3 , guarantees that the promise of a U b to deliver b is eventually fulfilled. Using this automaton AaUb , we proceed to Step 2. To combine the automaton
AaUb with the model of the system M shown in Figure 3.33, it is convenient first to redraw M with two versions of q3 ; see Figure 3.35(left). It is an equivalent system; all ways into q3 now
non-deterministically choose q3 or q3 , and which ever one we choose leads to the same successors. But it allows us to superimpose it on AaUb and select the transitions common to both, obtaining the
combined system of Figure 3.35(right). Step 3 now asks whether there is a path from q of the combined automaton. As can be seen, there are two kinds of path in the combined system: q3 , (q4 , q3 , )∗
q2 , q2 . . . , and q3 , q4 , (q3 , q4 , )∗ q3 , q1 , q2 , q2 , . . . where (q3 , q4 )∗ denotes either the empty string or q3 , q4 or q3 , q4 , q3 , q4 etc. Thus, according
3 Verification by model checking
236 q1
q2 ab
q3 ab q3
q2 abφ
q3 abφ
Figure 3.35. Left: the system M of Figure 3.33, redrawn with an expanded state space; right: the expanded M and AaUb combined.
to Step 3, and as we expected, ¬(a U b) is not satisfied in all paths of the original system M. Constructing the automaton Let us look in more detail at how the automaton is constructed. Given an LTL
formula φ, we wish to construct an automaton Aφ such that Aφ accepts precisely those runs on which φ holds. We assume that φ contains only the temporal connectives U and X; recall that the other
temporal connectives can be written in terms of these two. Define the closure C(φ) of formula φ as the set of subformulas of φ and their complements, identifying ¬¬ψ and ψ. For example, C(a U b) = {a,
b, ¬a, ¬b, a U b, ¬(a U b)}. The states of Aφ , denoted by q, q etc., are the maximal subsets of C(φ) which satisfy the following conditions: r r r r r
For all (non-negated) ψ ∈ C(φ), either ψ ∈ q or ¬ψ ∈ q, but not both. ψ1 ∨ ψ2 ∈ q holds iff ψ1 ∈ q or ψ2 ∈ q, whenever ψ1 ∨ ψ2 ∈ C(φ). Conditions for other boolean combinations are similar. If ψ1 U ψ2
∈ q, then ψ2 ∈ q or ψ1 ∈ q. If ¬(ψ1 U ψ2 ) ∈ q, then ¬ψ2 ∈ q.
Intuitively, these conditions imply that the states of Aφ are capable of saying which subformulas of φ are true.
3.6 Model-checking algorithms
The initial states of Aφ are those states containing φ. For transition relation δ of Aφ we have (q, q ) ∈ δ iff all of the following conditions hold: r r r r
if X ψ ∈ q then ψ ∈ q ; if ¬X ψ ∈ q then ¬ψ ∈ q ; If ψ1 U ψ2 ∈ q and ψ2 ∈ / q then ψ1 U ψ2 ∈ q ; If ¬(ψ1 U ψ2 ) ∈ q and ψ1 ∈ q then ¬(ψ1 U ψ2 ) ∈ q .
These last two conditions are justified by the recursion laws ψ1 U ψ2 = ψ2 ∨ (ψ1 ∧ X (ψ1 U ψ2 )) ¬(ψ1 U ψ2 ) = ¬ψ2 ∧ (¬ψ1 ∨ X ¬(ψ1 U ψ2 )) . In particular, they ensure that whenever some state
contains ψ1 U ψ2 , subsequent states contain ψ1 for as long as they do not contain ψ2 . As we have defined Aφ so far, not all paths through Aφ satisfy φ. We use additional acceptance conditions to
guarantee the ‘eventualities’ ψ promised by the formula ψ1 U ψ2 , namely that Aφ cannot stay for ever in states satisfying ψ1 without ever obtaining ψ2 . Recall that, for the automaton of Figure 3.34
for a U b, we stipulated the acceptance condition that the path through the automaton should not end q3 , q3 , . . . . The acceptance conditions of Aφ are defined so that they ensure that every state
containing some formula χ U ψ will eventually be followed by some state containing ψ. Let χ1 U ψ1 , . . . , χk U ψk be all subformulas of this form in C(φ). We stipulate the following acceptance
condition: a run is accepted if, for every i such that 1 ≤ i ≤ k, the run has infinitely many states satisfying ¬(χi U ψi ) ∨ ψi . To understand why this condition has the desired effect, imagine the
circumstances in which it is false. Suppose we have a run having only finitely many states satisfying ¬(χi U ψi ) ∨ ψi . Let us advance through all those finitely many states, taking the suffix of the
run none of whose states satisfies ¬(χi U ψi ) ∨ ψi , i.e., all of whose states satisfy (χi U ψi ) ∧ ¬ψi . That is precisely the sort of run we want to eliminate. If we carry out this construction on
a U b, we obtain the automaton shown in Figure 3.34. Another example is shown in Figure 3.36, for the formula (p U q) ∨ (¬p U q). Since that formula has two U subformulas, there are two sets specified
in the acceptance condition, namely, the states satisfying p U q and the states satisfying ¬p U q. How LTL model checking is implemented in NuSMV In the sections above, we described an algorithm for
LTL model checking. Given an LTL formula φ and a system M and a state s of M, we may check whether M, s φ holds by constructing the automaton A¬φ , combining it with M,
3 Verification by model checking
p U q, ¬(¬p U q), p, ¬q, φ
¬(p U q), ¬(¬p U q), ¬p, ¬q, ¬φ
p U q, ¬p U q, ¬p, q, φ
p U q, ¬p U q, p, q, φ
¬(p U q), ¬(¬p U q), p, ¬q, ¬φ
¬(p U q), ¬p U q, ¬p, ¬q, φ
Figure 3.36. Automaton accepting precisely traces satisfying φ = (p U q) ∨ (¬p U q). The transitions with no arrows can be taken in either direction. The acceptance condition asserts that every run
must pass infinitely often through the set {q1 , q3 , q4 , q5 , q6 }, and also the set {q1 , q2 , q3 , q5 , q6 }. def
and checking whether there is a path of the resulting system which satisfies the acceptance condition of A¬φ . It is possible to implement the check for such a path in terms of CTL model checking, and
this is in fact what NuSMV does. The combined system M × A¬φ is represented as the system to be model checked in NuSMV, and the formula to be checked is simply EG . Thus, we ask the question: does
the combined system have a path. The acceptance conditions of A¬φ are represented as implicit fairness conditions for the CTL model-checking procedure. Explicitly, this amounts to asserting ‘FAIRNESS
¬(χ U ψ) ∨ ψ’ for each formula χ U ψ occurring in C(φ).
3.7 The fixed-point characterisation of CTL On page 227, we presented an algorithm which, given a CTL formula φ and a model M = (S, →, L), computes the set of states s ∈ S satisfying φ. We write this
set as [[φ]]. The algorithm works recursively on the structure of φ. For formulas φ of height 1 (⊥, or p), [[φ]] is computed directly. Other
3.7 The fixed-point characterisation of CTL
formulas are composed of smaller subformulas combined by a connective of CTL. For example, if φ is ψ1 ∨ ψ2 , then the algorithm computes the sets [[ψ1 ]] and [[ψ2 ]] and combines them in a certain
way (in this case, by taking the union) in order to obtain [[ψ1 ∨ ψ2 ]]. The more interesting cases arise when we deal with a formula such as EX ψ, involving a temporal operator. The algorithm
computes the set [[ψ]] and then computes the set of all states which have a transition to a state in [[ψ]]. This is in accord with the semantics of EX ψ: M, s EX ψ iff there is a state s with s → s
and M, s ψ. For most of these logical operators, we may easily continue this discussion to see that the algorithms work just as expected. However, the cases EU, AF and EG (where we needed to iterate
a certain labelling policy until it stabilised) are not so obvious to reason about. The topic of this section is to develop the semantic insights into these operators that allow us to provide a
complete proof for their termination and correctness. Inspecting the pseudocode in Figure 3.28, we see that most of these clauses just do the obvious and correct thing according to the semantics of
CTL. For example, try out what SAT does when you call it with φ1 → φ2 . Our aim in this section is to prove the termination and correctness of SATAF and SATEU . In fact, we will also write a
procedure SATEG and prove its termination and correctness1 . The procedure SATEG is given in Figure 3.37 and is based on the intuitions given in Section 3.6.1: note how deleting the label if none of
the successor states is labelled is coded as intersecting the labelled set with the set of states which have a labelled successor. The semantics of EG φ says that s0 EG φ holds iff there exists a
computation path s0 → s1 → s2 → . . . such that si φ holds for all i ≥ 0. We could instead express it as follows: EG φ holds if φ holds and EG φ holds in one of the successor states to the current
state. This suggests the equivalence EG φ ≡ φ ∧ EX EG φ which can easily be proved from the semantic definitions of the connectives. Observing that [[EX ψ]] = pre∃ ([[ψ]]) we see that the equivalence
above can be written as [[EG φ]] = [[φ]] ∩ pre∃ ([[EG φ]]). This does not look like a very promising way of calculating EG φ, because we need to know EG φ in order to work out the right-hand side.
Fortunately, there is a way around this apparent circularity, known as computing fixed points, and that is the subject of this section. 1
Section 3.6.1 handles EG φ by translating it into ¬AF ¬φ, but we already noted in Section 3.6.1 that EG could be handled directly.
3 Verification by model checking
function SATEG (φ) /* determines the set of states satisfying EG φ */ local var X, Y begin Y := SAT (φ); X := ∅; repeat until X = Y begin X := Y ; Y := Y ∩ pre∃ (Y ) end return Y end Figure 3.37. The
pseudo-code for SATEG .
3.7.1 Monotone functions Definition 3.22 Let S be a set of states and F : P(S) → P(S) a function on the power set of S. 1. We say that F is monotone iff X ⊆ Y implies F (X) ⊆ F (Y ) for all subsets X
and Y of S. 2. A subset X of S is called a fixed point of F iff F (X) = X. def
For an example, let S = {s0 , s1 } and F (Y ) = Y ∪ {s0 } for all subsets Y of S. Since Y ⊆ Y implies Y ∪ {s0 } ⊆ Y ∪ {s0 }, we see that F is monotone. The fixed points of F are all subsets of S
containing s0 . Thus, F has two fixed points, the sets {s0 } and {s0 , s1 }. Notice that F has a least (= {s0 }) and a greatest (= {s0 , s1 }) fixed point. An example of a function G : P(S) → P(S),
which is not monotone, is given by def
G(Y ) = if Y = {s0 } then {s1 } else {s0 }. So G maps {s0 } to {s1 } and all other sets to {s0 }. The function G is not monotone since {s0 } ⊆ {s0 , s1 } but G({s0 }) = {s1 } is not a subset of G({s0
, s1 }) = {s0 }. Note that G has no fixed points whatsoever. The reasons for exploring monotone functions on P(S) in the context of proving the correctness of SAT are: 1. that monotone functions
always have a least and a greatest fixed point; 2. that the meanings of EG, AF and EU can be expressed via greatest, respectively least, fixed points of monotone functions on P(S);
3.7 The fixed-point characterisation of CTL
3. that these fixed-points can be easily computed, and; 4. that the procedures SATEU and SATAF code up such fixed-point computations, and are correct by item 2.
Notation 3.23 F i (X) means F (F (. . . F (X) . . . )) i times
Thus, the function F i is just ‘F applied i many times.’ def
For example, for the function F (Y ) = Y ∪ {s0 }, we obtain F 2 (Y ) = F (F (Y )) = (Y ∪ {s0 }) ∪ {s0 } = Y ∪ {s0 } = F (Y ). In this case, F 2 = F and therefore F i = F for all i ≥ 1. It is not
always the case that the sequence of functions (F 1 , F 2 , F 3 , . . . ) stabilises in such a way. For example, this won’t happen for the function G defined above (see Exercise 1(d) on page 253). The
following fact is a special case of a fundamental insight, often referred to as the Knaster–Tarski Theorem. Theorem 3.24 Let S be a set {s0 , s1 , . . . , sn } with n + 1 elements. If F : P(S) → P(S)
is a monotone function, then F n+1 (∅) is the least fixed point of F and F n+1 (S) is the greatest fixed point of F . PROOF: Since ∅ ⊆ F (∅), we get F (∅) ⊆ F (F (∅)), i.e., F 1 (∅) ⊆ F 2 (∅), for F is
monotone. We can now use mathematical induction to show that F 1 (∅) ⊆ F 2 (∅) ⊆ F 3 (∅) ⊆ . . . ⊆ F i (∅) def
for all i ≥ 1. In particular, taking i = n + 1, we claim that one of the expressions F k (∅) above is already a fixed point of F . Otherwise, F 1 (∅) needs to contain at least one element (for then ∅
= F (∅)). By the same token, F 2 (∅) needs to have at least two elements since it must be bigger than F 1 (∅). Continuing this argument, we see that F n+2 (∅) would have to contain at least n + 2
many elements. The latter is impossible since S has only n + 1 elements. Therefore, F (F k (∅)) = F k (∅) for some 0 ≤ k ≤ n + 1, which readily implies that F n+1 (∅) is a fixed point of F as well.
Now suppose that X is another fixed point of F . We need to show that n+1 (∅) is a subset of X; but, since ∅ ⊆ X, we conclude F (∅) ⊆ F (X) = F X, for F is monotone and X a fixed point of F . By
induction, we obtain def F i (∅) ⊆ X for all i ≥ 0. So, for i = n + 1, we get F n+1 (∅) ⊆ X. The proof of the statements about the greatest fixed point is dual to the one above. Simply replace ⊆ by ⊇,
∅ by S and ‘bigger’ by ‘smaller.’ 2
3 Verification by model checking
This theorem about the existence of least and greatest fixed points of monotone functions F : P(S) → P(S) not only asserted the existence of such fixed points; it also provided a recipe for computing
them, and correctly so. For example, in computing the least fixed point of F , all we have to do is apply F to the empty set ∅ and keep applying F to the result until the latter becomes invariant
under the application of F . The theorem above further ensures that this process is guaranteed to terminate. Moreover, we can specify an upper bound n + 1 to the worst-case number of iterations
necessary for reaching this fixed point, assuming that S has n + 1 elements.
3.7.2 The correctness of SATEG We saw at the end of the last section that [[EG φ]] = [[φ]] ∩ pre∃ ([[EG φ]]). This implies that EG φ is a fixed point of the function F (X) = [[φ]] ∩ pre∃ (X). In fact,
F is monotone, EG φ is its greatest fixed point and therefore EG φ can be computed using Theorem 3.24. Theorem 3.25 Let F be as defined above and let S have n + 1 elements. Then F is monotone, [[EG φ]]
is the greatest fixed point of F , and [[EG φ]] = F n+1 (S). PROOF: 1. In order to show that F is monotone, we take any two subsets X and Y of S such that X ⊆ Y and we need to show that F (X) is a
subset of F (Y ). Given s0 such that there is some s1 ∈ X with s0 → s1 , we certainly have s0 → s1 , where s1 ∈ Y , for X is a subset of Y . Thus, we showed pre∃ (X) ⊆ pre∃ (Y ) from which we readily
conclude that F (X) = [[φ]] ∩ pre∃ (X) ⊆ [[φ]] ∩ pre∃ (Y ) = F (Y ). 2. We have already seen that [[EG φ]] is a fixed point of F . To show that it is the greatest fixed point, it suffices to show here
that any set X with F (X) = X has to be contained in [[EG φ]]. So let s0 be an element of such a fixed point X. We need to show that s0 is in [[EG φ]] as well. For that we use the fact that s0 ∈ X = F
(X) = [[φ]] ∩ pre∃ (X) to infer that s0 ∈ [[φ]] and s0 → s1 for some s1 ∈ X; but, since s1 is in X, we may apply that same argument to s1 ∈ X = F (X) = [[φ]] ∩ pre∃ (X) and we get s1 ∈ [[φ]] and s1 →
s2 for some s2 ∈ X. By mathematical induction, we can therefore construct an infinite path s0 → s1 → · · · → sn → sn+1 → . . . such that si ∈ [[φ]] for all i ≥ 0. By the definition of [[EG φ]], this
entails s0 ∈ [[EG φ]]. 3. The last item is now immediately accessible from the previous one and Theorem 3.24. 2
3.7 The fixed-point characterisation of CTL
Now we can see that the procedure SATEG is correctly coded and terminates. First, note that the line Y := Y ∩ pre∃ (Y ) in the procedure SATEG (Figure 3.37) could be changed to Y := SAT(φ) ∩ pre∃ (Y
) without changing the effect of the procedure. To see this, note that the first time round the loop, Y is SAT(φ); and in subsequent loops, Y ⊆ SAT(φ), so it doesn’t matter whether we intersect with Y
or SAT(φ)2 . With the change, it is clear that SATEG is calculating the greatest fixed point of F ; therefore its correctness follows from Theorem 3.25.
3.7.3 The correctness of SATEU Proving the correctness of SATEU is similar. We start by noting the equivalence E[φ U ψ] ≡ ψ ∨ (φ ∧ EX E[φ U ψ]) and we write it as [[E[φ U ψ]]] = [[ψ]] ∪ ([[φ]] ∩ pre∃
[[E[φ U ψ]]]). That tells us that [[E[φ U ψ]]] is a fixed point of the function G(X) = [[ψ]] ∪ ([[φ]] ∩ pre∃ (X)). As before, we can prove that this function is monotone. It turns out that [[E[φ U
ψ]]] is its least fixed point and that the function SATEU is actually computing it in the manner of Theorem 3.24. Theorem 3.26 Let G be defined as above and let S have n + 1 elements. Then G is
monotone, [[E(φ U ψ)]] is the least fixed point of G, and we have [[E(φ U ψ)]] = Gn+1 (∅). 2
If you are sceptical, try computing the values Y0 , Y1 , Y2 , . . . , where Yi represents the value of Y after i iterations round the loop. The program before the change computes as follows: Y0 = SAT
(φ) Y1 = Y0 ∩ pre∃ (Y0 ) Y2 = Y1 ∩ pre∃ (Y1 ) = Y0 ∩ pre∃ (Y0 ) ∩ pre∃ (Y0 ∩ pre∃ (Y0 )) = Y0 ∩ pre∃ (Y0 ∩ pre∃ (Y0 )).
The last of these equalities follows from the monotonicity of pre∃ . Y3 = Y2 ∩ pre∃ (Y2 ) = Y0 ∩ pre∃ (Y0 ∩ pre∃ (Y0 )) ∩ pre∃ (Y0 ∩ pre∃ (Y0 ∩ pre∃ (Y0 ))) = Y0 ∩ pre∃ (Y0 ∩ pre∃ (Y0 ∩ pre∃ (Y0 ))).
Again the last one follows by monotonicity. Now look at what the program does after the change: Y0 = SAT(φ) Y1 = SAT(φ) ∩ pre∃ (Y0 ) = Y0 ∩ pre∃ (Y0 ) Y2 = Y0 ∩ pre∃ (Y1 ) Y3 = Y0 ∩ pre∃ (Y1 ) = Y0 ∩
pre∃ (Y0 ∩ pre∃ (Y0 )). A formal proof would follow by induction on i.
3 Verification by model checking
PROOF: 1. Again, we need to show that X ⊆ Y implies G(X) ⊆ G(Y ); but that is essentially the same argument as for F , since the function which sends X to pre∃ (X) is monotone and all that G now does
is to perform the intersection and union of that set with constant sets [[φ]] and [[ψ]]. 2. If S has n + 1 elements, then the least fixed point of G equals Gn+1 (∅) by Theorem 3.24. Therefore it
suffices to show that this set equals [[E(φ U ψ)]]. Simply observe what kind of states we obtain by iterating G on the empty set ∅: G1 (∅) = [[ψ]] ∪ ([[φ]] ∩ pre∃ ([[∅]])) = [[ψ]] ∪ ([[φ]] ∩ ∅) = [[ψ]]
∪ ∅ = [[ψ]], which are all states s0 ∈ [[E(φ U ψ)]], where we chose i = 0 according to the definition of Until. Now, G2 (∅) = [[ψ]] ∪ ([[φ]] ∩ pre∃ (G1 (∅))) tells us that the elements of G2 (∅) are
all those s0 ∈ [[E(φ U ψ)]] where we chose i ≤ 1. By mathematical induction, we see that Gk+1 (∅) is the set of all states s0 for which we chose i ≤ k to secure s0 ∈ [[E(φ U ψ)]]. Since this holds
for all k, we see that [[E(φ U ψ)]] is nothing but the union of all sets Gk+1 (∅) with k ≥ 0; but, since Gn+1 (∅) is a fixed point of G, we see that this union is just Gn+1 (∅). 2
The correctness of the coding of SATEU follows similarly to that of SATEG . We change the line Y := Y ∪ (W ∩ pre∃ (Y )) into Y := SAT(ψ) ∪ (W ∩ pre∃ (Y )) and observe that this does not change the
result of the procedure, because the first time round the loop, Y is SAT(ψ); and, since Y is always increasing, it makes no difference whether we perform a union with Y or with SAT(ψ). Having made that
change, it is then clear that SATEU is just computing the least fixed point of G using Theorem 3.24. We illustrate these results about the functions F and G above through an example. Consider the
system in Figure 3.38. We begin by computing the set [[EF p]]. By the definition of EF this is just def def [[E( U p)]]. So we have φ1 = and φ2 = p. From Figure 3.38, we obtain [[p]] = {s3 } and of
course [[]] = S. Thus, the function G above equals G(X) = {s3 } ∪ pre∃ (X). Since [[E( U p)]] equals the least fixed point of G, we need to iterate G on ∅ until this process stabilises. First, G1 (∅)
= {s3 } ∪ pre∃ (∅) = {s3 }. Second, G2 (∅) = G(G1 (∅)) = {s3 } ∪ pre∃ ({s3 }) = {s1 , s3 }. Third, G3 (∅) = G(G2 (∅)) = {s3 } ∪ pre∃ ({s1 , s3 }) = {s0 , s1 , s2 , s3 }. Fourth, G4 (∅) = G(G3 (∅)) =
{s3 } ∪ pre∃ ({s0 , s1 , s2 , s3 }) = {s0 , s1 , s2 , s3 }. Therefore, {s0 , s1 , s2 , s3 } is the least fixed point of G, which equals [[E( U p)]] by Theorem 3.20. But then [[E( U p)]] = [[EF p]].
3.8 Exercises s0
s3 s4
Figure 3.38. A system for which we compute invariants.
The other example we study is the computation of the set [[EG q]]. By Theorem 3.25, that set is the greatest fixed point of the function F above, def where φ = q. From Figure 3.38 we see that [[q]] =
{s0 , s4 } and so F (X) = [[q]] ∩ pre∃ (X) = {s0 , s4 } ∩ pre∃ (X). Since [[EG q]] equals the greatest fixed point of F , we need to iterate F on S until this process stabilises. First, F 1 (S) = {s0
, s4 } ∩ pre∃ (S) = {s0 , s4 } ∩ S since every s has some s with s → s . Thus, F 1 (S) = {s0 , s4 }. Second, F 2 (S) = F (F 1 (S)) = {s0 , s4 } ∩ pre∃ ({s0 , s4 }) = {s0 , s4 }. Therefore, {s0 , s4 }
is the greatest fixed point of F , which equals [[EG q]] by Theorem 3.25.
3.8 Exercises Exercises 3.1 1. Read Section 2.7 in case you have not yet done so and classify Alloy and its constraint analyser according to the classification criteria for formal methods proposed on
page 172. 2. Visit and browse the websites3 and4 to find formal methods that interest you for whatever reason. Then classify them according to the criteria from page 172.
Exercises 3.2 1. Draw parse trees for the LTL formulas: (a) F p ∧ G q → p W r (b) F (p → G r) ∨ ¬q U p (c) p W (q W r) (d) G F p → F (q ∨ s) 3 4
www.afm.sbu.ac.uk www.cs.indiana.edu/formal-methods-education/
3 Verification by model checking q2
q1 ab
q4 Figure 3.39. A model M.
2. Consider the system of Figure 3.39. For each of the formulas φ: (a) G a (b) a U b (c) a U X (a ∧ ¬b) (d) X ¬b ∧ G (¬a ∨ ¬b) (e) X (a ∧ b) ∧ F (¬a ∧ ¬b) (i) Find a path from the initial state q3
which satisfies φ. (ii) Determine whether M, q3 φ. 3. Working from the clauses of Definition 3.1 (page 175), prove the equivalences: φ U ψ ≡ φ W ψ ∧ Fψ φ W ψ ≡ φ U ψ ∨ Gφ φ W ψ ≡ ψ R (φ ∨ ψ) φ R ψ ≡ ψ
W (φ ∧ ψ) . 4. Prove that φ U ψ ≡ ψ R (φ ∨ ψ) ∧ F ψ. 5. List all subformulas of the LTL formula ¬p U (F r ∨ G ¬q → q W ¬r). 6. ‘Morally’ there ought to be a dual for W. Work out what it might mean,
and then pick a symbol based on the first letter of the meaning. 7. Prove that for all paths π of all models, π φ W ψ ∧ F ψ implies π φ U ψ. That is, prove the remaining half of equivalence (3.2) on
page 185. 8. Recall the algorithm NNF on page 62 which computes the negation normal form of propositional logic formulas. Extend this algorithm to LTL: you need to add program clauses for the
additional connectives X, F, G and U, R and W; these clauses have to animate the semantic equivalences that we presented in this section.
3.8 Exercises
Exercises 3.3 1. Consider the model in Figure 3.9 (page 193). * (a) Verify that G(req -> F busy) holds in all initial states. (b) Does ¬(req U ¬busy) hold in all initial states of that model? (c)
NuSMV has the capability of referring to the next value of a declared variable v by writing next(v). Consider the model obtained from Figure 3.9 by removing the self-loop on state !req & busy. Use
the NuSMV feature next(...) to code that modified model as an NuSMV program with the specification G(req -> F busy). Then run it. 2. Verify Remark 3.11 from page 190. * 3. Draw the transition system
described by the ABP program. Remarks: There are 28 reachable states of the ABP program. (Looking at the program, you can see that the state is described by nine boolean variables, namely S.st,
S.message1, S.message2, R.st, R.ack, R.expected, msg chan.output1, msg chan.output2 and finally ack chan.output. Therefore, there are 29 = 512 states in total. However, only 28 of them can be reached
from the initial state by following a finite path.) If you abstract away from the contents of the message (e.g., by setting S.message1 and msg chan.output1 to be constant 0), then there are only 12
reachable states. This is what you are asked to draw.
Exercises 3.4 1. Write the parse trees for the following CTL formulas: * (a) EG r * (b) AG (q → EG r) * (c) A[p U EF r] * (d) EF EG p → AF r, recall Convention 3.13 (e) A[p U A[q U r]] (f) E[A[p U q]
U r] (g) AG (p → A[p U (¬p ∧ A[¬p U q])]). 2. Explain why the following are not well-formed CTL formulas: * (a) F G r (b) X X r (c) A¬G ¬p (d) F [r U q] (e) EX X r * (f) AEF r * (g) AF [(r U q) ∧ (p
U r)]. 3. State which of the strings below are well-formed CTL formulas. For those which are well-formed, draw the parse tree. For those which are not well-formed, explain why not.
3 Verification by model checking
p, q
p, t, r
q, r
Figure 3.40. A model with four states. (a) ¬(¬p) ∨ (r ∧ s) (b) X q * (c) ¬AX q (d) p U (AX ⊥) * (e) E[(AX q) U (¬(¬p) ∨ ( ∧ s))] * (f) (F r) ∧ (AG q) (g) ¬(AG q) ∨ (EG q). * 4. List all subformulas
of the formula AG (p → A[p U (¬p ∧ A[¬p U q])]). 5. Does E[req U ¬busy] hold in all initial states of the model in Figure 3.9 on page 193? 6. Consider the system M in Figure 3.40. (a) Beginning from
state s0 , unwind this system into an infinite tree, and draw all computation paths up to length 4 (= the first four layers of that tree). (b) Determine whether M, s0 φ and M, s2 φ hold and justify
your answer, where φ is the LTL or CTL formula: * (i) ¬p → r (ii) F t *(iii) ¬EG r (iv) E (t U q) (v) F q (vi) EF q (vii) EG r (viii) G (r ∨ q). 7. Let M = (S, →, L) be any model for CTL and let
[[φ]] denote the set of all s ∈ S such that M, s φ. Prove the following set identities by inspecting the clauses of Definition 3.15 from page 211. * (a) [[]] = S, (b) [[⊥]] = ∅
3.8 Exercises
p, q
q, r
p, t
Figure 3.41. Another model with four states. (c) [[¬φ]] = S − [[φ]], (d) [[φ1 ∧ φ2 ]] = [[φ1 ]] ∩ [[φ2 ]] (e) [[φ1 ∨ φ2 ]] = [[φ1 ]] ∪ [[φ2 ]] * (f) [[φ1 → φ2 ]] = (S − [[φ1 ]]) ∪ [[φ2 ]] * (g) [[AX
φ]] = S − [[EX ¬φ]] (h) [[A(φ2 U φ2 )]] = [[¬(E(¬φ1 U (¬φ1 ∧ ¬φ2 )) ∨ EG ¬φ2 )]]. 8. Consider the model M in Figure 3.41. Check whether M, s0 φ and M, s2 φ hold for the CTL formulas φ: (a) AF q (b)
AG (EF (p ∨ r)) (c) EX (EX r) (d) AG (AF q). * 9. The meaning of the temporal operators F, G and U in LTL and AU, EU, AG, EG, AF and EF in CTL was defined to be such that ‘the present includes the
future.’ For example, EF p is true for a state if p is true for that state already. Often one would like corresponding operators such that the future excludes the present. Use suitable connectives of
the grammar on page 208 to define such (six) modified connectives as derived operators in CTL. 10. Which of the following pairs of CTL formulas are equivalent? For those which are not, exhibit a model
of one of the pair which is not a model of the other: (a) EF φ and EG φ * (b) EF φ ∨ EF ψ and EF (φ ∨ ψ) * (c) AF φ ∨ AF ψ and AF (φ ∨ ψ) (d) AF ¬φ and ¬EG φ * (e) EF ¬φ and ¬AF φ (f) A[φ1 U A[φ2 U
φ3 ]] and A[A[φ1 U φ2 ] U φ3 ], hint: it might make it simpler if you think first about models that have just one path (g) and AG φ → EG φ * (h) and EG φ → AG φ. 11. Find operators to replace the ?,
to make the following equivalences:
3 Verification by model checking
* (a) AG (φ ∧ ψ) ≡ AG φ ? AG ψ (b) EF ¬φ ≡ ¬??φ 12. State explicitly the meaning of the temporal connectives AR etc., as defined on page 217. 13. Prove the equivalences (3.6) on page 216. * 14. Write
pseudo-code for a recursive function TRANSLATE which takes as input an arbitrary CTL formula φ and returns as output an equivalent CTL formula ψ whose only operators are among the set {⊥, ¬, ∧, AF ,
EU , EX }.
Exercises 3.5 1. Express the following properties in CTL and LTL whenever possible. If neither is possible, try to express the property in CTL*: * (a) Whenever p is followed by q (after finitely many
steps), then the system enters an ‘interval’ in which no r occurs until t. (b) Event p precedes s and t on all computation paths. (You may find it easier to code the negation of that specification
first.) (c) After p, q is never true. (Where this constraint is meant to apply on all computation paths.) (d) Between the events q and r, event p is never true. (e) Transitions to states satisfying p
occur at most twice. * (f) Property p is true for every second state along a path. 2. Explain in detail why the LTL and CTL formulas for the practical specification patterns of pages 183 and 215
capture the stated ‘informal’ properties expressed in plain English. 3. Consider the set of LTL/CTL formulas F = {F p → F q, AF p → AF q, AG (p → AF q)}. (a) Is there a model such that all formulas
hold in it? (b) For each φ ∈ F, is there a model such that φ is the only formula in F satisfied in that model? (c) Find a model in which no formula of F holds. 4. Consider the CTL formula AG (p → AF
(s ∧ AX (AF t))). Explain what exactly it expresses in terms of the order of occurrence of events p, s and t. 5. Extend the algorithm NNF from page 62 which computes the negation normal form of
propositional logic formulas to CTL*. Since CTL* is defined in terms of two syntactic categories (state formulas and path formulas), this requires two separate versions of NNF which call each other in
a way that is reflected by the syntax of CTL* given on page 218. 6. Find a transition system which distinguishes the following pairs of CTL* formulas, i.e., show that they are not equivalent: (a) AF G
p and AF AG p * (b) AG F p and AG EF p (c) A[(p U r) ∨ (q U r)] and A[(p ∨ q) U r)]
3.8 Exercises
* (d) A[X p ∨ X X p] and AX p ∨ AX AX p (e) E[G F p] and EG EF p. 7. The translation from CTL with boolean combinations of path formulas to plain CTL introduced in Section 3.5.1 is not complete.
Invent CTL equivalents for: * (a) E[F p ∧ (q U r)] * (b) E[F p ∧ G q]. In this way, we have dealt with all formulas of the form E[φ ∧ ψ]. Formulas of the form E[φ ∨ ψ] can be rewritten as E[φ] ∨ E[ψ]
and A[φ] can be written ¬E[¬φ]. Use this translation to write the following in CTL: (c) E[(p U q) ∧ F p] * (d) A[(p U q) ∧ G p] * (e) A[F p → F q]. 8. The aim of this exercise is to demonstrate the
expansion given for AW at the end of the last section, i.e., A[p W q] ≡ ¬E[¬q U ¬(p ∨ q)]. (a) Show that the following LTL formulas are valid (i.e., true in any state of any model): (i) ¬q U (¬p ∧
¬q) → ¬G p (ii) G ¬q ∧ F ¬p → ¬q U (¬p ∧ ¬q). (b) Expand ¬((p U q) ∨ G p) using de Morgan rules and the LTL equivalence ¬(φ U ψ) ≡ (¬ψ U (¬φ ∧ ¬ψ)) ∨ ¬F ψ. (c) Using your expansion and the facts (i)
and (ii) above, show ¬((p U q) ∨ G p) ≡ ¬q U ¬(p ∧ q) and hence show that the desired expansion of AW above is correct.
Exercises 3.6 * 1. Verify φ1 to φ4 for the transition system given in Figure 3.11 on page 198. Which of them require the fairness constraints of the SMV program in Figure 3.10? 2. Try to write a CTL
formula that enforces non-blocking and no-strict-sequencing at the same time, for the SMV program in Figure 3.10 (page 196). * 3. Apply the labelling algorithm to check the formulas φ1 , φ2 , φ3 and
φ4 of the mutual exclusion model in Figure 3.7 (page 188). 4. Apply the labelling algorithm to check the formulas φ1 , φ2 , φ3 and φ4 of the mutual exclusion model in Figure 3.8 (page 191). 5. Prove
that (3.8) on page 228 holds in all models. Does your proof require that for every state s there is some state s with s → s ? 6. Inspecting the definition of the labelling algorithm, explain what
happens if you perform it on the formula p ∧ ¬p (in any state, in any model). 7. Modify the pseudo-code for SAT on page 227 by writing a special procedure for AG ψ1 , without rewriting it in terms of
other formulas5 . 5
Question: will your routine be more like the routine for AF, or more like that for EG on page 224? Why?
3 Verification by model checking
* 8. Write the pseudo-code for SATEG , based on the description in terms of deleting labels given in Section 3.6.1. * 9. For mutual exclusion, draw a transition system which forces the two processes
to enter their critical section in strict sequence and show that φ4 is false of its initial state. 10. Use the definition of between states and CTL formulas to explain why s AG AF φ means that φ is
true infinitely often along every path starting at s. * 11. Show that a CTL formula φ is true on infinitely many states of a computation path s0 → s1 → s2 → . . . iff for all n ≥ 0 there is some m ≥ n
such that sm φ. 12. Run the NuSMV system on some examples. Try commenting out, or deleting, some of the fairness constraints, if applicable, and see the counter examples NuSMV generates. NuSMV is
very easy to run. 13. In the one-bit channel, there are two fairness constraints. We could have written this as a single one, inserting ‘&’ between running and the long formula, or we could have
separated the long formula into two and made it into a total of three fairness constraints. In general, what is the difference between the single fairness constraint φ1 ∧ φ2 ∧ · · · ∧ φn and the n
fairness constraints φ1 , φ2 , . . . , φn ? Write an SMV program with a fairness constraint a & b which is not equivalent to the two fairness constraints a and b. (You can actually do it in four
lines of SMV.) 14. Explain the construction of formula φ4 , used to express that the processes need not enter their critical section in strict sequence. Does it rely on the fact that the safety
property φ1 holds? * 15. Compute the EC G labels for Figure 3.11, given the fairness constraints of the code in Figure 3.10 on page 196.
Exercises 3.7 1. Consider the functions H1 , H2 , H3 : P({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}) → P({1, 2, 3, 4, 5, 6, 7, 8, 9, 10}) defined by H1 (Y ) = Y − {1, 4, 7} def
H2 (Y ) = {2, 5, 9} − Y def
H3 (Y ) = {1, 2, 3, 4, 5} ∩ ({2, 4, 8} ∪ Y ) def
for all Y ⊆ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. * (a) Which of these functions are monotone; which ones aren’t? Justify your answer in each case. * (b) Compute the least and greatest fixed points of H3
using the iterations H3i with i = 1, 2, . . . and Theorem 3.24.
3.8 Exercises
Figure 3.42. Another system for which we compute invariants. (c) Does H2 have any fixed points? (d) Recall G : P({s0 , s1 }) → P({s0 , s1 }) with G(Y ) = if Y = {s0 } then {s1 } else {s0 } . def
* 2.
4. * 5.
7. 8. 9.
Use mathematical induction to show that Gi equals G for all odd numbers i ≥ 1. What does Gi look like for even numbers i? Let A and B be two subsets of S and let F : P(S) → P(S) be a monotone
function. Show that: def (a) F1 : P(S) → P(S) with F1 (Y ) = A ∩ F (Y ) is monotone; def (b) F2 : P(S) → P(S) with F2 (Y ) = A ∪ (B ∩ F (Y )) is monotone. Use Theorems 3.25 and 3.26 to compute the
following sets (the underlying model is in Figure 3.42): (a) [[EF p]] (b) [[EG q]]. Using the function F (X) = [[φ]] ∪ pre∀ (X) prove that [[AF φ]] is the least fixed point of F . Hence argue that the
procedure SATAF is correct and terminates. One may also compute AG φ directly as a fixed point. Consider the function H : P(S) → P(S) with H(X) = [[φ]] ∩ pre∀ (X). Show that H is monotone and that
[[AG φ]] is the greatest fixed point of H. Use that insight to write a procedure SATAG . Similarly, one may compute A[φ1 U φ2 ] directly as a fixed point, using K : P(S) → P(S), where K(X) = [[φ2 ]] ∪
([[φ1 ]] ∩ pre∀ (X)). Show that K is monotone and that [[A[φ1 U φ2 ]]] is the least fixed point of K. Use that insight to write a procedure SATAU . Can you use that routine to handle all calls of the
form AF φ as well? Prove that [[A[φ1 U φ2 ]]] = [[φ2 ∨ (φ1 ∧ AX (A[φ1 U φ2 ]))]]. Prove that [[AG φ]] = [[φ ∧ AX (AG φ)]]. Show that the repeat-statements in the code for SATEU and SATEG always
terminate. Use this fact to reason informally that the main program SAT terminates for all valid CTL formulas φ. Note that some subclauses, like the one for AU, call SAT recursively and with a more
complex formula. Why does this not affect termination?
3 Verification by model checking
3.9 Bibliographic notes Temporal logic was invented by the philosopher A. Prior in the 1960s; his logic was similar to what we now call LTL. The first use of temporal logic for reasoning about
concurrent programs was by A. Pnueli [Pnu81]. The logic CTL was invented by E. Clarke and E. A. Emerson (during the early 1980s); and CTL* was invented by E. A. Emerson and J. Halpern (in 1986) to
unify CTL and LTL. CTL model checking was invented by E. Clarke and E. A. Emerson [CE81] and by J. Quielle and J. Sifakis [QS81]. The technique we described for LTL model checking was invented by M.
Vardi and P. Wolper [VW84]. Surveys of some of these ideas can be found in [CGL93] and [CGP99]. The theorem about adequate sets of CTL connectives is proved in [Mar01]. The original SMV system was
written by K. McMillan [McM93] and is available with source code from Carnegie Mellon University6 . NuSMV7 is a reimplementation, developed in Trento by A. Cimatti, and M. Roveri and is aimed at
being customisable and extensible. Extensive documentation about NuSMV can be found at that site. NuSMV supports essentially the same system description language as CMU SMV, but it has an improved
user interface and a greater variety of algorithms. For example, whereas CMU SMV checks only CTL specification, NuSMV supports LTL and CTL. NuSMV implements bounded model checking [BCCZ99]. Cadence
SMV8 is an entirely new model checker focused on compositional systems and abstraction as ways of addressing the state explosion problem. It was also developed by K. McMillan and its description
language resembles but much extends the original SMV. A website which gathers frequently used specification patterns in various frameworks (such as CTL, LTL and regular expressions) is maintained by
M. Dwyer, G. Avrunin, J. Corbett and L. Dillon9 . Current research in model checking includes attempts to exploit abstractions, symmetries and compositionality [CGL94, Lon83, Dam96] in order to
reduce the impact of the state explosion problem. The model checker Spin, which is geared towards asynchronous systems and is based on the temporal logic LTL, can be found at the Spin website10 . A
model checker called FDR2 based on the process algebra CSP is available11 . www.cs.cmu.edu/~modelcheck/ nusmv.irst.itc.it www-cad.eecs.berkeley.edu/~kenmcmil/ 9 patterns.projects.cis.ksu.edu/ 10
netlib.bell-labs.com/netlib/spin/whatispin.html 11 www.fsel.com.fdr2 download.html 6 7 8
3.9 Bibliographic notes
The Edinburgh Concurrency Workbench12 and the Concurrency Workbench of North Carolina13 are similar software tools for the design and analysis of concurrent systems. An example of a customisable and
extensible modular model checking frameworks for the verification of concurrent software is Bogor14 . There are many textbooks about verification of reactive systems; we mention [MP91, MP95, Ros97,
Hol90]. The SMV code contained in this chapter can be downloaded from www.cs.bham.ac.uk/research/lics/. 12 13 14
www.dcs.ed.ac.uk/home/cwb www.cs.sunysb.edu/~cwb http://bogor.projects.cis.ksu.edu/
4 Program verification
The methods of the previous chapter are suitable for verifying systems of communicating processes, where control is the main issue, but there are no complex data. We relied on the fact that those
(abstracted) systems are in a finite state. These assumptions are not valid for sequential programs running on a single processor, the topic of this chapter. In those cases, the programs may
manipulate non-trivial data and – once we admit variables of type integer, list, or tree – we are in the domain of machines with infinite state space. In terms of the classification of verification
methods given at the beginning of the last chapter, the methods of this chapter are Proof-based. We do not exhaustively check every state that the system can get in to, as one does with model
checking; this would be impossible, given that program variables can have infinitely many interacting values. Instead, we construct a proof that the system satisfies the property at hand, using a proof
calculus. This is analogous to the situation in Chapter 2, where using a suitable proof calculus avoided the problem of having to check infinitely many models of a set of predicate logic formulas in
order to establish the validity of a sequent. Semi-automatic. Although many of the steps involved in proving that a program satisfies its specification are mechanical, there are some steps that involve
some intelligence and that cannot be carried out algorithmically by a computer. As we will see, there are often good heuristics to help the programmer complete these tasks. This contrasts with the
situation of the last chapter, which was fully automatic. Property-oriented. Just like in the previous chapter, we verify properties of a program rather than a full specification of its behaviour.
4.1 Why should we specify and verify code?
Application domain. The domain of application in this chapter is sequential transformational programs. ‘Sequential’ means that we assume the program runs on a single processor and that there are no
concurrency issues. ‘Transformational’ means that the program takes an input and, after some computation, is expected to terminate with an output. For example, methods of objects in Java are often
programmed in this style. This contrasts with the previous chapter which focuses on reactive systems that are not intended to terminate and that react continually with their environment. Pre/
post-development. The techniques of this chapter should be used during the coding process for small fragments of program that perform an identifiable (and hence, specifiable) task and hence should be
used during the development process in order to avoid functional bugs.
4.1 Why should we specify and verify code? The task of specifying and verifying code is often perceived as an unwelcome addition to the programmer’s job and a dispensable one. Arguments in favour of
verification include the following: r Documentation: The specification of a program is an important component in its documentation and the process of documenting a program may raise or resolve
important issues. The logical structure of the formal specification, written as a formula in a suitable logic, typically serves as a guiding principle in trying to write an implementation in which it
holds. r Time-to-market: Debugging big systems during the testing phase is costly and time-consuming and local ‘fixes’ often introduce new bugs at other places. Experience has shown that verifying
programs with respect to formal specifications can significantly cut down the duration of software development and maintenance by eliminating most errors in the planning phase and helping in the
clarification of the roles and structural aspects of system components. r Refactoring: Properly specified and verified software is easier to reuse, since we have a clear specification of what it is meant
to do. r Certification audits: Safety-critical computer systems – such as the control of cooling systems in nuclear power stations, or cockpits of modern aircrafts – demand that their software be
specified and verified with as much rigour and formality as possible. Other programs may be commercially critical, such as accountancy software used by banks, and they should be delivered with a
warranty: a guarantee for correct performance within proper use. The proof that a program meets its specifications is indeed such a warranty.
4 Program verification
The degree to which the software industry accepts the benefits of proper verification of code depends on the perceived extra cost of producing it and the perceived benefits of having it. As verification
technology improves, the costs are declining; and as the complexity of software and the extent to which society depends on it increase, the benefits are becoming more important. Thus, we can expect
that the importance of verification to industry will continue to increase over the next decades. Microsoft’s emergent technology A# combines program verification, testing, and model-checking techniques
in an integrated in-house development environment. Currently, many companies struggle with a legacy of ancient code without proper documentation which has to be adapted to new hardware and network
environments, as well as ever-changing requirements. Often, the original programmers who might still remember what certain pieces of code are for have moved, or died. Software systems now often have
a longer life-expectancy than humans, which necessitates a durable, transparent and portable design and implementation process; the year-2000 problem was just one such example. Software verification
provides some of this.
4.2 A framework for software verification Suppose you are working for a software company and your task is to write programs which are meant to solve sophisticated problems, or computations.
Typically, such a project involves an outside customer – a utility company, for example – who has written up an informal description, in plain English, of the real-world task that is at hand. In this
case, it could be the development and maintenance of a database of electricity accounts with all the possible applications of that – automated billing, customer service etc. Since the informality of
such descriptions may cause ambiguities which eventually could result in serious and expensive design flaws, it is desirable to condense all the requirements of such a project into formal
specifications. These formal specifications are usually symbolic encodings of real-world constraints into some sort of logic. Thus, a framework for producing the software could be: r Convert the
informal description R of requirements for an application domain into an ‘equivalent’ formula φR of some symbolic logic; r Write a program P which is meant to realise φ in the programming environment
R supplied by your company, or wanted by the particular customer; r Prove that the program P satisfies the formula φ . R
This scheme is quite crude – for example, constraints may be actual design decisions for interfaces and data types, or the specification may ‘evolve’
4.2 A framework for software verification
and may partly be ‘unknown’ in big projects – but it serves well as a first approximation to trying to define good programming methodology. Several variations of such a sequence of activities are
conceivable. For example, you, as a programmer, might have been given only the formula φR , so you might have little if any insight into the real-world problem which you are supposed to solve.
Technically, this poses no problem, but often it is handy to have both informal and formal descriptions available. Moreover, crafting the informal requirements R is often a mutual process between the
client and the programmer, whereby the attempt at formalising R can uncover ambiguities or undesired consequences and hence lead to revisions of R. This ‘going back and forth’ between the realms of
informal and formal specifications is necessary since it is impossible to ‘verify’ whether an informal requirement R is equivalent to a formal description φR . The meaning of R as a piece of natural
language is grounded in common sense and general knowledge about the real-world domain and often based on heuristics or quantitative reasoning. The meaning of a logic formula φR , on the other hand,
is defined in a precise mathematical, qualitative and compositional way by structural induction on the parse tree of φR – the first three chapters contain examples of this. Thus, the process of finding
a suitable formalisation φR of R requires the utmost care; otherwise it is always possible that φR specifies behaviour which is different from the one described in R. To make matters worse, the
requirements R are often inconsistent; customers usually have a fairly vague conception of what exactly a program should do for them. Thus, producing a clear and coherent description R of the
requirements for an application domain is already a crucial step in successful programming; this phase ideally is undertaken by customers and project managers around a table, or in a video
conference, talking to each other. We address this first item only implicitly in this text, but you should certainly be aware of its importance in practice. The next phase of the software development
framework involves constructing the program P and after that the last task is to verify that P satisfies φR . Here again, our framework is oversimplifying what goes on in practice, since often proving
that P satisfies its specification φR goes hand-in-hand with inventing a suitable P . This correspondence between proving and programming can be stated quite precisely, but that is beyond the scope of
this book.
4.2.1 A core programming language The programming language which we set out to study here is the typical core language of most imperative programming languages. Modulo trivial
4 Program verification
syntactic variations, it is a subset of Pascal, C, C++ and Java. Our language consists of assignments to integer- and boolean-valued variables, ifstatements, while-statements and sequential
compositions. Everything that can be computed by large languages like C and Java can also be computed by our language, though perhaps not as conveniently, because it does not have any objects,
procedures, threads or recursive data structures. While this makes it seem unrealistic compared with fully blown commercial languages, it allows us to focus our discussion on the process of formal
program verification. The features missing from our language could be implemented on top of it; that is the justification for saying that they do not add to the power of the language, but only to the
convenience of using it. Verifying programs using those features would require non-trivial extensions of the proof calculus we present here. In particular, dynamic scoping of variables presents hard
problems for program-verification methods, but this is beyond the scope of this book. Our core language has three syntactic domains: integer expressions, boolean expressions and commands – the latter
we consider to be our programs. Integer expressions are built in the familiar way from variables x, y, z, . . . , numerals 0, 1, 2, . . . , −1, −2, . . . and basic operations like addition (+) and
multiplication (∗). For example, 5 x 4 + (x − 3) x + (x ∗ (y − (5 + z))) are all valid integer expressions. Our grammar for generating integer expressions is E ::= n | x | (−E) | (E + E) | (E − E) |
(E ∗ E)
where n is any numeral in {. . . , −2, −1, 0, 1, 2, . . . } and x is any variable. Note that we write multiplication in ‘mathematics’ as 2 · 3, whereas our core language writes 2 ∗ 3 instead.
Convention 4.1 In the grammar above, negation − binds more tightly than multiplication ∗, which binds more tightly than subtraction − and addition +. Since if-statements and while-statements contain
conditions in them, we also need a syntactic domain B of boolean expressions. The grammar in
4.2 A framework for software verification
Backus Naur form B ::= true | false | (!B) | (B & B) | (B || B) | (E < E)
uses ! for the negation, & for conjunction and || for disjunction of boolean expressions. This grammar may be freely expanded by operators which are definable in terms of the above. For example, the
test for equality1 E1 == E2 may be expressed via !(E1 < E2 ) & !(E2 < E1 ). We generally make use of shorthand notation whenever this is convenient. We also write (E1 ! = E2 ) to abbreviate !(E1 ==
E2 ). We will also assume the usual binding priorities for logical operators stated in Convention 1.3 on page 5. Boolean expressions are built on top of integer expressions since the last clause of
(4.2) mentions integer expressions. Having integer and boolean expressions at hand, we can now define the syntactic domain of commands. Since commands are built from simpler commands using assignments
and the control structures, you may think of commands as the actual programs. We choose as grammar for commands C
::= x = E | C; C | if B {C} else {C} | while B {C}
where the braces { and } are to mark the extent of the blocks of code in the if-statement and the while-statement, as in languages such as C and Java. They can be omitted if the blocks consist of a
single statement. The intuitive meaning of the programming constructs is the following: 1. The atomic command x = E is the usual assignment statement; it evaluates the integer expression E in the
current state of the store and then overwrites the current value stored in x with the result of that evaluation. 2. The compound command C1 ; C2 is the sequential composition of the commands C1 and
C2 . It begins by executing C1 in the current state of the store. If that execution terminates, then it executes C2 in the storage state resulting from the execution of C1 . Otherwise – if the
execution of C1 does not terminate – the run of C1 ; C2 also does not terminate. Sequential composition is an example of a control structure since it implements a certain policy of flow of control in
a computation.
In common with languages like C and Java, we use a single equals sign = to mean assignment and a double sign == to mean equality. Earlier languages like Pascal used := for assignment and simple = for
equality; it is a great pity that C and its successors did not keep this convention. The reason that = is a bad symbol for assignment is that assignment is not symmetric: if we interpret x = y as the
assignment, then x becomes y which is not the same thing as y becoming x; yet, x = y and y = x are the same thing if we mean equality. The two dots in := helped remind the reader that this is an
asymmetric assignment operation rather than a symmetric assertion of equality. However, the notation = for assignment is now commonplace, so we will use it.
4 Program verification
3. Another control structure is if B {C1 } else {C2 }. It first evaluates the boolean expression B in the current state of the store; if that result is true, then C1 is executed; if B evaluated to
false, then C2 is executed. 4. The third control construct while B {C} allows us to write statements which are executed repeatedly. Its meaning is that: a the boolean expression B is evaluated in the
current state of the store; b if B evaluates to false, then the command terminates, c otherwise, the command C will be executed. If that execution terminates, then we resume at step (a) with a
re-evaluation of B as the updated state of the store may have changed its value. The point of the while-statement is that it repeatedly executes the command C as long as B evaluates to true. If B
never becomes false, or if one of the executions of C does not terminate, then the while-statement will not terminate. While-statements are the only real source of non-termination in our core
programming language.
Example 4.2 The factorial n! of a natural number n is defined inductively by def
0! = 1 (4.4)
(n + 1)! = (n + 1) · n! def
For example, unwinding this definition for n being 4, we get 4! = 4 · 3! = · · · = 4 · 3 · 2 · 1 · 0! = 24. The following program Fac1: y = 1; z = 0; while (z != x) { z = z + 1; y = y * z; } is
intended to compute the factorial2 of x and to store the result in y. We will prove that Fac1 really does this later in the chapter.
4.2.2 Hoare triples Program fragments generated by (4.3) commence running in a ‘state’ of the machine. After doing some computation, they might terminate. If they do, then the result is another,
usually different, state. Since our programming 2
Please note the difference between the formula x! = y, saying that the factorial of x is equal to y, and the piece of code x != y which says that x is not equal to y.
4.2 A framework for software verification
language does not have any procedures or local variables, the ‘state’ of the machine can be represented simply as a vector of values of all the variables used in the program. What syntax should we
use for φR , the formal specifications of requirements for such programs? Because we are interested in the output of the program, the language should allow us to talk about the variables in the state
after the program has executed, using operators like = to express equality and < for less than. You should be aware of the overloading of =. In code, it represents an assignment instruction; in
logical formulas, it stands for equality, which we write == within program code. For example, if the informal requirement R says that we should Compute a number y whose square is less than the input
then an appropriate specification may be y · y < x. But what if the input x is −4? There is no number whose square is less than a negative number, so it is not possible to write the program in a way
that it will work with all possible inputs. If we go back to the client and say this, he or she is quite likely to respond by saying that the requirement is only that the program work for positive
numbers; i.e., he or she revises the informal requirement so that it now says If the input x is a positive number, compute a number whose square is less than x.
This means we need to be able to talk not just about the state after the program executes, but also about the state before it executes. The assertions we make will therefore be triples, typically
looking like φ P ψ (4.5) which (roughly) means: If the program P is run in a state that satisfies φ, then the state resulting from P ’s execution will satisfy ψ.
The specification of the program P , to calculate a number whose square is less than x, now looks like this: x>0 P y·y 0, then the resulting state will be such that y · y < x. It does not tell us what
happens if we run P in a state in which x ≤ 0, the client required nothing for non-positive values of x. Thus, the programmer is free to do what he or she wants in that case. A program which produces
‘garbage’ in the case that x ≤ 0 satisfies the specification, as long as it works correctly for x > 0.
4 Program verification
Let us make these notions more precise.
Definition 4.3 1. The form φ P ψ of our specification is called a Hoare triple, after the computer scientist C. A. R. Hoare. 2. In (4.5), the formula φ is called the precondition of P and ψ is called
the postcondition. 3. A store or state of core programs is a function l that assigns to each variable x an integer l(x). 4. For a formula φ of predicate logic with function symbols − (unary), +, −,
and ∗ (binary); and a binary predicate symbols < and =, we say that a state l satisfies φ or l is a φ-state – written l φ – iff M l φ from page 128 holds, where l is viewed as a look-up table and the
model M has as set A all integers and interprets the function and predicate symbols in their standard manner. 5. For Hoare triples in (4.5), we demand that quantifiers in φ and ψ only bind variables
that do not occur in the program P .
Example 4.4 For any state l for which l(x) = −2, l(y) = 5, and l(z) = −1, the relation 1. l ¬(x + y < z) holds since x + y evaluates to −2 + 5 = 3, z evaluates to l(z) = −1, and 3 is not strictly
less than −1; 2. l y − x ∗ z < z does not hold, since the lefthand expression evaluates to 5 − (−2) · (−1) = 3 which is not strictly less than l(z) = −1; 3. l ∀u (y < u → y ∗ z < u ∗ z) does not
hold; for u being 7, l y < u holds, but l y ∗ z < u ∗ z does not.
Often, we do not want to put any constraints on the initial state; we simply wish to say that, no matter what state we start the program in, the resulting state should satisfy ψ. In that case the
precondition can be set to , which is – as in previous chapters – a formula which is true in any state. Note that the triple in (4.6) does not specify a unique program P , or a unique behaviour. For
example, the program which simply does y = 0; satisfies the specification – since 0 · 0 is less than any positive number – as does the program y = 0; while (y * y < x) { y = y + 1; } y = y - 1; This
program finds the greatest y whose square is less than x; the whilestatement overshoots a bit, but then we fix it after the while-statement.3 3
We could avoid this inelegance by using the repeat construct of exercise 3 on page 299.
4.2 A framework for software verification
Note that these two programs have different behaviour. For example, if x is 22, the first one will compute y = 0 and the second will render y = 4; but both of them satisfy the specification. Our agenda,
then, is to develop a notion of proof which allows us to prove that a program P satisfies the specification given by a precondition φ and a postcondition ψ in (4.5). Recall that we developed proof
calculi for propositional and predicate logic where such proofs could be accomplished by investigating the structure of the formula one wanted to prove. For example, for proving an implication φ → ψ
one had to assume φ and manage to show ψ; then the proof could be finished with the proof rule for implies-introduction. The proof calculi which we are about to develop follow similar lines. Yet, they
are different from the logics we previously studied since they prove triples which are built from two different sorts of things: logical formulas φ and ψ versus a piece of code P . Our proof calculi
have to address each of these appropriately. Nonetheless, we retain proof strategies which are compositional, but now in the structure of P . Note that this is an important advantage in the
verification of big projects, where code is built from a multitude of modules such that the correctness of certain parts will depend on the correctness of certain others. Thus, your code might call
subroutines which other members of your project are about to code, but you can already check the correctness of your code by assuming that the subroutines meet their own specifications. We will
explore this topic in Section 4.5.
4.2.3 Partial and correctness total Our explanation of when the triple φ P ψ holds was rather informal. In particular, it did not say what we should conclude if P does not terminate. In fact there
are two ways of handling this situation. Partial correctness means that we do not require the program to terminate, whereas in total correctness we insist upon its termination. Definition 4.5
(Partial correctness) We say that the triple φ P ψ is satisfied under partial correctness if, for all states which satisfy φ, the state resulting from P ’s execution satisfies the postcondition ψ,
provided that P actually terminates. In this case, the relation par φ P ψ holds. We call par the satisfaction relation for partial correctness. Thus, we insist on ψ being true of the resulting state
only if the program P has terminated on an input satisfying φ. Partial correctness is rather a weak requirement, since any program which does not terminate at all satisfies its
4 Program verification
specification. In particular, the program while true { x = 0; } – which endlessly ‘loops’ and never terminates – satisfies all specifications, since partial correctness only says what must happen if the
program terminates. Total correctness, on the other hand, requires that the program terminates in order for it to satisfy a specification. Definition 4.6 (Total correctness) We say that the triple φ P
ψ is satisfied under total correctness if, for all states in which P is executed which satisfy the precondition φ, P is guaranteed to terminate and the resulting state satisfies the postcondition ψ. In
this case, we say that tot φ P ψ holds and call tot the satisfaction relation of total correctness. A program which ‘loops’ forever on all input does not satisfy any specification under total
correctness. Clearly, total correctness is more useful than partial correctness, so the reader may wonder why partial correctness is introduced at all. Proving total correctness usually benefits from
proving partial correctness first and then proving termination. So, although our primary interest is in proving total correctness, it often happens that we have to or may wish to split this into
separate proofs of partial correctness and of termination. Most of this chapter is devoted to the proof of partial correctness, though we return to the issue of termination in Section 4.4. Before we
delve into the issue of crafting sound and complete proof calculi for partial and total correctness, let us briefly give examples of typical sorts of specifications which we would like to be able to
prove. Examples 4.7 1. Let Succ be the program a = x + 1; if (a - 1 == 0) { y = 1; } else { y = a; } The program Succ satisfies the specification Succ y = (x + 1) under partial and total correctness,
so if we think of x as input and y as output, then Succ computes the successor function. Note that this code is far from optimal.
4.2 A framework for software verification
In fact, it is a rather roundabout way of implementing the successor function. Despite this non-optimality, our proof rules need to be able to prove this program behaviour. 2. The program Fac1 from
Example 4.2 terminates only if x is initially nonnegative – why? Let us look at what properties of Fac1 we expect to be able to prove. We should be able to prove that tot x ≥ 0 Fac1 y = x! holds. It
states that, provided x ≥ 0, Fac1 terminates with the result y = x!. However, the stronger statement that tot Fac1 y = x! holds should not be provable, because Fac1 does not terminate for negative
values of x. Fac1 and x ≥ 0 y = x! For partial correctness, both statements par par Fac1 y = x! should be provable since they hold.
Definition 4.8 1. If the partial correctness of triples φ P ψ can be proved in the partial-correctness calculus we develop in this chapter, we say that the sequent par φ P ψ is valid. 2. Similarly,
if it can be proved in the total-correctness calculus to be developed in this chapter, we say that the sequent tot φ P ψ is valid.
Thus, par φ P ψ holds if P is partially correct, while the validity of par φ P ψ means that P can be proved to be partially-correct by our calculus. The first one means it is actually correct, while
the second one means it is provably correct according to our calculus. If our calculus is any good, then the relation par should be contained in par ! More precisely, we will say that our calculus is
sound if, whenever it tells us something can be proved, that thing is indeed true. Thus, it is sound if it doesn’t tell us that false things can be proved. Formally, we write that par is sound if par
φ P ψ holds whenever par φ P ψ is valid for all φ, ψ and P ; and, similarly, tot is sound if tot φ P ψ holds whenever tot φ P ψ is valid for all φ, ψ and P . We say that a calculus is complete if it
is able to prove everything that is true. Formally, par is complete if par φ P ψ is valid whenever par φ P ψ holds for all φ, ψ and P ; and similarly for tot being complete. In Chapters 1 and 2, we
said that soundness is relatively easy to show, since typically the soundness of individual proof rules can be established independently of the others. Completeness, on the other hand, is harder to
4 Program verification
show since it depends on the entire set of proof rules cooperating together. The same situation holds for the program logic we introduce in this chapter. Establishing its soundness is simply a matter
of considering each rule in turn – done in exercise 3 on page 303 – whereas establishing its (relative) completeness is harder and beyond the scope of this book.
4.2.4 Program variables and logical variables The variables which we have seen so far in the programs that we verify are called program variables. They can also appear in the preconditions and
postconditions of specifications. Sometimes, in order to formulate specifications, we need to use other variables which do not appear in programs. Examples 4.9 1. Another version of the factorial
program might have been Fac2: y = 1; while (x != 0) { y = y * x; x = x - 1; } Unlike the previous version, it ‘consumes’ the input x. Nevertheless, it correctly calculates the factorial of x and
stores the value in y; and we would like a Hoare triple. However, it is not a good idea to write to express that as x ≥ 0 Fac2 y = x! because, if the program terminates, then x will be 0 and y will
be the factorial of the initial value of x. We need a way of remembering the initial value of x, to cope with the fact that it is modified by the program. Logical variables achieve just that: in the
specification x = x0 ∧ x ≥ 0 Fac2 y = x0 ! the x0 is a logical variable and we read it as being universally quantified in the precondition. Therefore, this specification reads: for all integers x0 , if
x equals x0 , x ≥ 0 and we run the program such that it terminates, then the resulting state will satisfy y equals x0 !. This works since x0 cannot be modified by Fac2 as x0 does not occur in Fac2. 2.
Consider the program Sum: z = 0; while (x > 0) { z = z + x; x = x - 1; } This program and stores the result in z. adds up the first x integers Thus, x = 3 Sum z = 6 , x = 8 Sum z = 36 etc. We know
from Theorem 1.31 on page 41 that 1 + 2 + · · · + x = x(x + 1)/2 for all x ≥ 0, so
4.3 Proof calculus for partial correctness
we would like to express, as a Hoare triple, that the value of z upon termination is x0 (x0 + 1)/2 where x0 is the initial value of x. Thus, we write x = x0 ∧ x ≥ 0 Sum z = x0 (x0 + 1)/2 .
Variables like x0 in these examples are called logical variables, because they occur only in the logical formulas that constitute the precondition and postcondition; they do not occur in the code to
be verified. The state of the system gives a value to each program variable, but not for the logical variables. Logical variables take a similar role to the dummy variables of the rules for ∀i and ∃e
in Chapter 2. Definition 4.10 For a Hoare triple φ P ψ , its set of logical variables are those variables that are free in φ or ψ; and don’t occur in P .
4.3 Proof calculus for partial correctness The proof calculus which we now present goes back to R. Floyd and C. A. R. Hoare. In the next subsection, we specify proof rules for each of the grammar
clauses for commands. We could go on to use these proof rules directly, but it turns out to be more convenient to present them in a different form, suitable for the construction of proofs known as
proof tableaux. This is what we do in the subsection following the next one.
4.3.1 Proof rules The proof rules for our calculus are given in Figure 4.1. They should be interpreted as rules that allow us to pass from simple assertions of the form φ P ψ to more complex ones.
The rule for assignment is an axiom as it has no premises. This allows us to construct some triples out of nothing, to get the proof going. Complete proofs are trees, see page 274 for an example.
Composition. Given specifications for the program fragments C1 and C2 , say φ C1 η η C2 ψ , and where the postcondition of C1 is also the precondition of C2 , the proof rule for sequential composition
shown in Figure 4.1 allows us to derive a specification for C1 ; C2 , namely φ C1 ; C2 ψ .
4 Program verification
φ C1 η η C2 ψ Composition φ C1 ; C2 ψ
Assignment ψ[E/x] x = E ψ
φ ∧ B C1 ψ φ ∧ ¬B C2 ψ If-statement φ if B {C1 } else {C2 } ψ
ψ∧B C ψ Partial-while ψ while B {C} ψ ∧ ¬B AR φ → φ
φ C ψ φ C ψ
AR ψ → ψ
Figure 4.1. Proof rules for partial correctness of Hoare triples.
Thus, if we know that C1 takes φ-states to η-states and C2 takes η-states to ψ-states, then running C1 and C2 in that sequence will take φ-states to ψ-states. Using the proof rules of Figure 4.1 in
program verification, we have to read them bottom-up: e.g. in order φ to prove C1 ; C 2 ψ , we need to find an appropriate η and prove φ C1 η and η C2 ψ . If C1 ; C2 runs on input satisfying φ and we
need to show that the store satisfies ψ after its execution, then we hope to show this by splitting the problem into two. After the execution of C1 , we have a store satisfying η which, considered as
input for C2 , should result in an output satisfying ψ. We call η a midcondition. Assignment. The rule for assignment has no premises and is therefore an axiom of our logic. It tells us that, if we
wish to show that ψ holds in the state after the assignment x = E, we must show that ψ[E/x] holds before the assignment; ψ[E/x] denotes the formula obtained by taking ψ and replacing all free
occurrences of x with E as defined on page 105. We read the stroke as ‘in place of;’ thus, ψ[E/x] is ψ with E in place of x. Several explanations may be required to understand this rule. r At first
sight, it looks as if the rule has been stated in reverse; one might expect that, if ψ holds in a state in which we perform the assignment x = E, then surely
4.3 Proof calculus for partial correctness
ψ[E/x] holds in the resulting state, i.e. we just replace x by E. This is wrong. It is true that the assignment x = E replaces the value of x in the starting state by E, but that does not mean that
we replace occurrences of x in a condition on the starting state by E. For example, let ψ be x = 6 and E be 5. Then ψ x = 5 ψ[x/E] does not hold: given a state in which x equals 6, the execution of x
= 5 results in a state in which x equals 5. But ψ[x/E] is the formula 5 = 6 which holds in no state. The right way to understand the Assignment rule is to think about what you would have to prove
about the initial state in order to prove that ψ holds in the resulting state. Since ψ will – in general – be saying something about the value of x, whatever it says about that value must have been
true of E, since in the resulting state the value of x is E. Thus, ψ with E in place of x – which says whatever ψ says about x but applied to E – must be true in the initial state. r The axiom ψ[E/x]
x = E ψ is best applied backwards than forwards in the verification That is to say, if we know ψ and we wish to find φ such process. that φ x = E ψ , it is easy: we simply be ψ[E/x]; but, if we know
set φ to φ and we want to find ψ such that φ x = E ψ , there is no easy way of getting a suitable ψ. This backwards characteristic of the assignment and the composition rule will be important when we
look at how to construct proofs; we will work from the end of a program to its beginning. r If we apply this axiom in this backwards fashion, then it is completely mechanical to apply. It just
involves doing a substitution. That means we could get a computer to do it for us. Unfortunately, that is not true for all the rules; application of the rule for while-statements, for example,
requires ingenuity. Therefore a computer can at best assist us in performing a proof by carrying out the mechanical steps, such as application of the assignment axiom, while leaving the steps that
involve ingenuity to the programmer. r Observe that, in computing ψ[E/x] from ψ, we replace all the free occurrences of x in ψ. Note that there cannot be problems caused by bound occurrences, as seen
in Example 2.9 on page 106, provided that preconditions and postconditions quantify over logical variables only. For obvious reasons, this is recommended practice.
Examples 4.11 1. Suppose P is the program x = 2. The following are instances of axiom Assignment: a 2 = 2 P x = 2 b 2 = 4 P x = 4 c 2 = y P x = y d 2>0 P x>0 .
4 Program verification These are all correct statements. Reading them backwards, we see that they say: a If you want to prove x = 2 after the assignment x = 2, then we must be able to prove that 2 =
2 before it. Of course, 2 is equal to 2, so proving it shouldn’t present a problem. b If you wanted to prove that x = 4 after the assignment, the only way in which it work is if 2 = 4; however,
unfortunately it is not. More generally, would ⊥ x = E ψ holds for any E and ψ – why? c If you want to prove x = y after the assignment, you will need to prove that 2 = y before it. d To prove x > 0,
we’d better have 2 > 0 prior to the execution of P .
2. Suppose P is x = x + 1. By choosing various postconditions, we obtain the following instances of the assignment axiom: a x + 1 = 2 P x = 2 b x + 1 = y P x = y c x + 1 + 5 = y P x + 5 = y d x+1>0∧y
>0 P x>0∧y >0 . Note that the precondition obtained by performing the substitution can often be simplified. The proof rule for implications below will allow such simplifications which are needed to
make preconditions appreciable by human consumers.
If-statements. The proof rule for if-statements allows us to prove a triple of the form φ if B {C1 } else {C2 } ψ by decomposing it into two triples, subgoals corresponding to the cases of B
evaluating to true and to false. Typically, the precondition φ will not tell us anything about the value of the boolean expression B, so we have to consider both cases. If B is true in the state we
start in, then C1 is executed and hence C1 will have to translate φ states to ψ states; alternatively, if B is false, then C2 will and will have to do that job. Thus, be executed we have to prove
that φ ∧ B C1 ψ and φ ∧ ¬B C2 ψ . Note that the preconditions are augmented by the knowledge that B is true and false, respectively. This additional information is often crucial for completing the
respective subproofs. While-statements. The rule for while-statements given in Figure 4.1 is arguably the most complicated one. The reason is that the while-statement is the most complicated
construct in our language. It is the only command that ‘loops,’ i.e. executes the same piece of code several times. Also, unlike as the for-statement in languages like Java we cannot generally
predict how
4.3 Proof calculus for partial correctness
many times while-statements will ‘loop’ around, or even whether they will terminate at all. The key ingredient in the proof rule for Partial-while is the ‘invariant’ ψ. In general, the body C of the
command while (B) {C} changes the values of the variables it manipulates; but the invariant expresses a relationship between those values which is preserved by any execution of C. In the proof rule,
ψ expresses this invariant; the rule’s premise, ψ ∧ B C ψ , states that, if ψ and B are true before we execute C, and C terminates, then ψ will be true after it. The conclusion of Partial-while
states that, no matter how many times the body C is executed, if ψ is true initially and the whilestatement terminates, then ψ will be true at the end. Moreover, since the while-statement has
terminated, B will be false. Implied. One final rule is required in our the rule Implied of Figure calculus: 4.1. It tells us that, if we have proved φ P ψ and we have a formula φ which implies φ and
another one is implied by ψ, then we should ψ which also be allowed to prove that φ P ψ . A sequent AR φ → φ is valid iff there is a proof of φ in the natural deduction calculus for predicate logic,
where φ and standard laws of arithmetic – e.g. ∀x (x = x + 0) – are premises. Note that the rule Implied allows the precondition to be strengthened (thus, we assume more than we need to), while the
postcondition is weakened (i.e. we conclude less than we are entitled to). If we tried to do it the other way around, weakening the precondition or strengthening the postcondition, then we would
conclude things which are incorrect – see exercise 9(a) on page 300. The rule Implied acts as a link between program logic and a suitable extension of predicate logic. It allows us to import proofs
in predicate logic enlarged with the basic facts of arithmetic, which are required for reasoning about integer expressions, into the proofs in program logic.
4.3.2 Proof tableaux The proof rules presented in Figure 4.1 are not in a form which is easy to use in examples. To illustrate this point, we present of a an example proof in Figure 4.2; it is a
proof of the triple Fac1 y = x! where Fac1 is the factorial program given in Example 4.2. This proof abbreviates rule names; and drops the bars and names for Assignment as well as sequents for AR in
all applications of the Implied rule. We have not yet presented enough information for the reader to complete such a proof on her own, but she can at least use the proof rules in Figure 4.1 to check
whether all rule instances of that proof are permissible, i.e. match the required pattern.
0 y=
1; = 0; wh il e
x) {z
z+ 1;
y* z}
y* z}
y* z
y* z}
z+ 1;
z+ 1; z+ 1;
z+ 1
z+ 1 ∧z
wh il e
wh il e
1) =
Figure 4.2. A partial-correctness proof for Fac1 in tree form.
y* z
274 4 Program verification
4.3 Proof calculus for partial correctness
It should be clear that proofs in this form are unwieldy to work with. They will tend to be very wide and a lot of information is copied from one line to the next. Proving properties of programs
which are longer than Fac1 would be very difficult in this style. In Chapters 1, 2 and 5 we abandon representation of proofs as trees for similar reasons. The rule for sequential composition suggests a
more convenient way of presenting proofs in program logic, called proof tableaux. We can think of any program of our core programming language as a sequence C1 ; C2 ; · · · Cn where none of the
commands Ci is a composition of smaller programs, i.e. all of the Ci above are either assignments, if-statements or while-statements. Of course, we allow the if-statements and while-statements to
have embedded compositions. Let P stand for the program C1; C2; . . . ; Cn−1 ; Cn . Suppose that we want to show the validity of par φ0 P φn for a precondition φ0 and a postconto dition φn . Then, we
may split this problem into smaller ones by trying find formulas φj (0 < j < n) and prove the validity of par φi Ci+1 φi+1 for i = 0, 1, . . . , n − 1. This suggests that we should design a proof
calculus which presents a proof of par φ0 P ψn by interleaving formulas with code as in φ0 C1 ; φ1
C2 ; · · ·
Cn ; φn
justification justification
4 Program verification
Against each formula, we write a justification, whose nature will be clarified shortly. Proof tableaux thus consist of the program code interleaved with formulas, which we call midconditions, that
should hold at the point they are written. Each of the transitions φi Ci+1 φi+1 will appeal to one of the rules of Figure 4.1, depending on whether Ci+1 is an assignment, an if-statement or a
while-statement. Note that this notation for proofs makes the proof rule for composition in Figure 4.1 implicit. How should the intermediate formulas φi be found? In principle, it seems as though one
could start from φ0 and, using C1 , obtain φ1 and continue working downwards. However, because the assignment rule works backwards, it turns out that it is more convenient to start with φn and work
upwards, using Cn to obtain φn−1 etc. Definition 4.12 The process of obtaining φi from Ci+1 and φi+1 is called computing the weakest precondition of Ci+1 , given the postcondition φi+1 . That is to
say, we are looking for the logically weakest formula whose truth at the beginning of the execution of Ci+1 is enough to guarantee φi+1 4 . The construction of a proof tableau for φ C1 ; . . . ; Cn ψ
typically consists of starting with the postcondition ψ and pushing it upwards through Cn , then Cn−1 , . . . , until a formula φ emerges at the top. Ideally, the formula φ represents the weakest
precondition which guarantees that the ψ will hold if the composed program C1 ; C2 ; . . . ; Cn−1 ; Cn is executed and terminates. The weakest precondition φ is then checked to see whether it follows
from the given precondition φ. Thus, we appeal to the Implied rule of Figure 4.1. Before a discussion of how to find invariants for while-statement, we now look at the assignment and the if-statement
to see how the weakest precondition is calculated for each one. Assignment. The assignment axiom is easily adapted to work for proof tableaux. We write it thus: 4
φ is weaker than ψ means that φ is implied by ψ in predicate logic enlarged with the basic facts about arithmetic: the sequent AR ψ → φ is valid. We want the weakest formula, because we want to
impose as few constraints as possible on the preceding code. In some cases, especially those involving while-statements, it might not be possible to extract the logically weakest formula. We just
need one which is sufficiently weak to allow us to complete the proof at hand.
4.3 Proof calculus for partial correctness
x = E ψ
The justification is written against the ψ, since, once the proof has been constructed, we want to read it in a forwards direction. The construction itself proceeds in a backwards direction, because
that is the way the assignment axiom facilitates. Implied. In tableau form, the Implied rule allows us to write one formula φ2 directly underneath another one φ1 with no code in between, provided
that φ1 implies φ2 in that the sequent AR φ1 → φ2 is valid. Thus, the Implied rule acts as an interface between predicate logic with arithmetic and program logic. This is a surprising and crucial
insight. Our proof calculus for partial correctness is a hybrid system which interfaces with another proof calculus via the Implied proof rule only. When we appeal to the Implied rule, we will
usually not explicitly write out the proof of the implication in predicate logic, for this chapter focuses on the program logic. Mostly, the implications we typically encounter will be easy to
verify. The Implied rule is often used to simplify formulas that are generated by applications of the other rules. It is also used when the weakest precondition φ emerges by pushing the postcondition
upwards through the whole program. We use the Implied rule to show that the given precondition implies the weakest precondition. Let’s look at some examples of this. Examples 4.13 1. We show that par
y = 5 x = y + 1 x = 6 is valid:
x = y + 1 x=6
Implied Assignment
The proof is constructed from the bottom upwards. We start with x = 6 and, using the assignment axiom, we push it upwards through x = y + 1. This means substituting y + 1 for all occurrences of x,
resulting in y + 1 = 6 . Now, we compare this with the given precondition y = 5 . The given precondition and the arithmetic fact 5 + 1 = 6 imply it, so we have finished the proof.
4 Program verification
Although the proof is constructed bottom-up, its justifications make sense when read top-down: the second line is implied by the first and the fourth follows from the second by the assignment.
intervening 2. We prove the validity of par y < 3 y = y + 1 y < 4 :
y 0) { y = y * a; a = a - 1; } 19. Why can, or can’t, you prove the validity of par Copy1 x = y ? 20. Let all while-statements while (B) {C} in P be annotated with invariant candidates η at the and
of their bodies, and η ∧ B at the beginning of their body. (a) Explain how a proof of par φ P ψ can be automatically reduced to showing the validity of some AR ψ1 ∧ · · · ∧ ψn . (b) Identify such a
sequent AR ψ1 ∧ · · · ∧ ψn for the proof in Example 4.17 on page 287. 21. Given n = 5 test the correctness of Min Sum on the arrays below: * (a) [−3, 1, −2, 1, −8] (b) [1, 45, −1, 23, −1] * (c) [−1,
−2, −3, −4, 1097]. 22. If we swap the first and second assignment in the while-statement of Min Sum, so that it first assigns to s and then to t, is the program still correct? Justify your answer. *
23. Prove the partial correctness of S2 for Min Sum. 24. The program Min Sum does not reveal where a minimal-sum section may be found in an input array. Adapt Min Sum to achieve that. Can you do this
with a single pass through the array? 25. Consider the proof rule φ C ψ1 φ C ψ2 Conj φ C ψ1 ∧ ψ2 7
You may have to strengthen your invariant.
4.6 Exercises
for Hoare triples. (a) Show that this proof rule is sound for par . (b) Derive this proof rule from the ones on page 270. (c) Explain how this rule, or its derived version, is used to establish the
overall correctness of Min Sum. 26. The maximal-sum problem is to compute the maximal sum of all sections on an array. (a) Adapt the program from page 289 so that it computes the maximal sum of these
sections. (b) Prove the partial correctess of your modified program. (c) Which aspects of the correctness proof given in Figure 4.3 (page 291) can be ‘re-used?’
Exercises 4.4 1. Prove the of the total-correctness sequents: validity following * (a) tot x ≥ 0 Copy1 x = y * (b) tot y ≥ 0 Multi1 z = x·y z = x · y (c) tot (y = y0 ) ∧ (y ≥ 0) Multi2 0 * (d) tot x
≥ 0 Downfac y =x! * (e) tot x ≥ 0 Copy2 x = y , does your invariant have an active part in securing correctness? (f) tot ¬(y = 0) Div (x = d · y + r) ∧ (r < y) . 2. Prove total correctness of S1 and
S2 for Min Sum. 3. Prove that par is sound for par . Just like in Section 1.4.3, it suffices to assume that the premises of proof rules are instances of par . Then, you need to prove that their
respective conclusion must be an instance of par as well. 4. Prove that tot is sound for tot . 5. Implement program Collatz in a programming language of your choice such that the value of x is the
program’s input and the final value of c its output. Test your program on a range of inputs. Which is the biggest integer for which your program terminates without raising an exception or dumping the
core? 6. A function over integers f : I → I is affine iff there are integers a and b such that f (x) = a · x + b for all x ∈ I. The else-branch of the program Collatz assigns to c the value f (c), where
f is an affine function with a = 3 and b = 1. (a) Write an parameterized implementation of Collatz in which you can initially specify the values of a and b either statically or through keyboard input
such that the else-branch assigns to c the value of f (c). def (b) Determine for which pairs (a, b) ∈ I × I the set Pos = {x ∈ I | 0 < x} is invariant under the affine function f (x) = a · x + b: for
all x ∈ Pos, f (x) ∈ Pos. def * (c) Find an affine function that leaves Pos invariant, but not the set Odd = {x ∈ I | ∃y ∈ I : x = 2 · y + 1}, such that there is an input drawn from Pos whose
4 Program verification
execution with the modified Collatz program eventually enters a cycle, and therefore does not terminate.
Exercises 4.5 1. Consider methods of the form boolean certify V(c : Certificate) which return true iff the certificate c is judged valid by the verifier V, a class in which method certify V resides. *
(a) Discuss how programming by contract can be used to delegate the judgment of a certificate to another verifier. * (b) What potential problems do you see in this context if the resulting
methoddependency graph is circular? * 2. Consider the method boolean withdraw(amount: int) { if (amount < 0 && isGood(amount)) { balance = balance - amount; return true; } else { return false; } }
named withdraw which attempts to withdraw amount from an integer field balance of the class within which method withdraw lives. This method makes use of another method isGood which returns true iff the
value of balance is greater or equal to the value of amount. (a) Write a contract for method isGood. (b) Use that contract to show the validity of the contract for withdraw: method name: withdraw
input: amount of Type int assumes: 0 i). We describe how nodes of layer i (i.e. xi -nodes) are being handled. Definition 6.8 Given a non-terminal node n in a BDD, we define lo(n) to be the node
pointed to via the dashed line from n. Dually, hi(n) is the node pointed to via the solid line from n. Let us describe how the labelling is done. Given an xi -node n, there are three ways in which it
may get its label:
6.2 Algorithms for reduced OBDDs #4
#0 0
Reduce =⇒
#2 x3
Figure 6.14. An example execution of the algorithm reduce. r If the label id(lo(n)) is the same as id(hi(n)), then we set id(n) to be that label. That is because the boolean function represented at n
is the same function as the one represented at lo(n) and hi(n). In other words, node n performs a redundant test and can be eliminated by reduction C2. r If there is another node m such that n and m
have the same variable x , and i id(lo(n)) = id(lo(m)) and id(hi(n)) = id(hi(m)), then we set id(n) to be id(m). This is because the nodes n and m compute the same boolean function (compare with
reduction C3). r Otherwise, we set id(n) to the next unused integer label.
Note that only the last case creates a new label. Consider the OBDD in left side of Figure 6.14; each node has an integer label obtained in the manner just described. The algorithm reduce then
finishes by redirecting edges bottom-up as outlined in C1–C3. The resulting reduced OBDD is in right of Figure 6.14. Since there are efficient bottom-up traversal algorithms for dags, reduce is an
efficient operation in the number of nodes of an OBDD.
6.2.2 The algorithm apply Another procedure at the heart of OBDDs is the algorithm apply. It is used to implement operations on boolean functions such as +, · , ⊕ and complementation (via f ⊕ 1).
Given OBDDs Bf and Bg for boolean formulas f and g, the call apply (op, Bf , Bg ) computes the reduced OBDD of the boolean formula f op g, where op denotes any function from {0, 1} × {0, 1} to {0,
6 Binary decision diagrams
The intuition behind the apply algorithm is fairly simple. The algorithm operates recursively on the structure of the two OBDDs: 1. let v be the variable highest in the ordering (=leftmost in the
list) which occurs in Bf or Bg . 2. split the problem into two subproblems for v being 0 and v being 1 and solve recursively; 3. at the leaves, apply the boolean operation op directly.
The result will usually have to be reduced to make it into an OBDD. Some reduction can be done ‘on the fly’ in step 2, by avoiding the creation of a new node if both branches are equal (in which case
return the common result), or if an equivalent node already exists (in which case, use it). Let us make all this more precise and detailed. Definition 6.9 Let f be a boolean formula and x a variable.
1. We denote by f [0/x] the boolean formula obtained by replacing all occurrences of x in f by 0. The formula f [1/x] is defined similarly. The expressions f [0/x] and f [1/x] are called restrictions
of f . 2. We say that two boolean formulas f and g are semantically equivalent if they represent the same boolean function (with respect to the boolean variables that they depend upon). In that case,
we write f ≡ g. def
For example, if f (x, y) = x · (y + x), then f [0/x](x, y) equals 0 · (y + 0), which is semantically equivalent to 0. Similarly, f [1/y](x, y) is x · (1 + x), which is semantically equivalent to x.
Restrictions allow us to perform recursion on boolean formulas, by decomposing boolean formulas into simpler ones. For example, if x is a variable in f , then f is equivalent to x · f [0/x] + x · f
[1/x]. To see this, consider the case x = 0; the expression computes to f [0/x]. When x = 1 it yields f [1/x]. This observation is known as the Shannon expansion, although it can already be found in
G. Boole’s book ‘The Laws of Thought’ from 1854. Lemma 6.10 (Shannon expansion) For all boolean formulas f and all boolean variables x (even those not occurring in f ) we have f ≡ x · f [0/x] + x · f
The function apply is based on the Shannon expansion for f op g: f op g = xi · (f [0/xi ] op g[0/xi ]) + xi · (f [1/xi ] op g[1/xi ]).
This is used as a control structure of apply which proceeds from the roots
6.2 Algorithms for reduced OBDDs R1
R6 0
S5 0
Figure 6.15. An example of two arguments for a call apply (+, Bf , Bg ).
of Bf and Bg downwards to construct nodes of the OBDD Bf op g . Let rf be the root node of Bf and rg the root node of Bg . 1. If both rf and rg are terminal nodes with labels lf and lg , respectively
(recall that terminal labels are either 0 or 1), then we compute the value lf op lg and let the resulting OBDD be B0 if that value is 0 and B1 otherwise. 2. In the remaining cases, at least one of
the root nodes is a non-terminal. Suppose that both root nodes are xi -nodes. Then we create an xi -node n with a dashed line to apply (op, lo(rf ), lo(rg )) and a solid line to apply (op, hi(rf ),
hi(rg )), i.e. we call apply recursively on the basis of (6.2). 3. If rf is an xi -node, but rg is a terminal node or an xj -node with j > i, then we know that there is no xi -node in Bg because the
two OBDDs have a compatible ordering of boolean variables. Thus, g is independent of xi (g ≡ g[0/xi ] ≡ g[1/xi ]). Therefore, we create an xi -node n with a dashed line to apply (op, lo(rf ), rg )
and a solid line to apply (op, hi(rf ), rg ). 4. The case in which rg is a non-terminal, but rf is a terminal or an xj -node with j > i, is handled symmetrically to case 3.
The result of this procedure might not be reduced; therefore apply finishes by calling the function reduce on the OBDD it constructed. An example of apply (where op is +) can be seen in Figures
6.15–6.17. Figure 6.16 shows the recursive descent control structure of apply and Figure 6.17 shows the final result. In this example, the result of apply (+, Bf , Bg ) is Bf . Figure 6.16 shows that
numerous calls to apply occur several times with the same arguments. Efficiency could be gained if these were evaluated only
6 Binary decision diagrams
(R1 , S1 ) x1
(R2 , S3 )
(R3 , S2 )
(R4 , S3 )
(R3 , S3 )
(R5 , S4 )
(R6 , S5 )
(R4 , S3 )
(R6 , S5 )
(R6 , S3 )
(R5 , S4 )
(R6 , S5 )
(R5 , S4 )
(R4 , S3 )
(R6 , S5 )
(R6 , S4 )
(R6 , S5 )
Figure 6.16. The recursive call structure of apply for the example in Figure 6.15 (without memoisation). x1
Figure 6.17. The result of apply (+, Bf , Bg ), where Bf and Bg are given in Figure 6.15.
6.2 Algorithms for reduced OBDDs
the first time and the result remembered for future calls. This programming technique is known as memoisation. As well as being more efficient, it has the advantage that the resulting OBDD requires less
reduction. (In this example, using memoisation eliminates the need for the final call to reduce altogether.) Without memoisation, apply is exponential in the size of its arguments, since each non-leaf
call generates a further two calls. With memoisation, the number of calls to apply is bounded by 2 · |Bf | · |Bg |, where |B| is the size of the BDD. This is a worst-time complexity; the actual
performance is often much better than this.
6.2.3 The algorithm restrict Given an OBDD Bf representing a boolean formula f , we need an algorithm restrict such that the call restrict(0, x, Bf ) computes the reduced OBDD representing f [0/x]
using the same variable ordering as Bf . The algorithm for restrict(0, x, Bf ) works as follows. For each node n labelled with x, incoming edges are redirected to lo(n) and n is removed. Then we call
reduce on the resulting OBDD. The call restrict (1, x, Bf ) proceeds similarly, only we now redirect incoming edges to hi(n).
6.2.4 The algorithm exists A boolean function can be thought of as putting a constraint on the values of its argument variables. For example, the function x + (y · z) evaluates to 1 only if x is 1;
or y is 0 and z is 1 – this is a constraint on x, y, and z. It is useful to be able to express the relaxation of the constraint on a subset of the variables concerned. To allow this, we write ∃x. f
for the boolean function f with the constraint on x relaxed. Formally, ∃x. f is defined as f [0/x] + f [1/x]; that is, ∃x. f is true if f could be made true by putting x def to 0 or to 1. Given that
∃x. f = f [0/x] + f [1/x] the exists algorithm can be implemented in terms of the algorithms apply and restrict as apply (+, restrict (0, x, Bf ), restrict (1, x, Bf )) . def
Consider, for example, the OBDD Bf for the function f = x1 · y1 + x2 · y2 + x3 · y3 , shown in Figure 6.19. Figure 6.20 shows restrict(0, x3 , Bf ) and restrict(1, x3 , Bf ) and the result of
applying + to them. (In this case the apply function happens to return its second argument.) We can improve the efficiency of this algorithm. Consider what happens during the apply stage of (6.3). In
that case, the apply algorithm works on two BDDs which are identical all the way down to the level of the x-nodes;
6 Binary decision diagrams
Figure 6.18. An example of a BDD which is not a read-1-BDD.
x1 y1 x2 y2 x3 y3
Figure 6.19. A BDD Bf to illustrate the exists algorithm.
therefore the returned BDD also has that structure down to the x-nodes. At the x-nodes, the two argument BDDs differ, so the apply algorithm will compute the apply of + to these two subBDDs and return
that as the subBDD of the result. This is illustrated in Figure 6.20. Therefore, we can compute the OBDD for ∃x. f by taking the OBDD for f and replacing each node labelled with x by the result of
calling apply on + and its two branches. This can easily be generalised to a sequence of exists operations. We ˆ denotes (x1 , x2 , . . . , xn ). write ∃ˆ x. f to mean ∃x1 .∃x2 . . . . ∃xn . f ,
where x
6.2 Algorithms for reduced OBDDs x1
Figure 6.20. restrict(0, x3 , Bf ) and restrict(1, x3 , Bf ) and the result of applying + to them.
x1 y1
x1 y1
x3 y3
y3 1
Figure 6.21. OBDDs for f , ∃x3 . f and ∃x2 .∃x3 . f .
The OBDD for this boolean function is obtained from the OBDD for f by replacing every node labelled with an xi by the + of its two branches. Figure 6.21 shows the computation of ∃x3 . f and ∃x2 .∃x3
. f (which is semantically equivalent to x1 · y1 + y2 + y3 ) in this way. The boolean quantifier ∀ is the dual of ∃: def
∀x.f = f [0/x] · f [1/x] asserting that f could be made false by putting x to 0 or to 1. The translation of boolean formulas into OBDDs using the algorithms of this section is summarised in Figure
6 Binary decision diagrams
Boolean formula f
Representing OBDD Bf
B0 (Fig. 6.6)
B1 (Fig. 6.6)
Bx (Fig. 6.6)
swap the 0- and 1-nodes in Bf
f +g
apply (+, Bf , Bg )
f ·g
apply (· , Bf , Bg )
f ⊕g
apply (⊕, Bf , Bg )
f [1/x]
restrict (1, x, Bf )
f [0/x]
restrict (0, x, Bf )
apply (+, Bf [0/x] , Bf [1/x] )
apply (· , Bf [0/x] , Bf [1/x] )
Figure 6.22. Translating boolean formulas f to OBDDs Bf , given a fixed, global ordering on boolean variables.
Algorithm Input OBDD(s) Output OBDD reduce
Bf , Bg (reduced) Bf op g (reduced)
restrict Bf (reduced) ∃
Bf (reduced)
reduced B
Time-complexity O(|B| · log |B|) O(|Bf | · |Bg |)
Bf [0/x] or Bf [1/x] (reduced) O(|Bf | · log |Bf |) B∃x1 .∃x2 ....∃xn .f (reduced)
Figure 6.23. Upper bounds in terms of the input OBDD(s) for the worst-case running times of our algorithms needed in our implementation of boolean formulas.
6.2.5 Assessment of OBDDs Time complexities for computing OBDDs We can measure the complexity of the algorithms of the preceding section by giving upper bounds for the running time in terms of the
sizes of the input OBDDs. The table in Figure 6.23 summarises these upper bounds (some of those upper bounds may require more sophisticated versions of the algorithms than the versions presented in
this chapter). All the operations except nested boolean quantification are practically efficient in the size of the participating OBDDs. Thus, modelling very large systems with this approach will work
if the OBDDs
6.2 Algorithms for reduced OBDDs
which represent the systems don’t grow too large too fast. If we can somehow control the size of OBDDs, e.g. by using good heuristics for the choice of variable ordering, then these operations are
computationally feasible. It has already been shown that OBDDs modelling certain classes of systems and networks don’t grow excessively. The expensive computational operations are the nested boolean
quantifications ∃z1 . . . . ∃zn .f and ∀z1 . . . . ∀zn .f . By exercise 1 on page 406, the computation of the OBDD for ∃z1 . . . . ∃zn .f , given the OBDD for f , is an NPcomplete problem2 ; thus, it
is unlikely that there exists an algorithm with a feasible worst-time complexity. This is not to say that boolean functions modelling practical systems may not have efficient nested boolean
quantifications. The performance of our algorithms can be improved by using further optimisation techniques, such as parallelisation. Note that the operations apply, restrict, etc. are only efficient in
the size of the input OBDDs. So if a function f does not have a compact representation as an OBDD, then computing with its OBDD will not be efficient. There are such nasty functions; indeed, one of
them is integer multiplication. Let bn−1 bn−2 . . . b0 and an−1 an−2 . . . a0 be two n-bit integers, where bn−1 and an−1 are the most significant bits and b0 and a0 are the least significant bits. The
multiplication of these two integers results in a 2n-bit integer. Thus, we may think of multiplication as 2n many boolean functions fi in 2n variables (n bits for input b and n bits for input a),
where fi denotes the ith output bit of the multiplication. The following negative result, due to R. E. Bryant, shows that OBDDs cannot be used for implementing integer multiplication. Theorem 6.11
Any OBDD representation of fn−1 has at least a number of vertices proportional to 1.09n , i.e. its size is exponential in n. Extensions and variations of OBDDs There are many variations and
extensions to the OBDD data structure. Many of them can implement certain operations more efficiently than their OBDD counterparts, but it seems that none of them perform as well as OBDDs overall. In
particular, one feature which many of the variations lack is the canonical form; therefore they lack an efficient algorithm for deciding when two objects denote the same boolean function. One kind of
variation allows non-terminal nodes to be labelled with binary operators as well as boolean variables. Parity OBDDs are like OBDDs in that there is an ordering on variables and every variable may
occur at 2
Another NP-complete problem is to decide the satisfiability of formulas of propositional logic.
6 Binary decision diagrams
most once on a path; but some non-terminal nodes may be labelled with ⊕, the exclusive-or operation. The meaning is that the function represented by that node is the exclusive-or of the boolean
functions determined by its children. Parity OBDDs have similar algorithms for apply, restrict, etc. with the same performance, but they do not have a canonical form. Checking for equivalence cannot
be done in constant time. There is, however, a cubic algorithm for determining equivalence; and there are also efficient probabilistic tests. Another variation of OBDDs allows complementation nodes,
with the obvious meaning. Again, the main disadvantage is the lack of canonical form. One can also allow non-terminal nodes to be unlabelled and to branch to more than two children. This can then be
understood either as nondeterministic branching, or as probabilistic branching: throw a pair of dice to determine where to continue the path. Such methods may compute wrong results; one then aims at
repeating the test to keep the (probabilistic) error as small as desired. This method of repeating probabilistic tests is called probabilistic amplification. Unfortunately, the satisfiability problem
for probabilistic branching OBDDs is NP-complete. On a good note, probabilistic branching OBDDs can verify integer multiplication. The development of extensions or variations of OBDDS which are
customised to certain classes of boolean functions is an important area of ongoing research.
6.3 Symbolic model checking The use of BDDs in model checking resulted in a significant breakthrough in verification in the early 1990s, because they have allowed systems with much larger state spaces
to be verified. In this section, we describe in detail how the model-checking algorithm presented in Chapter 3 can be implemented using OBDDs as the basic data structure. The pseudo-code presented in
Figure 3.28 on page 227 takes as input a CTL formula φ and returns the set of states of the given model which satisfy φ. Inspection of the code shows that the algorithm consists of manipulating
intermediate sets of states. We show in this section how the model and the intermediate sets of states can be stored as OBDDs; and how the operations required in that pseudo-code can be implemented
in terms of the operations on OBDDs which we have seen in this chapter. We start by showing how sets of states are represented with OBDDs, together with some of the operations required. Then, we
extend that to the representation of the transition system; and finally, we show how the remainder of the required operations is implemented.
6.3 Symbolic model checking
Model checking using OBDDs is called symbolic model checking. The term emphasises that individual states are not represented; rather, sets of states are represented symbolically, namely, those which
satisfy the formula being checked.
6.3.1 Representing subsets of the set of states Let S be a finite set (we forget for the moment that it is a set of states). The task is to represent the various subsets of S as OBDDs. Since OBDDs
encode boolean functions, we need somehow to code the elements of S as boolean values. The way to do this in general is to assign to each element s ∈ S a unique vector of boolean values (v1 , v2 , .
. . , vn ), each vi ∈ {0, 1}. Then, we represent a subset T by the boolean function fT which maps (v1 , v2 , . . . , vn ) onto 1 if s ∈ T and maps it onto 0 otherwise. There are 2n boolean vectors
(v1 , v2 , . . . , vn ) of length n. Therefore, n should be chosen such that 2n−1 < |S| ≤ 2n , where |S| is the number of elements in S. If |S| is not an exact power of 2, there will be some vectors
which do not correspond to any element of S; they are just ignored. The function fT : {0, 1}n → {0, 1} which tells us, for each s, represented by (v1 , v2 , . . . , vn ), whether it is in the set T
or not, is called the characteristic function of T . In the case that S is the set of states of a transition system M = (S, →, L) (see Definition 3.4), there is a natural way of choosing the
representation of S as boolean vectors. The labelling function L : S → P(Atoms) (where P(Atoms) is the set of subsets of Atoms) gives us the encoding. We assume a fixed ordering on the set Atoms, say
x1 , x2 , . . . , xn , and then represent s ∈ S by the vector (v1 , v2 , . . . , vn ), where, for each i, vi equals 1 if xi ∈ L(s) and vi is 0 otherwise. In order to guarantee that each s has a
unique representation as a boolean vector, we require that, for all s1 , s2 ∈ S, L(s1 ) = L(s2 ) implies s1 = s2 . If this is not the case, perhaps because 2|Atoms| < |S|, we can add extra atomic
propositions in order to make enough distinctions (Cf. introduction of the turn variable for mutual exclusion in Section 3.3.4.) From now on, we refer to a state s ∈ S by its representing boolean
vector (v1 , v2 , . . . , vn ), where vi is 1 if xi ∈ L(s) and 0 otherwise. As an OBDD, this state is represented by the OBDD of the boolean function l1 · l2 · · · · · ln , where li is xi if xi ∈ L
(s) and xi otherwise. The set of states {s1 , s2 , . . . , sm } is represented by the OBDD of the boolean function (l11 · l12 · · · · · l1n ) + (l21 · l22 · · · · · l2n ) + · · · + (lm1 · lm2 · · · ·
· lmn ) where li1 · li2 · · · · · lin represents state si .
6 Binary decision diagrams
s1 s0
Figure 6.24. A simple CTL model (Example 6.12).
set of states ∅ {s0 } {s1 } {s2 } {s0 , s1 } {s0 , s2 } {s1 , s2 } S
representation by boolean values (1, 0) (0, 1) (0, 0) (1, 0), (1, 0), (0, 1), (1, 0),
(0, 1) (0, 0) (0, 0) (0, 1), (0, 0)
representation by boolean function 0 x1 · x2 x1 · x2 x1 · x2 x1 · x2 + x1 · x2 x1 · x2 + x1 · x2 x1 · x2 + x1 · x2 x1 · x2 + x1 · x2 + x1 · x2
Figure 6.25. Representation of subsets of states of the model of Figure 6.24.
The key point which makes this representation interesting is that the OBDD representing a set of states may be quite small. Example 6.12 Consider the CTL model in Figure 6.24, given by: def
S = {s0 , s1 , s2 } def → = {(s0 , s1 ), (s1 , s2 ), (s2 , s0 ), (s2 , s2 )} def L(s0 ) = {x1 } def L(s1 ) = {x2 } def L(s2 ) = ∅. Note that it has the property that, for all states s1 and s2 , L(s1
) = L(s2 ) implies s1 = s2 , i.e. a state is determined entirely by the atomic formulas true in it. Sets of states may be represented by boolean values and by boolean formulas with the ordering [x1 ,
x2 ], as shown in Figure 6.25. Notice that the vector (1, 1) and the corresponding function x1 · x2 are unused. Therefore, we are free to include it in the representation of a subset
6.3 Symbolic model checking
x2 0
Figure 6.26. Two OBDDs for the set {s0 , s1 } (Example 6.12).
of S or not; so we may choose to include it or not in order to optimise the size of the OBDD. For example, the subset {s0 , s1 } is better represented by the boolean function x1 + x2 , since its OBDD
is smaller than that for x1 · x2 + x1 · x2 (Figure 6.26). In order to justify the claim that the representation of subsets of S as OBDDs will be suitable for the algorithm presented in Section 3.6.1,
we need to look at how the operations on subsets which are used in that algorithm can be implemented in terms of the operations we have defined on OBDDs. The operations in that algorithm are: r
Intersection, union and complementation of subsets. It is clear that these are represented by the boolean functions ·, + and ¯ respectively. The implementation via OBDDs of · and + uses the apply
algorithm (Section 6.2.2). r The functions pre∃ (X) = {s ∈ S | exists s , (s → s and s ∈ X)} pre∀ (X) = {s | for all s , (s → s implies s ∈ X)}.
The function pre∃ (instrumental in SATEX and SATEU ) takes a subset X of states and returns the set of states which can make a transition into X. The function pre∀ , used in SATAF , takes a set X and
returns the set of states which can make a transition only into X. In order to see how these are implemented in terms of OBDDs, we need first to look at how the transition relation itself is
6.3.2 Representing the transition relation The transition relation → of a model M = (S, →, L) is a subset of S × S. We have already seen that subsets of a given finite set may be represented as OBDDs
by considering the characteristic function of a binary encoding. Just like in the case of subsets of S, the binary encoding is naturally given by the labelling function L. Since → is a subset of S ×
S, we need two copies of the boolean vectors. Thus, the link s → s is represented by the pair of
6 Binary decision diagrams
x1 x2 x1 x2 → 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0
x1 x1 x2 x2 → 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 0
Figure 6.27. The truth table for the transition relation of Figure 6.24 (see Example 6.13). The left version shows the ordering of variables [x1 , x2 , x1 , x2 ], while the right one orders the
variables [x1 , x1 , x2 , x2 ] (the rows are ordered lexicographically).
boolean vectors ((v1 , v2 , . . . , vn ), (v1 , v2 , . . . , vn )), where vi is 1 if pi ∈ L(s) and 0 otherwise; and similarly, vi is 1 if pi ∈ L(s ) and 0 otherwise. As an OBDD, the link is
represented by the OBDD for the boolean function (l1 · l2 · · · · · ln ) · (l1 · l2 · · · · · ln ) and a set of links (for example, the entire relation →) is the OBDD for the + of such formulas.
Example 6.13 To compute the OBDD for the transition relation of Figure 6.24, we first show it as a truth table (Figure 6.27 (left)). Each 1 in the final column corresponds to a link in the transition
relation and each 0 corresponds to the absence of a link. The boolean function is obtained by taking the disjunction of the rows having 1 in the last column and is f → = x1 · x2 · x1 · x2 + x1 · x2 ·
x1 · x2 + x1 · x2 · x1 · x2 + x1 · x2 · x1 · x2 . (6.5) It turns out that it is usually more efficient to interleave unprimed and primed variables in the OBDD variable ordering for →. We therefore use
6.3 Symbolic model checking
x1 x1
x1 x2
Figure 6.28. An OBDD for the transition relation of Example 6.13.
[x1 , x1 , x2 , x2 ] rather than [x1 , x2 , x1 , x2 ]. Figure 6.27 (right) shows the truth table redrawn with the interleaved ordering of the columns and the rows reordered lexicographically. The
resulting OBDD is shown in Figure 6.28.
6.3.3 Implementing the functions pre∃ and pre∀ It remains to show how an OBDD for pre∃ (X) and pre∀ (X) can be computed, given OBDDs BX for X and B→ for the transition relation →. First we observe
that pre∀ can be expressed in terms of complementation and pre∃ , as follows: pre∀ (X) = S − pre∃ (S − X), where we write S − Y for the set of all s ∈ S which are not in Y . Therefore, we need only
explain how to compute the OBDD for pre∃ (X) in terms of BX and B→ . Now (6.4) suggests that one should proceed as follows: 1. Rename the variables in BX to their primed versions; call the resulting
OBDD BX . 2. Compute the OBDD for exists(ˆ x , apply(·, B→ , BX )) using the apply and exists algorithms (Sections 6.2.2 and 6.2.4).
6.3.4 Synthesising OBDDs The method used in Example 6.13 for producing an OBDD for the transition relation was to compute first the truth table and then an OBDD which might not be in its fully reduced
form; hence the need for a final call to
6 Binary decision diagrams
the reduce function. However, this procedure would be unacceptable if applied to realistically sized systems with a large number of variables, for the truth table’s size is exponential in the number
of boolean variables. The key idea and attraction of applying OBDDs to finite systems is therefore to take a system description in a language such as SMV and to synthesise the OBDD directly, without
having to go via intermediate representations (such as binary decision trees or truth tables) which are exponential in size. SMV allows us to define the next value of a variable in terms of the
current values of variables (see the examples of code in Section 3.3.2)3 . This can be compiled into a set of boolean functions fi , one for each variable xi , which define the next value of xi in
terms of the current values of all the variables. In order to cope with non-deterministic assignment (such as the assignment to status in the example on page 192), we extend the set of variables by
adding unconstrained variables which model the input. Each xi is a deterministic function of this enlarged set of variables; thus, xi ↔ fi , where f ↔ g = 1 if, and only if, f and g compute the same
values, i.e. it is a shorthand for f ⊕ g. The boolean function representing the transition relation is therefore of the form xi ↔ fi , (6.6) 1≤i≤n
ranges where 1≤i≤n gi is a shorthand for g1 · g2 · . . . · gn . Note that the only over the non-input variables. So, if u is an input variable, the boolean function does not contain any u ↔ fu .
Figure 6.22 showed how the reduced OBDD could be computed from the parse tree of such a boolean function. Thus, it is possible to compile SMV programs into OBDDs such that their specifications can be
executed according to the pseudo-code of the function SAT, now interpreted over OBDDs. On page 396 we will see that this OBDD implementation can be extended to simple fairness constraints.
Modelling sequential circuits As a further application of OBDDs to verification, we show how OBDDs representing circuits may be synthesised. Synchronous circuits. Suppose that we have a design of a
sequential circuit such as the one in Figure 6.29. This is a synchronous circuit (meaning that 3
SMV also allows next values to be defined in terms of next values, i.e. the keyword next to appear in expressions on the right-hand side of :=. This is useful for describing synchronisations, for
example, but we ignore that feature here.
6.3 Symbolic model checking
x2 Figure 6.29. A simple synchronous circuit with two registers.
all the state variables are updated synchronously in parallel) whose functionality can be described by saying what the values of the registers x1 and x2 in the next state of the circuit are. The
function f → coding the possible next states of the circuits is (x1 ↔ x1 ) · (x2 ↔ x1 ⊕ x2 ).
This may now be translated into an OBDD by the methods summarised in Figure 6.22. Asynchronous circuits. The symbolic encoding of synchronous circuits is in its logical structure very similar to the
encoding of f → for CTL models; compare the codings in (6.7) and (6.6). In asynchronous circuits, or processes in SMV, the logical structure of f → changes. As before, we can construct functions fi
which code the possible next state in the local component, or the SMV process, i. For asynchronous systems, there are two principal ways of composing these functions into global system behaviour: r
In a simultaneous model, a global transition is one in which any number of components may make their local transition. This is modelled as f→ =
((xi ↔ fi ) + (xi ↔ xi )) .
r In an interleaving model, exactly one local component makes a local transition;
6 Binary decision diagrams
all other local components remain in their local state: n def (xi ↔ fi ) · (xj ↔ xj ) . f→ = i=1
Observe the duality in these approaches: the simultaneous model has an outer product, whereas the interleaving model has an outer sum. The latter, if used in ∃ˆ x .f (‘for some next state’), can be
optimised since sums distribute over existential quantification; in Chapter 2 this was the equivalence ∃x.(φ ∨ ψ) ≡ ∃x.φ ∨ ∃x.ψ. Thus, global states reachable in one step are the ‘union’ of all the
states reachable in one step in the local components; compare the formulas in (6.8) and (6.9) with (6.6).
6.4 A relational mu-calculus We saw in Section 3.7 that evaluating the set of states satisfying a CTL formula in a model may involve the computation of a fixed point of an operator. For example, [[EF
φ]] is the least fixed point of the operator F : P(S) → P(S) given by F (X) = [[φ]] ∪ pre∃ (X). In this section, we introduce a syntax for referring to fixed points in the context of boolean formulas.
Fixed-point invariants frequently occur in all sorts of applications (for example, the common-knowledge operator CG in Chapter 5), so it makes sense to have an intermediate language for expressing
such invariants syntactically. This language also provides a formalism for describing interactions and dependences of such invariants. We will see shortly that symbolic model checking in the presence
of simple fairness constraints exhibits such more complex relationships between invariants.
6.4.1 Syntax and semantics Definition 6.14 The formulas of the relational mu-calculus are given by the grammar v ::= x | Z f ::= 0 | 1 | v | f | f1 + f2 | f1 · f2 | f1 ⊕ f2 | ∃x.f | ∀x.f | µZ.f |
νZ.f | f [ˆ x :=
x ˆ ]
where x and Z are boolean variables, and x ˆ is a tuple of variables. In the formulas µZ.f and νZ.f , any occurrence of Z in f is required to fall within an even number of complementation symbols ¯;
such an f is said to be formally monotone in Z. (In exercise 7 on page 410 we consider what happens if we do not require formal monotonicity.)
6.4 A relational mu-calculus
Convention 6.15 The binding priorities for the grammar in (6.10) are that ¯, and [ˆ x := x ˆ ] have the highest priority; followed by ∃x and ∀y; then µZ and νZ; followed by · . The operators + and ⊕
have the lowest binding priority. The symbols µ and ν are called least fixed-point and greatest fixed-point operators, respectively. In the formula µZ.f , the interesting case is that in which f
contains an occurrence of Z. In that case, f can be thought of as a function, taking Z to f . The formula µZ.f is intended to mean the least fixed point of that function. Similarly, νZ.f is the
greatest fixed point of the function. We will see how this is done in the semantics. The formula f [ˆ x := x ˆ ] expresses an explicit substitution which forces f to be evaluated using the values of
xi rather than xi . (Recall that the primed variables refer to the next state.) Thus, this syntactic form is not a metaoperation denoting a substitution, but an explicit syntactic form in its own
right. The substitution will be made on the semantic side, not the syntactic side. This difference will become clear when we present the semantics of . A valuation ρ for f is an assignment of values 0
or 1 to all variables v. We define a satisfaction relation ρ f inductively over the structure of such formulas f , given a valuation ρ. Definition 6.16 Let ρ be a valuation and v a variable. We write
ρ(v) for the value of v assigned by ρ. We define ρ[v → 0] to be the updated valuation which assigns 0 to v and ρ(w) to all other variables w. Dually, ρ[v → 1] assigns 1 to v and ρ(w) to all other
variables w. For example, if ρ is the valuation represented by (x, y, Z) ⇒ (1, 0, 1) – meaning that ρ(x) = 1, ρ(y) = 0, ρ(Z) = 1 and ρ(v) = 0 for all other variables v – then ρ[x → 0] is represented
by (x, y, Z) ⇒ (0, 0, 1), whereas ρ[Z → 0] is (x, y, Z) ⇒ (1, 0, 0). The assumption that valuations assign values to all variables is rather mathematical, but avoids some complications which have to
be addressed in implementations (see exercise 3 on page 409). Updated valuations allow us to define the satisfaction relation for all formulas without fixed points: Definition 6.17 We define a
satisfaction relation ρ f for formulas f without fixed-point subformulas with respect to a valuation ρ by structural induction: r ρ 0 r ρ1 r ρ v iff ρ(v) equals 1
392 r r r r r r r
6 Binary decision diagrams
ρ f iff ρ f ρ f + g iff ρ f or ρ g ρ f · g iff ρ f and ρ g ρ f ⊕ g iff ρ (f · g + f · g) ρ ∃x.f iff ρ[x → 0] f or ρ[x → 1] f ρ ∀x.f iff ρ[x → 0] f and ρ[x → 1] f ρ f [ˆ x := x ˆ ] iff ρ[ˆ x := x ˆ ] f ,
where ρ[ˆ x := x ˆ ] is the valuation which assigns the same values as ρ, but for each xi it assigns ρ(xi ). The semantics of boolean quantification closely resembles the one for the quantifiers of
predicate logic. The crucial difference, however, is that boolean formulas are only interpreted over the fixed universe of values {0, 1}, whereas predicate formulas may take on values in all sorts of
finite or infinite models. Example 6.18 Let ρ be such that ρ(x1 ) equals 0 and ρ(x2 ) is 1. We evaluate x := x ˆ ] which holds iff ρ[ˆ x := x ˆ ] (x1 + x2 ). Thus, we need ρ (x1 + x2 )[ˆ ρ[ˆ x := x ˆ ]
x1 or ρ[ˆ x := x ˆ ] x2 to be the case. Now, ρ[ˆ x := x ˆ ] x1 cannot be, for this would mean that ρ(x1 ) equals 1. Since ρ[ˆ x := x ˆ ] x2 would x := x ˆ ] x2 because ρ(x2 ) equals imply that ρ[ˆ x
:= x ˆ ] x2 , we infer that ρ[ˆ x := x ˆ ]. 1. In summary, we demonstrated that ρ (x1 + x2 )[ˆ We now extend the definition of to the fixed-point operators µ and ν. Their semantics will have to reflect
their meaning as least, respectively greatest, fixed-point operators. We define the semantics of µZ.f via its syntactic approximants which unfold the meaning of µZ.f : def
µ0 Z.f = 0 def µm+1 Z.f = f [µm Z.f /Z]
(m ≥ 0).
The unfolding is achieved by a meta-operation [g/Z] which, when applied to a formula f , replaces all free occurrences of Z in f with g. Thus, we view µZ as a binding construct similar to the
quantifiers ∀x and ∃x, and [g/Z] is similar to the substitution [t/x] in predicate logic. For example, (x1 + ∃x2 .(Z · x2 ))[x1 /Z] is the formula x1 + ∃x2 .(x1 · x2 ), whereas ((µZ.x1 + Z) · (x1 +
∃x2 .(Z · x2 )))[x1 /Z] equals (µZ.x1 + Z) · (x1 + ∃x2 .(x1 · x2 )). See exercise 3 on page 409 for a formal account of this meta-operation. With these approximants we can define: ρ µZ.f iff (ρ µm Z.f
for some m ≥ 0).
6.4 A relational mu-calculus
Thus, to determine whether µZ.f is true with respect to a valuation ρ, we have to find some m ≥ 0 such that ρ µm Z.f holds. A sensible strategy is to try to prove this for the smallest such m
possible, if indeed such an m can be found. For example, in attempting to show ρ µZ.Z, we try ρ µ0 Z.Z, which fails since the latter formula is just 0. Now, µ1 Z.Z is defined to be Z[µ0 Z.Z/Z] which
is just µ0 Z.Z again. We can now use mathematical induction on m ≥ 0 to show that µm Z.Z equals µ0 Z.Z for all m ≥ 0. By (6.12), this implies ρ µZ.Z. The semantics for νZ.f is similar. First, let us
define a family of approximants ν0 Z.f , ν1 Z.f , . . . by def
ν0 Z.f = 1 def νm+1 Z.f = f [νm Z.f /Z]
(m ≥ 0).
Note that this definition only differs from the one for µm Z.f in that the first approximant is defined to be 1 instead of 0. Recall how the greatest fixed point for EG φ requires that φ holds on all
states of some path. Such invariant behaviour cannot be expressed with a condition such as in (6.12), but is adequately defined by demanding that ρ νZ.f iff (ρ νm Z.f for all m ≥ 0).
A dual reasoning to the above shows that ρ νZ.Z holds, regardless of the nature of ρ. One informal way of understanding the definitions in (6.12) and (6.14) is that ρ µZ.f is false until, and if, it
is proven to hold; whereas ρ νZ.f is true until, and if, it is proven to be false. The temporal aspect is encoded by the unfolding of the recursion in (6.11), or in (6.13). To prove that this
recursive way of specifying ρ f actually is well defined, one has to consider more general forms of induction which keep track not only of the height of f ’s parse tree, but also of the number of
syntactic approximants µm Z.g and νn Z.h, their ‘degree’ (in this case, m and n), as well as their ‘alternation’ (the body of a fixed point may contain a free occurrence of a variable for a recursion
higher up in the parse tree). This can be done, though we won’t discuss the details here.
6.4.2 Coding CTL models and specifications Given a CTL model M = (S, →, L), the µ and ν operators permit us to translate any CTL formula φ into a formula, f φ , of the relational mu-calculus such
that f φ represents the set of states s ∈ S with s φ. Since we already saw how to represent subsets of states as such formulas, we can then capture
6 Binary decision diagrams
the model-checking problem ?
M, I φ
of whether all initial states s ∈ I satisfy φ, in purely symbolic form: we φ answer in the affirmative if f I · f is unsatisfiable, where f I is the characφ teristic function of I ⊆ S. Otherwise, the
logical structure of f I · f may be exploited to extract debugging information for correcting the model M in order to make (6.15) true. Recall how we can represent the transition relation → as a
boolean formula f → (see Section 6.3.2). As before, we assume that states are coded as bit vectors (v1 , v2 , . . . , vn ) and so the free boolean variables of all functions f φ are subsumed by the
vector x ˆ. The coding of the CTL formula φ as a φ function f in the relational mu-calculus is now given inductively as follows: f x = x for variables x def
f⊥ = 0 def
f ¬φ = f φ def
f φ∧ψ = f φ · f ψ def
f EX φ = ∃ˆ x . (f → · f φ [ˆ x := x ˆ ]). def
The clause for EX deserves explanation. The variables xi refer to the current state, whereas xi refer to the next state. The semantics of CTL says that s EX φ if, and only if, there is some s with s
→ s and s φ. The boolean formula encodes this definition, computing 1 precisely when this is the case. If x ˆ models the current state s, then x ˆ models a possible successor → state if f , a function
in (ˆ x, x ˆ ), holds. We use the nested boolean quantifier ∃ˆ x in order to say ‘there is some successor state.’ Observe also the desired effect of [ˆ x := x ˆ ] performed on f φ , thereby ‘forcing’ φ
to be true at some next state4 . The clause for EF is more complicated and involves the µ operator. Recall the equivalence EF φ ≡ φ ∨ EX EF φ.
Exercise 6 on page 409 should give you a feel for how the semantics of f [ˆ x := x ˆ ] does not interfere with potential ∃ˆ x or ∀ˆ x quantifiers within f . For example, to evaluate ρ (∃ˆ x .f )[ˆ x
:= x ˆ ], we evaluate ρ[ˆ x := x ˆ ] ∃ˆ x .f , which is true if we can find some values (v1 , v2 , . . . , vn ) ∈ {0, 1}n such that ρ[ˆ x := x ˆ ][x1 → v1 ][x2 → v2 ] . . . [xn → vn ] f is true.
Observe that the resulting environment binds all xi to vi , but for all other values it binds them according to ρ[ˆ x := x ˆ ]; since the latter binds xi to ρ(xi ) which is the ‘old’ value of xi ,
this is exactly what we desire in order to prevent a clash of variable names with the intended semantics. Recall that an OBDD implementation synthesises formulas in a bottom-up fashion, so a reduced
OBDD for ∃ˆ x .f will not contain any xi nodes as its function does not depend on those variables. Thus, OBDDs also avoid such name clash problems.
6.4 A relational mu-calculus
Therefore, f EF φ has to be equivalent to f φ + f EX EF φ which in turn is equivx . (f → · f EF φ [ˆ x := x ˆ ]). Now, since EF involves computing alent to f φ + ∃ˆ the least fixed point of the
operator derived from the Equivalence (6.16), we obtain f EF φ = µZ. (f φ + ∃ˆ x . (f → · Z[ˆ x := x ˆ ])). def
Note that the substitution Z[ˆ x := x ˆ ] means that the boolean function Z should be made to depend on the xi variables, rather than the xi variables. This is because the evaluation of ρ Z[ˆ x := x
ˆ ] results in ρ[ˆ x := x ˆ ] Z, where the latter valuation satisfies ρ[ˆ x := x ˆ ](xi ) = ρ(xi ). Then, we use the modified valuation ρ[ˆ x := x ˆ ] to evaluate Z. Since EF φ is equivalent to E[ U
φ], we can generalise our coding of EF φ accordingly: x . (f → · Z[ˆ x := x ˆ ])). f E[φUψ] = µZ. (f ψ + f φ · ∃ˆ def
The coding of AF is similar to the one for EF in (6.17), except that ‘for some’ (boolean quantification ∃ˆ x ) gets replaced by ‘for all’ (boolean quantification ∀ˆ x ) and the ‘conjunction’ f → · Z[ˆ
x := x ˆ ] turns into the ‘implication’ f → + Z[ˆ x := x ˆ ]: f AF φ = µZ. (f φ + ∀ˆ x . (f → + Z[ˆ x := x ˆ ])). def
Notice how the semantics of µZ.f in (6.12) reflects the intended meaning of the AF connective. The mth approximant of f AF φ , which we write as AF φ , represents those states where all paths reach a
φ-state within m steps. fm This leaves us with coding EG, for then we have provided such a coding for an adequate fragment of CTL (recall Theorem 3.17 on page 216). Because EG involves computing
greatest fixed points, we make use of the ν operator: f EG φ = νZ. (f φ · ∃ˆ x . (f → · Z[ˆ x := x ˆ ])). def
Observe that this does follow the logical structure of the semantics of EG: we need to show φ in the present state and then we have to find some successor state satisfying EG φ. The crucial point is
that this obligation never ceases; this is exactly what we ensured in (6.14). Let us see these codings in action on the model of Figure 6.24. We want to perform a symbolic model check of the formula
EX (x1 ∨ ¬x2 ). You should verify, using e.g. the labelling algorithm from Chapter 3, that [[EX (x1 ∨ ¬x2 )]] = {s1 , s2 }. Our claim is that this set is computed symbolically by the resulting
formula f EX (x1 ∨¬x2 ) . First, we compute the formula
6 Binary decision diagrams
f → which represents the transition relation →: f → = (x1 ↔ x1 · x2 · u) · (x2 ↔ x1 ) where u is an input variable used to model the non-determinism (compare the form (6.6) for the transition
relation in Section 6.3.4). Thus, we obtain x := x ˆ ]) f EX (x1 ∨¬x2 ) = ∃x1 .∃x2 .(f → · f x1 ∨¬x2 [ˆ = ∃x1 .∃x2 .((x1 ↔ x1 · x2 · u) · (x2 ↔ x1 ) · (x1 + x2 )). To see whether s0 satisfies EX (x1 ∨
¬x2 ), we evaluate ρ0 f EX (x1 ∨¬x2 ) , where ρ0 (x1 ) = 1 and ρ0 (x2 ) = 0 (the value of ρ0 (u) does not matter). We find that this does not hold, whence s0 EX (x1 ∨ ¬x2 ). Likewise, we verify s1 EX
(x1 ∨ ¬x2 ) by showing ρ1 f EX (x1 ∨¬x2 ) ; and s2 EX (x1 ∨ ¬x2 ) by showing ρ2 f EX (x1 ∨¬x2 ) , where ρi is the valuation representing state si . As a second example, we compute f AF (¬x1 ∧¬x2 )
for the model in Figure 6.24. First, note that all three5 states satisfy AF (¬x1 ∧ ¬x2 ), if we apply the labelling algorithm to the explicit model. Let us verify that the symbolic encoding matches
this result. By (6.19), we have that f AF (¬x1 ∧¬x2 ) equals x := x ˆ ] . (6.21) µZ. (x1 · x2 ) + ∀x1 .∀x2 .(x1 ↔ x1 · x2 · u) · (x2 ↔ x1 ) · Z[ˆ AF (¬x ∧¬x )
1 2 for some m ≥ 0. By (6.12), we have ρ f AF (¬x1 ∧¬x2 ) iff ρ fm AF (¬x1 ∧¬x2 ) AF (¬x1 ∧¬x2 ) Clearly, we have ρ f0 . Now, f1 equals
x := x ˆ ])[0/Z]. ((x1 · x2 ) + ∀x1 .∀x2 .(x1 ↔ x1 · x2 · u) · (x2 ↔ x1 ) · Z[ˆ Since [0/Z] is a meta-operation, the latter formula is just x := x ˆ ]. (x1 · x2 ) + ∀x1 .∀x2 .(x1 ↔ x1 · x2 · u) · (x2
↔ x1 ) · 0[ˆ Thus, we need to evaluate the disjunction (x1 · x2 ) + ∀x1 .∀x2 .(x1 ↔ x1 · x2 · x := x ˆ ] at ρ. In particular, if ρ(x1 ) = 0 and ρ(x2 ) = 0, u) · (x2 ↔ x1 ) · 0[ˆ then ρ x1 · x2 and so
ρ (x1 · x2 ) + ∀x1 .∀x2 .(x1 ↔ x1 · x2 · u) · (x2 ↔ x1 ) · 0[ˆ x := x ˆ ]. Thus, s2 AF (¬x1 ∧ ¬x2 ) holds. Similar reasoning establishes that the formula in (6.21) renders a correct coding for the
remaining two states as well, which you are invited to verify as an exercise. Symbolic model checking with fairness In Chapter 3, we sketched how SMV could use fairness assumptions which were not
expressible entirely 5
Since we have added the variable u, there are actually six states; they all satisfy the formula.
6.4 A relational mu-calculus
within CTL and its semantics. The addition of fairness could be achieved by restricting the ordinary CTL semantics to fair computation paths, or fair states. Formally, we were given a set C = {ψ1 ,
ψ2 , . . . , ψk } of CTL formulas, called the fairness constraints, and we wanted to check whether s φ holds for a CTL formula φ and all initial states s, with the additional fairness constraints in
C. Since ⊥, ¬, ∧, EX, EU and EG form an adequate set of connectives for CTL, we may restrict this discussion to only these operators. Clearly, the propositional connectives won’t change their meaning
with the addition of fairness constraints. Therefore, it suffices to provide symbolic codings for the fair connectives EC X, EC U and EC G from Chapter 3. The key is to represent the set of fair states
symbolically as a boolean formula fair defined as def
fair = f EC G
which uses the (yet to be defined) function f EC G φ with as an instance. Assuming that the coding of f EC G φ is correct, we see that fair computes 1 in a state s if, and only if, there is a fair
path with respect to C that begins in s. We say that such an s is a fair state. As for EC X, note that s EC Xφ if, and only if, there is some next state s with s → s and s φ such that s is a fair
state. This immediately renders x .(f → · (f φ · fair)[ˆ x := x ˆ ]). f EC Xφ = ∃ˆ def
Similarly, we obtain x . (f → · Z[ˆ x := x ˆ ])). f EC [φ1 Uφ2 ] = µZ. (f φ2 · fair + f φ1 · ∃ˆ def
This leaves us with the task of coding f EC G φ . It is this last connective which reveals the complexity of fairness checks at work. Because the coding of f EC G φ is rather complex, we proceed in
steps. It is convenient to have the EX and EU functionality also at the level of boolean formulas directly. For example, if f is a boolean function in x ˆ, then checkEX (f ) codes the boolean formula
which computes 1 for those vectors x ˆ which have a next state x ˆ for which f computes 1: x .(f → · f [ˆ x := x ˆ ]). checkEX (f ) = ∃ˆ def
Thus, f EC Xφ equals checkEX (f φ · fair). We proceed in the same way for functions f and g in n arguments x ˆ to obtain checkEU (f, g) which computes
6 Binary decision diagrams
1 at x ˆ if there is a path that realises the f U g pattern: def
checkEU (f, g) = µY.g + (f · checkEX(Y )).
With this in place, we can code f EC Gφ quite easily: f EC Gφ = νZ.f φ · def
checkEX (checkEU (f φ , Z · f ψi ) · fair).
Note that this coding has a least fixed point (checkEU) in the body of a greatest fixed point. This is computationally rather involved since the call of checkEU contains Z, the recursion variable of
the outer greatest fixed point, as a free variable; thus these recursions are nested and inter-dependent; the recursions ‘alternate.’ Observe how this coding operates: to have a fair path from x ˆ on
which φ holds globally, we need φ to hold at x ˆ; and for ˆ , where the whole all fairness constraints ψi there has to be a next state x property is true again (enforced by the free Z) and each
fairness constraint is realised eventually on that path. The recursion in Z constantly reiterates this reasoning, so if this function computes 1, then there is a path on which φ holds globally and
where each ψi is true infinitely often.
6.5 Exercises Exercises 6.1 1. Write down the truth tables for the boolean formulas in Example 6.2 on page 359. In your table, you may use 0 and 1, or F and T, whatever you prefer. What truth value
does the boolean formula of item (4) on page 359 compute? def 2. ⊕ is the exclusive-or: x ⊕ y = 1 if the values of x and y are different; otherwise, def x ⊕ y = 0. Express this in propositional logic,
i.e. find a formula φ having the same truth table as ⊕. * 3. Write down a boolean formula f (x, y) in terms of ·, +, ¯, 0 and 1, such that f has the same truth table as p → q. 4. Write down a BNF for
the syntax of boolean formulas based on the operations in Definition 6.1.
Exercises 6.2 * 1. Suppose we swap all dashed and solid lines in the binary decision tree of Figure 6.2. Write out the truth table of the resulting binary decision tree and find a formula for it.
6.5 Exercises
* 2. Consider the following truth table: p T T T T F F F F
q T T F F T T F F
r φ T T F F T F F F T T F F T T F F
Write down a binary decision tree which represents the boolean function specified in this truth table. 3. Construct a binary decision tree for the boolean function specified in Figure 6.2, but now the
root should be a y-node and its two successors should be x-nodes. 4. Consider the following boolean function given by its truth table: x y z f (x, y, z) 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1
0 0 0 0 1 1 0 0 0 (a) Construct a binary decision tree for f (x, y, z) such that the root is an x-node followed by y- and then z-nodes. (b) Construct another binary decision tree for f (x, y, z), but
now let its root be a z-node followed by y- and then x-nodes. 5. Let T be a binary decision tree for a boolean function f (x1 , x2 , . . . , xn ) of n boolean variables. Suppose that every variable
occurs exactly once as one travels down on any path of the tree T . Use mathematical induction to show that T has 2n+1 − 1 nodes.
Exercises 6.3 * 1. Explain why all reductions C1–C3 (page 363) on a BDD B result in BDDs which still represent the same function as B. 2. Consider the BDD in Figure 6.7. * (a) Specify the truth table
for the boolean function f (x, y, z) represented by this BDD.
6 Binary decision diagrams
(b) Find a BDD for that function which does not have multiple occurrences of variables along any path. 3. Let f be the function represented by the BDD of Figure 6.3(b). Using also the BDDs B0 , B1
and Bx illustrated in Figure 6.6, find BDDs representing (a) f · x (b) x + f (c) f · 0 (d) f · 1.
Exercises 6.4 1. Figure 6.9 (page 367) shows a BDD with ordering [x, y, z]. * (a) Find an equivalent reduced BDD with ordering [z, y, x]. (Hint: find first the decision tree with the ordering [z, y,
x], and then reduce it using C1–C3.) (b) Carry out the same construction process for the variable ordering [y, z, x]. Does the reduced BDD have more or fewer nodes than the ones for the orderings [x,
y, z] and [z, y, x]? 2. Consider the BDDs in Figures 6.4–6.10. Determine which of them are OBDDs. If you find an OBDD, you need to specify a list of its boolean variables without double occurrences
which demonstrates that ordering. 3. Consider the following boolean formulas. Compute their unique reduced OBDDs with respect to the ordering [x, y, z]. It is advisable to first compute a binary
decision tree and then to perform the removal of redundancies. def (a) f (x, y) = x · y def * (b) f (x, y) = x + y def (c) f (x, y) = x ⊕ y def * (d) f (x, y, z) = (x ⊕ y) · (x + z). 4. Recall the
derived connective φ ↔ ψ from Chapter 1 saying that for all valuations φ is true if, and only if, ψ is true. (a) Define this operator for boolean formulas using the basic operations ·, +, ⊕ and ¯ from
Definition 6.1. def (b) Draw a reduced OBDD for the formula g(x, y) = x ↔ y using the ordering [y, x]. 5. Consider the even parity function introduced at the end of the last section. (a) Define the odd
parity function fodd (x1 , x2 , . . . , xn ). (b) Draw an OBDD for the odd parity function for n = 5 and the ordering [x3 , x5 , x1 , x4 , x2 ]. Would the overall structure of this OBDD change if you
changed the ordering? (c) Show that feven (x1 , x2 , . . . , xn ) and fodd (x1 , x2 , . . . , xn ) denote the same boolean function. 6. Use Theorem 6.7 (page 368) to show that, if the reductions
C1–C3 are applied until no more reduction is possible, the result is independent of the order in which they were applied.
6.5 Exercises
Exercises 6.5 1. Given the boolean formula f (x1 , x2 , x3 ) = x1 · (x2 + x3 ), compute its reduced OBDD for the following orderings: (a) [x1 , x2 , x3 ] (b) [x3 , x1 , x2 ] (c) [x3 , x2 , x1 ]. 2.
Compute the reduced OBDD for f (x, y, z) = x · (z + z) + y · x in any ordering you like. Is there a z-node in that reduced OBDD? def 3. Consider the boolean formula f (x, y, z) = (x + y + z) · (x + y
+ z) · (x + y). For the variable orderings below, compute the (unique) reduced OBDD Bf of f with respect to that ordering. It is best to write down the binary decision tree for that ordering and then
to apply all possible reductions. (a) [x, y, z]. (b) [y, x, z]. (c) [z, x, y]. (d) Find an ordering of variables for which the resulting reduced OBDD Bf has a minimal number of edges; i.e. there is
no ordering for which the corresponding Bf has fewer edges. (How many possible orderings for x, y and z are there?) 4. Given the truth table def
x y 1 1 1 1 1 0 1 0 0 1 0 1 0 0 0 0
z 1 0 1 0 1 0 1 0
f (x, y, z) 0 1 1 0 0 1 0 1
compute the reduced OBDD with respect to the following ordering of variables: (a) [x, y, z] (b) [z, y, x] (c) [y, z, x] (d) [x, z, y]. 5. Given the ordering [p, q, r], compute the reduced BDDs for p
∧ (q ∨ r) and (p ∧ q) ∨ (p ∧ r) and explain why they are identical. * 6. Consider the BDD in Figure 6.11 (page 370). (a) Construct its truth table. (b) Compute its conjunctive normal form. (c)
Compare the length of that normal form with the size of the BDD. What is your assessment?
6 Binary decision diagrams
Exercises 6.6 1. Perform the execution of reduce on the following OBDDs: (a) The binary decision tree for i. x ⊕ y ii. x · y iii. x + y iv. x ↔ y. (b) The OBDD in Figure 6.2 (page 361). * (c) The
OBDD in Figure 6.4 (page 363).
Exercises 6.7 1. Recall the Shannon expansion in (6.1) on page 374. Suppose that x does not occur in f at all. Why does (6.1) still hold? def 2. Let f (x, y, z) = y + z · x + z · y + y · x be a
boolean formula. Compute f ’s Shannon expansion with respect to (a) x (b) y (c) z. 3. Show that boolean formulas f and g are semantically equivalent if, and only if, the boolean formula (f + g) · (f
+ g) computes 1 for all possible assignments of 0s and 1s to their variables. 4. We may use the Shannon expansion to define formally how BDDs determine boolean functions. Let B be a BDD. It is
intuitively clear that B determines a unique boolean function. Formally, we compute a function fn inductively (bottom-up) for all nodes n of B: – If n is a terminal node labelled 0, then fn is the
constant 0 function. – Dually, if n is a terminal 1-node, then fn is the constant 1 function. – If n is a non-terminal node labelled x, then we already have defined the boolean functions flo(n) and
fhi(n) and set fn to be x · flo(n) + x · fhi(n) . If i is the initial node of B, then fi is the boolean function represented by B. Observe that we could apply this definition as a symbolic evaluation
of B resulting in a boolean formula. For example, the BDD of Figure 6.3(b) renders x · (y · 1 + y · 0) + x · 0. Compute the boolean formulas obtained in this way for the following BDDs: (a) the BDD
in Figure 6.5(b) (page 364) (b) the BDDs in Figure 6.6 (page 365) (c) the BDD in Figure 6.11 (page 370). * 5. Consider a ternary (= takes three arguments) boolean connective f → (g, h) which is
equivalent to g when f is true; otherwise, it is equivalent to h. (a) Define this connective using any of the operators +, ·, ⊕ or ¯. (b) Recall exercise 4. Use the ternary operator above to write fn
as an expression of flo(n) , fhi(n) and its label x.
6.5 Exercises
y x
Figure 6.30. The reduced OBDDs Bf and Bg (see exercises).
6. 7. 8.
* 10.
(c) Use mathematical induction (on what?) to prove that, if the root of fn is an x-node, then fn is independent of any y which comes before x in an assumed variable ordering. Explain why apply (op,
Bf , Bg ), where Bf and Bg have compatible ordering, produces an OBDD with an ordering compatible with that of Bf and Bg . Explain why the four cases of the control structure for apply are
exhaustive, i.e. there are no other possible cases in its execution. Consider the reduced OBDDs Bf and Bg in Figure 6.30. Recall that, in order to compute the reduced OBDD for f op g, you need to –
construct the tree showing the recursive descent of apply (op, Bf , Bg ) as done in Figure 6.16; – use that tree to simulate apply (op, Bf , Bg ); and – reduce, if necessary, the resulting OBDD.
Perform these steps on the OBDDs of Figure 6.30 for the operation ‘op’ being (a) + (b) ⊕ (c) · Let Bf be the OBDD in Figure 6.11 (page 370). Compute apply (⊕, Bf , B1 ) and reduce the resulting OBDD.
If you did everything correctly, then this OBDD should be isomorphic to the one obtained from swapping 0- and 1-nodes in Figure 6.11. Consider the OBDD Bc in Figure 6.31 which represents the ‘don’t
care’ conditions for comparing the boolean functions f and g represented in Figure 6.30. This means that we want to compare whether f and g are equal for all values of variables except those for
which c is true (i.e. we ‘don’t care’ when c is true). (a) Show that the boolean formula (f ⊕ g) + c is valid (always computes 1) if, and only if, f and g are equivalent on all values for which c
evaluates to 0.
6 Binary decision diagrams
y z
Figure 6.31. The reduced OBDD Bc representing the ‘don’t care’ conditions for the equivalence test of the OBDDs in Figure 6.30. (b) Proceed in three steps as in exercise 8 on page 403 to compute the
reduced OBDD for (f ⊕ g) + c from the OBDDs for f , g and c. Which call to apply needs to be first? 11. We say that v ∈ {0, 1} is a (left)-controlling value for the operation op, if either v op x = 1
or v op x = 0 for all values of x. We say that v is a controlling value if it is a left- and right-controlling value. (a) Define the notion of a right-controlling value. (b) Give examples of
operations with controlling values. (c) Describe informally how apply can be optimised when op has a controlling value. (d) Could one still do some optimisation if op had only a left- or
right-controlling value? 12. We showed that the worst-time complexity of apply is O(|Bf | · |Bg |). Show that this upper bound is hard, i.e. it cannot be improved: def (a) Consider the functions f
(x1 , x2 , . . . , x2n+2m ) = x1 · xn+m+1 + · · · + xn · def x2n+m and g(x1 , x2 , . . . , x2n+2m ) = xn+1 · x2n+m+1 + · · · + xn+m · x2n+2m which are in sum-of-product form. Compute the
sum-of-product form of f + g. (b) Choose the ordering [x1 , x2 , . . . , x2n+2m ] and argue that the OBDDs Bf and Bg have 2n+1 and 2m+1 edges, respectively. (c) Use the result from part (a) to
conclude that Bf +g has 2n+m+1 edges, i.e. 0.5 · |Bf | · |Bg |.
Exercises 6.8 1. Let f be the reduced OBDD represented in Figure 6.5(b) (page 364). Compute the reduced OBDD for the restrictions: (a) f [0/x] * (b) f [1/x]
6.5 Exercises
(c) f [1/y] * (d) f [0/z]. * 2. Suppose that we intend to modify the algorithm restrict so that it is capable of computing reduced OBDDs for a general composition f [g/x]. (a) Generalise Equation
(6.1) to reflect the intuitive meaning of the operation [g/x]. (b) What fact about OBDDs causes problems for computing this composition directly? (c) How can we compute this composition given the
algorithms discussed so far? 3. We define read-1-BDDs as BDDs B where each boolean variable occurs at most once on any evaluation path of B. In particular, read-1-BDDs need not possess an ordering on
their boolean variables. Clearly, every OBDD is a read-1-BDD; but not every read-1-BDD is an OBDD (see Figure 6.10). In Figure 6.18 we see a BDD which is not a read-1-BDD; the path for (x, y, z) ⇒
(1, 0, 1) ‘reads’ the value of x twice. Critically assess the implementation of boolean formulas via OBDDs to see which implementation details could be carried out for read-1-BDDs as well. Which
implementation aspects would be problematic? 4. (For those who have had a course on finite automata.) Every boolean function f in n arguments can be viewed as a subset Lf of {0, 1}n ; defined to be the
set of all those bit vectors (v1 , v2 , . . . , vn ) for which f computes 1. Since this is a finite set, Lf is a regular language and has therefore a deterministic finite automaton with a minimal
number of states which accepts Lf . Can you match some of the OBDD operations with those known for finite automata? How close is the correspondence? (You may have to consider non-reduced OBDDs.) 5.
(a) Show that every boolean function in n arguments can be represented as a boolean formula of the grammar f ::= 0 | x | f | f1 + f2 . (b) Why does this also imply that every such function can be
represented by a reduced OBDD in any variable ordering? n 6. Use mathematical induction on n to prove that there are exactly 2(2 ) many different boolean functions in n arguments.
Exercises 6.9 1. Use the exists algorithm to compute the OBDDs for (a) ∃x3 .f , given the OBDD for f in Figure 6.11 (page 370) (b) ∀y.g, given the OBDD for g in Figure 6.9 (page 367) (c) ∃x2 .∃x3 .x1
· y1 + x2 · y2 + x3 · y3 . 2. Let f be a boolean function depending on n variables. (a) Show:
6 Binary decision diagrams
i. The formula ∃x.f depends on all those variables that f depends upon, except x. ii. If f computes to 1 with respect to a valuation ρ, then ∃x.f computes 1 with respect to the same valuation. iii.
If ∃x.f computes to 1 with respect to a valuation ρ, then there is a valuation ρ for f which agrees with ρ for all variables other than x such that f computes to 1 under ρ . (b) Can the statements
above be shown for the function value 0? 3. Let φ be a boolean formula. * (a) Show that φ is satisfiable if, and only if, ∃x.φ is satisfiable. (b) Show that φ is valid if, and only if, ∀x.φ is valid.
(c) Generalise the two facts above to nested quantifications ∃ˆ x and ∀ˆ x. (Use induction on the number of quantified variables.) x.f are semantically equivalent. Use induction on the 4. Show that ∀ˆ
x.f and ∃ˆ number of arguments in the vector x ˆ.
Exercises 6.10 (For those who know about complexity classes.) 1. Show that 3SAT can be reduced to nested existential boolean quantification. Given an instance of 3SAT, we may think of it as a boolean
formula f in productof-sums form g1 · g2 · · · · · gn , where each gi is of the form (l1 + l2 + l3 ) with each lj being a boolean variable or its complementation. For example, f could be (x + y + z)
· (x5 + x + x7 ) · (x2 + z + x) · (x4 + x2 + x4 ). (a) Show that you can represent each function gi with an OBDD of no more than three non-terminals, independently of the chosen ordering.
(b) Introduce n new boolean variables z1 , z2 , . . . , zn . We write 1≤i≤n fi for the expression f1 + f2 + · · · + fn and 1≤i≤n fi for f1 · f2 · · · · · fn . Consider the boolean formula h, defined
as g i · zi · (6.28) zj . 1≤i≤n | {"url":"https://silo.pub/logic-in-computer-science.html","timestamp":"2024-11-07T19:02:12Z","content_type":"text/html","content_length":"720858","record_id":"<urn:uuid:354e0355-1cd6-450a-8d25-a21989ad8986>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00093.warc.gz"} |
Wave Equation
SOCR ≫ TCIU Website ≫ TCIU GitHub ≫
Problem setup
One approach to understand wave dynamics in spacekime is to plot the basic separable solution. Recall the 5D wave equation. \[ \Delta_x u = \Delta_t u, \text{ where } x=(x_1,x_2,x_3)\in\mathbb{R}^3,
t=(t_1,t_2)\in\mathbb{R}^2\] The equation is essentially a ultrahyperbolic equation, and in general, if we do not impose a proper condition, the equation does not permit stable and unique and global
solutions. One possible approach to resolve these instabilities of the potential functions is to impose Periodic Boundary conditions (PBC) and consider the corresponding base function solutions. In
general, the basis takes the form \[u(x,t)=e^{2\pi i(x_1\xi_1+x_2\xi_2+x_3\xi_3+t_1\eta_1+t_2\eta_2) }, \xi_1,\xi_2,\xi_3,\eta_1,\eta_2\in \mathbb{Z}\] For the periodic boundary conditions to hold,
we require that “multiples” of the spatial and temporal coordinates are integers. In the simplest scenario, we consider a “degenerated” hyperbolic equation to visualize the dynamics of wave equation
under Periodic boundary conditions on the spatial dimension, a simpler case in 1 spatial dimension and 1 time dimension is: \[ \Delta_x u = \Delta_t u, x\in \mathbb{R}, t\in \mathbb{R}, u(-1,t)=u
(1,t), u_x(-1,t)=u_x(1,t), u(x,0)=Cos(\pi x), u_t(x,0)=-\pi Sin(\pi x)\] | {"url":"https://www.socr.umich.edu/TCIU/HTMLs/Chapter3_Wave_Equation_in_Spacekime.html","timestamp":"2024-11-09T22:35:07Z","content_type":"text/html","content_length":"1048957","record_id":"<urn:uuid:3d14755b-5db1-4d19-920d-105ad3325b86>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00551.warc.gz"} |
What is an inverse relationship?
Definition. An inverse relationship is one in which the value of one parameter tends to decrease as the value of the other parameter in the relationship increases. It is often described as a negative
What is an example of a reverse relationship?
There are many real-world examples of inverse relationships. … The mathematical explanation is that if f(x) = x + 2 and y(x) = x 2, the relationship is reversed. Also, f(x) = x and f(x) = 1/x to
eliminate a zero value.
How do I show inverse relationships?
In such cases, an inverse relationship is the opposite of a direct relationship, where in y = f(x) y increases as x increases, or in x = f(y) x increases as y increases. In an inverse relationship,
given by y = f(x), y would decrease as x increases. These relationships can be represented graphically.
How do you know if a relationship is direct or reverse?
Live vs. In Direct Relationships, an increase in x causes a corresponding increase in y, and a decrease has the opposite effect. This creates a line chart. In the inverse relationships, an increase
in x causes a corresponding decrease in y, and a decrease in x causes an increase in y.
Is an inverse relationship positive or negative?
In statistics, correlation describes the relationship between two variables. … Inverse correlation is sometimes called negative correlation, which describes the same type of relationship between
What is the opposite of a reverse relationship?
The opposite of a reverse relationship is a direct relationship. Two or more physical quantities can have an inverse relationship or a direct relationship.
Which two factors have an inverse relationship?
Direct and Inverse Relationships The relationship between mass and acceleration is different. It’s a reverse relationship. In an inverse relationship, when one variable increases, the other variable
decreases. The greater the mass of an object, the less it will accelerate when a given force is applied.
What is the opposite of a reverse relationship?
The opposite of a reverse relationship is a direct relationship. Two or more physical quantities can have an inverse relationship or a direct relationship.
Why do speed and time have inverse relationships?
Speed and driving time are inversely proportional, because the faster you drive, the shorter the time. | {"url":"https://www.readersfact.com/what-is-an-inverse-relationship/","timestamp":"2024-11-10T19:20:23Z","content_type":"text/html","content_length":"104786","record_id":"<urn:uuid:5a3d0bb2-63f9-4d06-ab4d-bc3cb98a1e6e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00647.warc.gz"} |
Leonardo Balerdi called up to Argentina national team
Leonardo Balerdi has been called up to the Argentina national team.
Balerdi, who was not part of the initial list of 25 players, has been selected as the Olympique Marseille man will join the team for the two World Cup qualifiers this month. The 21 year old last
played for Argentina last year and was not selected for the October games.
Argentina national team coach Lionel Scaloni announced his list of players for the World Cup qualifiers against Paraguay and Peru on Friday.
52 Comments
1. Montiel is not injured. he is expected to start against Rosario central in the game will start in less than one hour.
2. Messi looked good today. He suffered minor discomfort so only played 2nd half.
Amazing dummy to Griezmann and 1st open play goal in a while… he just blasted that fucker in.
□ i forget to mention the powerful pen! It’s great to see him scoring pens lately with such confidence.
3. Scaloni have secret relations with the sister of Bardeli, Kanema and Perez … its the only explanation.
□ What the hell!! Are you serious? Have some decency man!
4. What a chip goal by Paredes Di maria* combination..
□ I know. And a second goal as well.
□ Z
5. Ever since the 2002 world cup early loss we have been very hard on our older players and very judgemental of our young players. Pekerman was our last coach to make a good balance of young and
I’m happy balerdi got the call. He’s young and rough on the edges, but he needs experience and I think he has a big future.
I’m still confused as to why people question DiMaria being called to the team . I am confident that almost every national team in the world would start him if they could. He a great player. Slot
of the members on this site are very fickle and many others don’t understand the sport it seems. I don’t mean to be negative but if you go thru all of the comments below you will find members
saying that certain players should be given a chance. I agree with some of them, but some of these names are of average players who would not get the start on the Mexican team. (All respect due
to my Mexican brothers)
□ I agree that we need to balance young and old, but when we have other young options (Romero, Senesi) that are FAR better, why not them? Why chose someone that is young and bad when you can
have someone that is young and good?
☆ > Why chose someone that is young and bad when you can have someone that is young and good?
Maybe to help promote players that are “young & bad + potential”??
When NT call players, league management pay attention. Romero and Sensei making a name for themselves and may not need the extra help while the others do. I’m pretty sure with their
current trajectory, we’ll see Romero and Sensi soon anyway.
○ If Balerdi has a little help that is fine with me. But I hope to see the other too being important members of the squad soon enough
☆ Romero is injured i think
6. Lo celso , di maria , Armani , pereyra these farmers don’t deserve to Argentina
□ I think you don’t deserve the tag of Argentine fan.. Such a brainless stupid you are!
After Reqilame lo celso is the best Argentine midfielder, who will rule the midfield up coming 4/5 years.
□ Calling professional footballers, “farmers” is very inappropriate. No matter u like them or not they worked hard and sacrificed to get to that position. They are way above your level. Who are
you? compared to these professionals? Nobody lol. So be humble. If you are not, thats upto you but it shows where you come from. And ppl are just laughing at you. This is not a comedy show
here in Mundo, you can share your pathetic jokes where you will blend in better and be acceptable. But here you will look like a clown in this thread. If that is ok with you then thats your
problem. Be safe.
□ Guys this is another romance king alt account. He is a troll and not an Argentina fan. Please ignore him or else you are giving him exactly what he wants – your reactions. The best you can do
is contact Roy about it so he bans him. Or even better – ask for an ignore button.
mundo . albiceleste @ gmail . com (without spaces)
☆ sensible advice, Vikin.
———Ignore and report.
□ The fact that you think Lo Celso is a farmer and you even use the word farmer has made me lose all respect for you. What an actual idiot. What are you doing here? Because if you are an actual
fan you would know Lo Celso is good. It is mind boggling to me that you can even have an account on this forum and not know that.
□ Using Messi’s name..and talking nonsense.
7. Ocampos misses a penalty, but it turns out that the goalkeeper was far off his line, so it is taken again and then he scores. I think before the penalty he’s had a better performance than most of
the season so far.
8. Does anyone know when ascasaibar will be back from injury?
□ If Ascacibar overcome, it will be very tough to gain his position in best 11. Guendusi from Arsenal ( france) allready stolen his position.
9. We have better options
10. Another farmer selected
□ This is the another nonsense chutiya farmer id
☆ Calm down bro this is my right to say everything you cannot pressurised me like this , you are a biggest chutiya on this forum,
○ Shameless Romance king…
11. The biggest problem of Argentina is we always rely on untelented players like lo celso, Armani, pereyra, and many others , we don’t utilize specialist winger, midfielder or goalkeeper, actually
we have lots of talented players like barco , thiago almada, buendia , goalkeeper like musso ,
12. Acuna who is in great form with an injury , great .
□ Yup..Acuna is injured. Now Milton casco confirmed😆😆😆
□ Damn really? Did it seem serious or minor? If he can’t make the international call we will need a Tagliafico backup. Could be the time to call up Licha Martínez at last.
☆ If scaloni sellect Lisandro Martinez for Acuna’s Back up, it will be a great outgiving for us. Lisandro deserve the call up. But i think scaloni will select Milton Casco fr sure😥
○ Ya we still don’t know how serious it is
☆ Long overdue for scaloni to call Lisandro. Lisandro for Acuan is like for like backup. Both can play multiple positions in mid and defense. Seems like the obvious replacement but doubt
Scaloni will call him.
13. Dont get me wrong but Balerdi is extremely raw. He has been very inconsistent and we have more polished defenders. Its baffling to see Scaloni’s propaganda towards CRomero,Senesi and Licha.
Romero has 2 full seasons of Europe experience,Balerdi has none. Already playing for one of the best teams in the world at the moment. We all know how promising Licha is. Senesi is Eredivise’s
best centre back and is a very reliable one. We have much better options than Balerdi at defence as of now.
14. I think Balerdi going to play as Right Back
□ I hope so…scaloni trying Balerdi as a right Back
15. Emiliano buendia v swansea
1 assist 5 total shots 6 big chances created 5 key passes created 5 total on targets pass acvuracy 90 percent…………. Name one good reason not to select him
□ One huge reason: only in second division.
☆ 2nd division is not a huge reason . Scaloni had called Nico Gonzalez when he was in Bundesliga 2nd division. Buendia already has proven very well in EPL last season with his assists &
creative play. Selecting a Salvio or Prereya over Buendia is a bad decision .
☆ ( 2nd Division)This is not the main reason bro.. Actually his position..we have allready Messi, Dybala, Ocampos, Di maria, j.correa, Lo celso, Papu( Right wing or Central Attacking
midfield). His height 1.70 is also another reason.
☆ He was on top in many of these categories in first devision last season.
16. Balerdi is improving.
□ Improving from the Bench of the weaskest team in Champions League history ? (Marseille lose their 11 last CL games. It’s the record !)
17. Nico Gonzalez with a good game in Stuttgard’s draw with one goal and one assist. A very good player in my opinion with great pace. Someone who surely we have to keep an eye on.
18. I don’t understand why Scaloni isn’t investing in more young goalkeepers. Why does he keep calling some 34 year old starter wtf?
□ Because most goalies don’t hit their prime till their 30 and can play at a high level til around 35. Young goalies have a lot of nerves and errors in them
☆ @ Leonardo….100% agreed
☆ That ruins the whole point of this then.
We need to give younger players experience.
19. Idk about this in terms of positions . he played a very good game yesterday
20. 3 CB at the back..? | {"url":"https://mundoalbiceleste.com/2020/11/07/leonardo-balerdi-called-up-to-argentina-national-team/","timestamp":"2024-11-08T12:40:34Z","content_type":"text/html","content_length":"244711","record_id":"<urn:uuid:73400d1e-6e5f-43a4-a6aa-77cb746ed126>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00104.warc.gz"} |
A module for radiation hydrodynamic calculations with ZEUS-2D using flux-limited diffusion
A module for the ZEUS-2D code is described that may be used to solve the equations of radiation hydrodynamics to order unity in v/c, in the flux-limited diffusion (FLD) approximation. In this
approximation, the factor Eddington tensor f, which closes the radiation moment equations, is chosen to be an empirical function of the radiation energy density. This is easier to implement and
faster than fulltransport techniques, in which f is computed by solving the transfer equation. However, FLD is less accurate when the flux has a component perpendicular to the gradient in radiation
energy density and in optically thin regions when the radiation field depends strongly on angle. The material component of the fluid is here assumed to be in local thermodynamic equilibrium. The
energy equations are operator split, with transport terms, radiation diffusion term, and other source terms evolved separately. Transport terms are applied using the same consistent transport
algorithm as in ZEUS-2D. The radiation diffusion term is updated using an alternating direction-implicit method with convergence checking. Remaining source terms are advanced together implicitly
using numerical root finding. However, when absorption opacity is zero, accuracy is improved by instead treating the compression and expansion source terms using a time-centered differencing scheme.
Results are discussed for test problems including radiation-damped linear waves, radiation fronts propagating in optically thin media, subcritical and supercritical radiating shocks, and an optically
thick shock in which radiation dominates downstream pressure.
All Science Journal Classification (ASJC) codes
• Astronomy and Astrophysics
• Space and Planetary Science
• Hydrodynamics
• Methods: numerical
• Radiative transfer
Dive into the research topics of 'A module for radiation hydrodynamic calculations with ZEUS-2D using flux-limited diffusion'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-module-for-radiation-hydrodynamic-calculations-with-zeus-2d-usi","timestamp":"2024-11-08T12:39:28Z","content_type":"text/html","content_length":"53482","record_id":"<urn:uuid:48e8db94-ea75-4634-857e-4f467e8084ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00700.warc.gz"} |
Banker’s algorithm | Board Infinity
The banker’s algorithm in the operating system is especially used to avoid deadlock and resource allocation problems that basically use the predefined simulation for all the resources.
Why this algorithm is named Banker’s algorithm?
This algorithm is internally used in the banking system to check whether a loan can be allotted to a particular person or not.
For example, consider a scenario in which there is x number of people who holds an account in a bank and all of them have T amount of money in total. Now suppose a person wants to take a loan of a
particular amount then the loan will be sanctioned only if the total amount left after deducing that loan amount is equal to or greater than T. This is done in order to ensure that if all people
having account in the bank come to withdraw their deposited amount then they can withdraw easily.
In simple words, we can say that the bank should never sanction loan amounts in such a way that it cannot allocate money to the depositors.
Let there exist ‘N’ number of processes in a system and ‘M’ be the number of available resources.
Available resources
• The number of available resources are represented in an 1- d array of size ‘M’.
• Available_Resources[i] = R represents there are R number of resource of i type.
• The maximum demand of each process is represented in a 2-d array of dimensions M x N.
• Maximum_Demand[i, j] = L represents the process of type i may request k instances resource type j.
• The number of resources of each type allocated to each process is defined in a 2-D array of size N x M.
• Allocation[i, j] = S represents that the process of ith type is currently allocated S instances of the resource of type j.
• The remaining resource need of each process is represented in a 2-D array of size N x M.
• Need[i, j] = Q represents the process of ith type requires Q instances of resource of jth type.
• Need[i, j] = Maximum_Demand[i, j] - Allocation[i, j]
The banker’s algorithm is itself composed of a safety algorithm and a resource request algorithm:
Safety Algorithm
The safety algorithm determines whether a system exists in a safe state or not and can be described using the following:
• Let us consider that Work and Finish be arrays of lengths M and N respectively. Let us initialize the Work as Available, Finish[i] = false; for i = 1, 2, 3,….n.
• Find an i such that both, Finish[i] = false and Need[i] <= Work if there doesn’t exists any such i then go to step 4.
• Work = Work + Allocation[i]
Then mark, Finish[i] = true, and then go to step 2.
• If the value of Finish [i] is equal to true for all i then the system is in a safe state.
Resource request algorithm
Let us consider that Request[i] represents the request array for the process of ith type (Pi). Then, Requesti [j] = R signifies that the process of ith type (i.e, Pi ) requires R resources of type Rj
. Generally, the following actions are taken when a process Pi resquest some resources:
1. If the ith need is greater than ith request, i.e, Request <= Needi then go to the step 2 otherwise, raise an error condition as the process has reached to its maximum claim.
2. If the ith request is lesser than or equal to ith available resource i.e, If Requesti <= Availablei the go to the step 3. Otherwise, the ith process (i.e, Pi) must due to the non-availability of
the resources.
3. Have the system pretend to have allocated the requested resources to process Pi by modifying the state as follows:
Availablei = Available – Requesti
Allocationi = Allocationi + Requesti
Needi = Needi – Requesti
In this article, we discussed in detail regarding the Banker’s algorithm in operating system. | {"url":"https://www.boardinfinity.com/blog/understanding-bankers-algorithm-in-operating-system/","timestamp":"2024-11-09T03:12:18Z","content_type":"text/html","content_length":"62424","record_id":"<urn:uuid:5614f947-0fba-43e1-8af0-416505aa0173>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00874.warc.gz"} |
The coalescence of coupling points in the theory of radio waves in the ionosphere
The present work considers the propagation of a radio wave obliquely or vertically incident on a horizontally stratified ionosphere in the case where two coupling points in the complex height plane
are close together or coalesce, in which case the Airy integral function approximation fails and must be modified. In particular, the case where the same two waves are coupled at the adjacent
coupling points is investigated. Two different kinds of coalescence are distinguished for this type of coupling. In coalescence of the first kind (C1), the coupling between the two waves remains
strong when the coupling points move close together and when they coalesce. In coalescence of the second kind (C2), the coupling becomes weaker when the coupling points move close together and
disappears completely when they coalesce. The conditions for coalescence are set forth, and it is shown how the equations for conditions near C1 can be reduced to Weber's equation. A different form
of Weber's equation must be used for C2, and the Stokes constants are shown to be different in this case. The application to 'crossover' in the theory of ion cyclotron whistlers is discussed.
Proceedings of the Royal Society of London Series A
Pub Date:
October 1974
□ Coupling Coefficients;
□ Ionospheric Propagation;
□ Plasma-Electromagnetic Interaction;
□ Radio Wave Refraction;
□ Wave Equations;
□ Airy Function;
□ Ion Cyclotron Radiation;
□ Points (Mathematics);
□ Reflected Waves;
□ Stokes Law Of Radiation;
□ Wentzel-Kramer-Brillouin Method;
□ Whistlers;
□ Communications and Radar | {"url":"https://ui.adsabs.harvard.edu/abs/1974RSPSA.341....1B/abstract","timestamp":"2024-11-14T20:17:46Z","content_type":"text/html","content_length":"39320","record_id":"<urn:uuid:791a9872-81c3-46c8-a33d-b270cb5b53d9>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00382.warc.gz"} |
This Is Why Scientists Will Never Exactly Solve General Relativity
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Even extremely simple configurations in General Relativity cannot be solved exactly. Here’s the science of why.
It’s difficult to appreciate how revolutionary of a transformation it is to consider the Universe from Einstein’s, rather than Newton’s, point of view. According to Newtonian mechanics and Newtonian
gravity, the Universe is a perfectly deterministic system. If you were to give a scientist who understood the masses, positions, and momenta of each and every particle in the Universe, they could
determine for you where any particle would be and what it would be doing at any point in the future.
In theory, Einstein’s equations are deterministic as well, so you can imagine something similar would occur: if you could only know the mass, position, and momentum of each particle in the Universe,
you could compute anything as far into the future as you were willing to look. But whereas you can write down the equations that would govern how these particles would behave in a Newtonian Universe,
we can’t practically achieve even that step in a Universe governed by General Relativity. Here’s why.
Newton’s law of Universal Gravitation has been superseded by Einstein’s General Relativity, but relied on the concept of an instantaneous action (force) at a distance, and is incredibly
straightforward. The gravitational constant in this equation, G, along with the values of the two masses and the distance between them, are the only factors in determining a gravitational force. G
also appears in Einstein’s theory. (WIKIMEDIA COMMONS USER DENNIS NILSSON)
In a Newtonian Universe, every massive object in the Universe exerts a well-defined gravitational force on every other object in the Universe. You can do this so long as you can determine the
gravitational force between every pair of masses that exists, and then just calculate the Newtonian gravitational force. That force also tells you how that mass is going to move (because F = ma), and
that’s how you can determine the Universe’s evolution.
But in General Relativity, the challenge is much greater. Even if you knew those same pieces of information—positions, masses, and momenta of each particle—plus the particular relativistic
reference frame in which they were valid, that wouldn’t be enough to determine how things evolve. The structure of Einstein’s greatest theory is too complex even for that.
Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been ‘straight’ lines to instead become curved by a specific amount. In General Relativity, we treat
space and time as continuous, but all forms of energy, including but not limited to mass, contribute to spacetime curvature. If we were to replace Earth with a denser version, up to and including a
singularity, the spacetime deformation shown here would be identical; only inside the Earth itself would a difference be notable. (CHRISTOPHER VITALE OF NETWORKOLOGIES AND THE PRATT INSTITUTE)
In General Relativity, it isn’t the net force acting on an object that determines how it moves and accelerates, but rather the curvature of space (and spacetime) itself. This immediately poses a
problem, because the entity that determines the curvature of space is all of the matter and energy present within the Universe, which includes a lot more than merely the positions and momenta of the
massive particles we have.
In General Relativity, unlike Newtonian gravity, the interaction of any mass you consider also plays a role: the fact that it also has energy means that it also deforms the fabric of spacetime. When
you have any two massive objects moving and/or accelerating relative to one another in space, it causes the emission of gravitational radiation, too. That radiation isn’t instantaneous, but only
propagates outwards at the speed of light. This is an enormously difficult factor to account for.
Ripples in spacetime are what gravitational waves are, and they travel through space at the speed of light in all directions. Although the constants of electromagnetism never appear in the equations
for Einstein’s General Relativity, the speed of gravity undoubtedly equals the speed of light. The existence of gravitational radiation, relative effects between moving masses, and many other subtle
effects make calculating anything in General Relativity an extraordinary challenge. (EUROPEAN GRAVITATIONAL OBSERVATORY, LIONEL BRET/EUROLIOS)
Whereas you can easily write down the equations that govern any system you can imagine in a Newtonian Universe, even that step is an enormous challenge in a Universe governed by General Relativity.
Because of how many things can affect how space itself is curved or otherwise evolves with time, we often cannot even write down the equations that describe the shape of even a simple, toy-model
Perhaps the most demonstrative example is to imagine the simplest Universe possible: one that was empty, with no matter or energy, and that never changed with time. That’s completely plausible, and
is the special case that gives us plain old special relativity and flat, euclidean space. It’s the simplest, most uninteresting case possible.
A representation of flat, empty space with no matter, energy or curvature of any type. With the exception of small quantum fluctuations, space in an inflationary Universe becomes incredibly flat like
this, except in a 3D grid rather than a 2D sheet. Space is stretched flat, and particles are rapidly driven away. (AMBER STUVER / LIVING LIGO)
Now go one step more complex: take a point mass and put it down anywhere in the Universe. All of a sudden, spacetime is tremendously different.
Instead of flat, euclidean space, we find that space is curved, no matter how far away you get from the mass. We find that the closer you get, the faster the space beneath you “flows” towards the
location of that point mass. We find that there’s a specific distance at which you’ll cross the event horizon: the point-of-no-return, where you cannot escape even if you were to move arbitrarily
close to the speed of light.
This spacetime is much more complicated than empty space, and all we did was add one mass. This was the first exact, non-trivial solution ever discovered in General Relativity: the Schwarzschild
solution, which corresponds to a non-rotating black hole.
Both inside and outside the event horizon of a Schwarzschild black hole, space flows like either a moving walkway or a waterfall, depending on how you want to visualize it. At the event horizon, even
if you ran (or swam) at the speed of light, there would be no overcoming the flow of spacetime, which drags you into the singularity at the center. Outside the event horizon, though, other forces
(like electromagnetism) can frequently overcome the pull of gravity, causing even infalling matter to escape. (ANDREW HAMILTON / JILA / UNIVERSITY OF COLORADO)
Over the past century, many other exact solutions have been found, but they’re not significantly more complicated. They include:
You might notice that these solutions are also extraordinarily simple, and don’t include the most basic gravitational system we consider all the time: a Universe where two masses are gravitationally
bound together.
Countless scientific tests of Einstein’s general theory of relativity have been performed, subjecting the idea to some of the most stringent constraints ever obtained by humanity. Einstein’s first
solution was for the weak-field limit around a single mass, like the Sun; he applied these results to our Solar System with dramatic success. We can view this orbit as Earth (or any planet) being in
free-fall around the Sun, traveling in a straight-line path in its own frame of reference. All masses and all sources of energy contribute to the curvature of spacetime, but we can only calculate the
Earth-Sun orbit approximately, not exactly. (LIGO SCIENTIFIC COLLABORATION / T. PYLE / CALTECH / MIT)
This problem—the two-body problem in General Relativity—cannot be solved exactly. There is no exact, analytical solution known for a spacetime with more than one mass in it, and it’s thought (but
not, to my knowledge, proven) that no such solution is possible.
Instead, all we can do is make assumptions and either tease out some higher-order approximate terms (the post-Newtonian expansion) or to examine the specific form of a problem and attempt to solve it
numerically. Advances in the science of numerical relativity, particularly in the 1990s and later, are what enabled astrophysicists to calculate and determine templates for a variety of gravitational
wave signatures in the Universe, including approximate solutions for two merging black holes. Whenever LIGO or Virgo make a detection, this is the theoretical work that makes it possible.
The gravitational wave signal from the first pair of detected, merging black holes from the LIGO collaboration. The raw data and the theoretical templates are incredible in how well they match up,
and clearly show a wave-like pattern. The theoretical template required enormous advances in numerical relativity in order to make this identification possible. (B. P. ABBOTT ET AL. (LIGO SCIENTIFIC
That said, there are an incredible number of problems we can solve, at least approximately, by taking advantage of the behaviors of solutions that we do understand. We can patch together what happens
in an inhomogeneous patch of an otherwise smooth, fluid-filled Universe to learn how overdense regions grow and underdense regions shrink.
We can extract how the behavior of a solvable system differs from Newtonian gravity and then apply those corrections to a more complicated system that perhaps we cannot solve.
Or we can develop novel numerical methods for solving problems that are entirely intractable from a theoretical point of view; so long as the gravitational fields are relatively weak (i.e., we aren’t
too close to too large a mass), this is a plausible approach.
In the Newtonian picture of gravity, space and time are absolute, fixed quantities, while in the Einsteinian picture, spacetime is a single, unified structure where the three dimensions of space and
the one dimension of time are inextricably linked. (NASA)
Still, General Relativity poses a unique set of challenges that don’t arise in a Newtonian Universe. The facts are as follows:
• the curvature of space is continuously changing,
• every mass has its own self-energy that also changes spacetime’s curvature,
• objects moving through curved space interact with it and emit gravitational radiation,
• all the gravitational signals generated only move at the speed of light,
• and the object’s velocity relative to any other object results in a relativistic (length contraction and time dilation) transformation that must be accounted for.
When you take all of these into account, it all adds up to most spacetimes that you can imagine, even relatively simple ones, leading to equations that are so complex that we cannot find a solution
to Einstein’s equations.
An animated look at how spacetime responds as a mass moves through it helps showcase exactly how, qualitatively, it isn’t merely a sheet of fabric but all of space itself gets curved by the presence
and properties of the matter and energy within the Universe. Note that spacetime can only be described if we include not only the position of the massive object, but where that mass is located
throughout time. Both instantaneous location and the past history of where that object was located determine the forces experienced by objects moving through the Universe. (LUCASVB)
One of the most valuable lessons I ever got in my life came during the first day of my first college math class on differential equations. The professor told us, “Most of the differential equations
that exist cannot be solved. And most of the differential equations that can be solved cannot be solved by you.” This is exactly what General Relativity is—a series of coupled differential
equations—and the difficulty that it presents to all those who study it.
We cannot even write down the Einstein field equations that describe most spacetimes or most Universes we can imagine. Most of the ones we can write down cannot be solved. And most of the ones that
can be solved cannot be solved by me, you, or anyone. But still, we can make approximations that allow us to extract some meaningful predictions and descriptions. In the grand scheme of the cosmos,
that’s as close as anyone’s ever gotten to figuring it all out, but there’s still much farther to go. May we never give up until we get there.
Ethan Siegel is the author of Beyond the Galaxy and Treknology. You can pre-order his third book, currently in development: the Encyclopaedia Cosmologica.
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Take it from Bezos, Musk, and Einstein — rethinking lines of inquiry can transform business, investing, and innovation strategy.
A recent study challenges the conventional thinking that says only young people can dream up successful new businesses. | {"url":"https://bigthink.com/starts-with-a-bang/this-is-why-scientists-will-never-exactly-solve-general-relativity/","timestamp":"2024-11-10T02:16:38Z","content_type":"text/html","content_length":"171413","record_id":"<urn:uuid:af4e5771-5554-4dc3-9b79-4d72fc50e01c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00405.warc.gz"} |
Fossil Capital - a review
It is a great read for those interested in the origins of fossil capital, and the implications of this for the ecological and climate crisis we face today.
It is not uncontroversial. Andreas Malm also enters into one of the key debates on the ecological left today, the Anthropocene. This is the matter of whether (or not) the current definition of the
geological epoch through which the planet is passing, the ‘Holocene’, should be changed to the ‘Anthropocene’ (or the age of humans) as a result of the impact of modern humans on its ecosystems.
Also the extent to which steam power and fossil capital, as the roots of global warming, has contributed to that impact.
I take a different view on the Anthropocene to Malm, as I will explain later in this review.
Both, however, for what it tells us about the origins on fossil capitalism, and its contribution to the debate on the Anthropocene, his book deserves to be widely read amongst those who are committed
to the struggle both for an end to capitalism and for a new society build on social justice and ecological sustainability.
The industrial revolution
The earlier sections (and biggest part of the book) present a highly detailed, and fascinating, account of how waterpower drove the early years of the industrial revolution, in Britain, and the
economic and social consequences that arose from them.
It describes, how, by the middle of the18th century, the dominant source of energy in the cotton mills of Lancashire, Derbyshire, and of Greenock in Scotland, was water. Abundant supplies of water
with, an adequate drop, were provided by the topography and climate of the Northwest of England and the West of Scotland.
Rivers such as the Derwent, the Irwell, and the Tame, powered machines such as Hargreaves’s spinning jenny, Arkwright’s water frame, and Crompton’s spinning mule. Water mills churned out textiles in
ever-greater quantities and the factory system of production was born. Since most water mills had to be built in remote locations where the waterpower was available, the mill owners had to build
housing and community structures in order to accommodate the workforce—Malm rightly calls them ‘colonies’.
Waterpower was highly profitable with high levels of capital investment producing rates of profit of up to 50%. By the early part of the 19th century, however, waterpower was in trouble. It was
struggling to compete with steam, despite its continued viability, and the high cost of coal.
Waterpower had many drawbacks, however, as far as the mill owners were concerned—apart from the remote locations. The rivers could freeze up or flood, and they were vulnerable to drought and water
shortage. With steam the mills could be built anywhere, including inside the cities with the workforce on the doorstep, and there was plenty of coal available. There were disputes between the mill
owners over the best supplies of water, particularly when it became scarce.
The book describes how great efforts were made, in engineering terms, to improve the efficiently, and availability of waterpower. Systems of reservoirs, sluices and channels were constructed to
ensure that each owner could have the water they needed. Disputes, however, continued between the mill owners as one accused the other of using more water than they were entitled, or of deliberately
sabotaging the production of a rival in a water grab.
With steam the mills could be built anywhere including in the cities and mill owners could be in complete control of their own power source. They would not have to supply accommodation for their
labour force. Instead the slums of the big cities would provide all the labour they needed.
The revolt against steam
By the early part of the 19th century steam had become the fuel of choice of the mill owners, eclipsing water within a few years. It led to even higher rates of profitability and exploitation than
water. It also led to considerably worse social conditions, and to much higher levels of industrial conflict.
The book paints a vivid picture of how the repeal of the Combination Act in 1824, and then the first major financial crash (or structural crisis as Malm describes it) of the capitalist era, led to an
explosion of uprisings, strikes, and unionisation struggles in Glasgow, Manchester, and the cotton towns of Lancashire.
The book records that: “Outbreaks of popular insurrection became more frequent as the crisis took hold: first the Lancashire rising of 1826, then the Swing Riots of 1830, the South Wales rebellion of
1832-4, The Reform crisis of 1831-2, all succeeded in 1838 by the supreme challenge of Chartism, culminating in the general strike of 1842 – the most critical near-revolutionary moment in the 19th
century, if not the entire modern history of Britain.” (Page 62)
The book recounts in remarkable detail how, faced with tumbling profits, the mill owners in Stalybridge and Ashton cut wages of their workforce by up to 25%: as a result thousands of workers
assembled under the banner of Chartism to immobilise the steam engines by ‘pulling the plugs’. It records how most mills in Manchester and Salford were closed down and how in Manchester rioters
successfully entered a police station and threw furniture out of the window.
It goes on: “in Bury, insurgents drew the plugs in all the factories; according to one of the many correspondents serving the government with daily updates from the field, a mob on the outskirts of
the town was also in the process of ‘breaking machinery and had nearly pulled down one mill’. From Burnley word came that ‘almost every mill and workshop in the town where steam is employed was
in a few hours effectively stopped.”
The fossil fuel economy
The date from which Britain became a fully fossil fuel economy—or put the other way around when it broke from renewable energy in favour of steam—Malm argues, was around 1830. The years between
1830 and 1854, he says, saw the rapid expansion of mass production from cotton and textiles to general engineering—driven by steam and fuelled by coal. By 1870 three times as much coal was used in
general engineering, and in iron and steel production, than was used for domestic consumption.
By 1850 Britain emitted nearly twice as much CO2 as the USA, Germany and Belgium combined. This opened up a new stage in the development of capitalism with a great expansion of productively and
growth—capitalism as we know it today.
The fossil fuel economy, Malm notes, liberated British capitalism from the constraints imposed upon it by all previous energy sources: water, biomass (wood), animal (horses and oxen) and human beings
and opened up the road for built-in growth. He points out that by 1900 a Europe without fossil fuels would have needed 2.7 times its land surface to sustain waterpower or biomass and 20 times its
land surface by 2,000.
For future generations, however, it ushered in a process that would threaten the viability of the planet itself in terms of its habitability for many of the species that currently inhabit it—via
global warming and climate change.
A succession of major fossil fuel technologies quickly followed steam: electricity, the internal combustion engine—the age of the motor can and the age of air travel.
Malm explains how global warming takes place. He points out that for many thousands of years (since the last ice age) CO2 in the atmosphere, which causes the greenhouse effect and therefore global
warming, was constant at around 285 parts per million (ppm). After the establishment of the fossil economy, first in Britain and then beyond, it rose rapidly. By 2013 it had reached 400 ppm and is
now rising by 2 ppm every year.
He explains that: “ Given that CO2 acts as a thermostat in regulating the temperature on earth, and given that the temperature sets the climatic conditions in which all life on earth exists, the
magnitude of the rise—from 285 ppm as late as the mid-19th century to the current 400 plus—upgrades Homo sapiens into a geological agent.” (Page 27)
He is certainly right about that, or at least that it is one of several factors that pitch homo sapiens in the role of such an agency.
The Anthropocene debate
Malm then enters more controversial territory, as mentioned above, by launching into an important current debate amongst environmentalists—the idea of the Anthropocene. This is the
proposition—first proposed by Paul Crutzen, a climate scientist and a Nobel Prize winner, in 2000 and now supported by a growing body of scientific opinion—that the impact of the human species
(Homo sapiens), or modern humans, on the biosphere of the planet is now so great that it defines the current geological epoch.
Advocates of the Anthropocene, which includes myself, argue that the current geological epoch—the Holocene (or interglacial period)—should be superseded by, or redefined as, the Anthropocene, or
the ‘age of humans’. This would involve a change to the official geological time scale—the chart that divides the Earth’s 4.5bn year history into eons, eras, periods, epochs and ages, with each
division of diminishing length and geological significance.
A more substantial explanation of the case for the Anthropocene proposition can be found in my review of The Anthropocene and the Global Environmental Crisis edited by Clive Hamilton, Cristophe
Bonneuil, and Francois Gemenne published by the Routledge Environmental Humanities Series in 2015. This can be found at here.
The Anthropocene was first proposed by Paul Crutzen, a climate scientist and a Nobel Prize winner, in 2000. Since then scientific opinion has increasingly supported his thesis.
Malm opposes the Anthropocene, branding it ‘species thinking’. He argues that it is wrong to attribute global warming to the human species (Homo sapiens) as such but to a small group of capitalists
within the human species. We should ‘not mistake capitalist for human beings’ he argues.
Steam engines, he argues, “were not adopted by some natural-born deputies of the human species. By the nature of the social order of things, they could only be installed by the owners of the means of
production. [Emphasis original]… Is there any reason to consider it any more truly representative of ‘the human enterprise’ than the Luddites or the plug drawers or the preachers of steam
demonology?” (Page 267)
He goes on: “Capitalists in a small corner of the Western world invested in steam, laying the foundations of the fossil economy; at no moment did the species vote for it either with feet or ballots,
or march in mechanical unison, or exercise any sort of shared authority under its own destiny and that of the earth system. It did not figure as an actor on the historical stage.”
The Anthropocene, he says: “might be a useful concept and narrative for polar bears and amphibians and birds who want to know what species is wreaking such terrible havoc on their habitats, but,
alas, they lack the capacity to scrutinize and stand up to the human actions; for those who may do so—other human beings—species thinking on climate change conduces to paralysis”. (Page 272)
It is certainly true (as Malm suggests at one point) that global warming is far too narrow a basis on which to declare the change from the Holocene to the Anthropocene. In fact it is only one part of
a much wider rationale for such a change. The most compelling single factor for the Anthropocene idea, in my view, is the biodiversity crisis.
In her excellent book The Sixth Extinction Elizabeth Kolbert argues (along with an increasing body of opinion) that we are facing the biggest mass extinction of species (the “sixth mass extinction”)
since the demise of the dinosaurs 65m years ago. She also argues strongly for a recognition of the Anthropocene and predicts an early decision on its recognition. My review of her book can be found
She points out that 40% of all mammal species are currently under a short to medium term threat of extinction against a background rate of one every 700 years. Amphibians are disappearing at
staggering 45,000 times the background rate. She argues that an extinction rate of this scale ultimately puts at risk all species on the planet, including, eventually, our own.
Marxism and class society
Malm argues that responsibility for the false conception of the Anthropocene, as he sees it, lies with the natural science community. The Anthropocene narrative, he argues, is: “an illogical and
ultimately self-defeating foray of the natural science community—responsible for the original discovery—into the domain of human affairs.” (Page 270)
His alternative to the Anthropocene is the ‘capitalocene’, or an epoch defined by capitalism. This designation, he argues, is based on: “the geology not of mankind, but of capital accumulation.” He
does not however expect to see a consensus gather behind his idea. (Page 391)
His conclusion appears to be that the Anthropocene, or any notion of assessing the environmental impact of modern humans on the planet as a species, runs counter to a class based (or Marxist)
analysis of capitalist society.
But why? The Anthropocene does not imply that modern human are all equally responsible for their impact on the planet. That would be ridiculous. The human species is class divided. The rich and the
powerful, and corporate interests, clearly bear the main responsibility for such impact. They are the driving force of it.
This does not, however, mean that we can ignore the overall impact of our species on the biosphere, on other species, and on the viability of the plant itself to sustain life. We ignore this at our
own peril.
As I argue in my review of The Anthropocene and the Global Environmental Crisis modern humans are the most, successful, resourceful, and effective species the planet has produced—by a very long
way—and they had a disproportionate impact on other species and their habitats from the outset.
As humans migrated out from their African homelands to other parts of the globe they eliminated most of the big land animals and flightless birds, who were defenseless against their hunting skills,
on the spot—often going far beyond their immediate needs. A fifth of all species were eliminated in this way. This was the case in Australia, New Zealand Madagascar, Indonesia, the Americas and
The ecological crisis did not start with the industrial revolution—although it clearly took it to a new level.
Another factor the book fails to address adequately is the challenge that industrialisation represented (and represents) to the ecology of the planet whatever mode of production emerged in the course
of it.
Capitalism is the most rapacious system of society—with its drive for growth and profit—that modern humans have produced, with the possible exception of Stalinism. But the challenge represented
by the invention of the steam engine (fuelled by coal) and the internal combustion engine (fuelled by oil) and the massive expansion of production and population made possible by these inventions,
and others that followed, to the ecology of the planet, whatever mode of production took control of it, was huge.
Surprisingly the Malm also does not mention ecosocialism (or at least I have been unable to find it)—which is crucial in my view in terms of a framework, from a Marxist standpoint, for a
sustainable socialist society.
It defines the kind of society that we want to build when we are in a position to build a socialist society. It is a declaration that we are looking towards a model of socialism that has not yet
existed and that few are even advocating.
The absence of capitalism does not necessarily resolve the problem as Malm recognises (on page 277). The Soviet Union and its satellites were arguably even more ecologically destructive than their
capitalist counterparts.
Ecosocialism is a signal that we don’t want to see a socialist revolution take place under conditions where there is nothing left, where the working class inherit a scorched earth with most other
species gone, where we would be struggling even to produce food because the basic biodiversity and fertile land (and water) necessary for food production no longer exist. It is a recognition that we
have to put the ecological crisis at the heart of our struggle for social change.
It is a declaration of all that we recognise that socialist revolution (and the end of capitalism) will not automatically resolve the ecological crisis. That the struggle to defend the ecosystems of
the planet will have to continue even after a socialist revolution has been completed, and that it will not be easy.
It is a vision of a society that is built on the idea working in harmony with nature, of being a part of nature, and not existing at the expense of nature.
The book is essential reading, including the debates it raises, for those interested in the ecological crisis, first in terms of the rise of fossil capital and also in terms of the current debates in
the movement. It adds to the body of knowledge as to the origins of the fossil capitalism and to the body of knowledge we will need to bring it to an end.
February 2016 | {"url":"https://internationalviewpoint.org/spip.php?article4430","timestamp":"2024-11-08T00:51:33Z","content_type":"application/xhtml+xml","content_length":"50363","record_id":"<urn:uuid:901fbb02-f2e5-4c98-85eb-3e28992c17f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00830.warc.gz"} |
160. A star shines for you
As explained the structure gives you all, it tells you of the cycles both of the stars and that of mankind, both as a whole as well as on an individual level. The stories told cannot be disconnected
from each other. The stars tell a story of the events of the heavens which reflects the story of man and vice versa. Science often tells you that the moon causes the seas to rise and fall “ebb and
flood” but as everything is connected you cannot isolate one thing as cause. Both micro and macro.
There are many religions that tell a similar story: Gilgamesh tells a similar story about a basket with a baby, like Moses, who is then raised by royalty, it talks about the dove etc. The same story
is told in India where he is called Manu, in Babylon he is called Nemo, in Crete he is called Minos and in Egypt he is called Mises.
I also told you about cycles within cycles repeating itself, look at the comparison:
Joseph – Jesus
12 brothers – 12 disciples
The miracle birth – the miracle birth
Judah suggest to sell Joseph – Judas suggests to sale Jesus
Sold for 20 pieces – sold for 30 pieces
Works at age 30 – works at age 30
You might recall that Jesus lived out gods plan, he physically expressed the major story line with his own life. Let’s have a closer look , from the summer to the winter solstice the sun reaches its
lowest point as it moves to the south where it literally dies for 3 days on the 22th, 23th , and 24th. During those 3 days the sun rises through the constellation called the southern cross or crux
and is reborn to move to the north again.
On a large scale let’s look at the zodiac, 12 signs or 12 tribes of is-ra-el, remember RA sun-god? 12 judges of Israel, 12 patriarchs, 12 OT prophets, 12 kings and Jesus at the age of 12 in the
temple, 12 signs of the zodiac. Moses moves from the bull or Taurus to Aries or goat and Jesus moved into Pisces or two fishes. When Jesus is asked where the new Passover will be, he answers in Luke
22:10: behold when ye are entered into the city there shall a man meet you bearing a pitcher of water, follow him into the house where he entereth, in Aquarius!
As you will know a great year is 25920 years and is a full cycle of 12 zodiacs, 12 times 2160 years. 12 of these cycles bring you to 31104 which is equal to 144 times 2160. While 25 times the eight
or 5760 is 144,000.
5760 itself is 40 times 144.
There are two of the eights or 576 which gives you 1152
And a third which makes 1728 another cycle.
576 light and 576 darkness or 40 days and 40 nights.
Now look closely at these mirrors 1296 and 6912.
1296 divided by 12 is 108 and 6912 divided by 12 is 576.
Those of you who have studied the other articles know the relationship between 18 and 576. 6912 x 25 is 172800.
1296 divided by 25 is 51,84 while 1296 times 25 is 32400.
And when you look where this is on the star of Bethlehem then you realize it is where the 5 and 6 pointed star come together to make 96 degrees. 1296 is halfway a turning point a full cycle of 25920.
Divided by 2 is 12960.
As you might also remember the ark 300×50×30 is 45(000).
The big cycle of 25920 divided by 576 is indeed 45 and getting out is reverse or 54 and 54 times 576 is indeed the creational date again of 31104.
Joseph was suggested to be sold for 20 pieces and Jesus for 30 pieces. 2160 divided by 20 is 10,8 and 2160 divided by 30 is 7,2 the difference of 3,6 36 or 360 or add them together 10,8 plus 7,2
makes 18.
Big cycles and small cycles 5760 minus 5184 is 576.
Now 51,84 is the angle of degrees also found in the structure of the pyramid at Giza in Egypt, but can be found in alchemy also. And just one last little thing: the keys were, 52,36 and 56,25 and
31,25 and 51,84.
So now you have all 4. Added up is 1917, a not familiar number yet? Let’s go back to the 6912 but now add a cycle 69012 and divide by 36. Yes indeed 1917 the sum of all keys, and a turning point.
Moshiya van den Broek | {"url":"https://www.truth-revelations.org/?page_id=1053","timestamp":"2024-11-09T17:36:18Z","content_type":"text/html","content_length":"30091","record_id":"<urn:uuid:32206087-a9dc-4ac1-b8b7-1d0c9a23334a>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00224.warc.gz"} |
Optimum Design of Torque Sensor Elastomer
Torque is one of the typical mechanical parameters that need to be detected in the industrial production process. The torque sensor is an important part of the torque test system, which is mainly
used to measure various torque, speed and power. Elastomer is the core component of torque sensor, and its structural design parameters are an important factor affecting the sensitivity of torque
sensor. Previous researchers have provided a lot of rationale for the design of torque sensors, but have not considered structural optimization of the elastomer itself.
For the elastomer, on the basis of the force analysis, the finite element analysis software ANSYS Workbench is used for static analysis, and the structure of the elastomer is optimized through Design
Exploration, which increases the strain of the elastomer and improves the mechanical structure, the sensitivity of the sensor.
1. Force Analysis of Elastomer
The torque sensor adopts a shear spoke structure, which has the advantages of good linearity, high output sensitivity, resistance to eccentric load and strong lateral force. The 4 elastic bodies are
symmetrically distributed between the inner and outer hubs. When subjected to external torque, the elastomer will deform accordingly. Paste the strain gauge in the direction of 45° to the neutral
plane of the elastomer to measure The strain proportional to the torque is finally converted into the corresponding voltage signal output through the bridge circuit. Where R is the radius of the
inner hub, l is the length of the elastomer, b is the width of the elastomer, k is the distance from the strain gauge to the inner hub, and T is the torque received by the torque sensor.
As shown in Figure 2, in order to facilitate the force analysis, one of the elastic bodies is used as the analysis object. The elastomer of the torque sensor can be regarded as converted from a
cantilever beam. Let the external torque be T, imagine that the shear force of the elastomer at the k section is Q, and the bending moment is M(k).
When the external torque is constant, the elastomer strain is determined by the material and the structure size. The selected material should have excellent comprehensive performance and good process
performance, and the reasonable structure size of the elastomer becomes the formula: E is Modulus of elasticity, μ is Poisson's ratio and δ is the thickness of the elastomer.
The larger the elastomer strain is, the greater the electrical signal output by the strain gauge will be, and the sensitivity of the torque sensor will be correspondingly improved. The more accurate
the measurement is: when the external torque is constant, the elastomer strain is determined by the material and structure size. The selected material should have excellent comprehensive performance
and good process performance, and the reasonable structural size of the elastomer will become the main factor determining the sensitivity of the torque sensor.
2. Elastomer Optimized Design
2.1 Optimizing design parameters
The optimal design variables are selected as the width, thickness and length of the elastomer, the maximum strain variable of the elastomer is selected as the optimization objective function, and the
constraint condition is selected to satisfy the strength condition of the elastomer, that is, the maximum stress is less than the allowable stress of the material. Therefore, the mathematical model
for the optimization of the elastomer structure is as follows:
X=[x[1], x[2], x[3]]=[b, δ, l]^T
Where: F(X) is the strain function of the elastomer; G(X) is the stress function of the elastomer; [σ] is the allowable stress of the elastomer.
The Design Explorer module in ANSYS Workbench software enables rapid optimization of product performance. The general process of optimal design includes: defining optimal design variables and
standard variables, sensitivity analysis, output response analysis, optimization analysis and solution verification.
Under the premise of ensuring the bearing capacity of the torque sensor, the elastomer is optimized to increase the strain. The width, thickness and length of the elastomer are selected as the
optimization design variables, the initial value is given, the variation range of each structural parameter of the elastomer is set to be ±10% of the initial value, and the maximum stress and maximum
strain of the elastomer are used as the target variables. The optimized design parameters are shown in Table 1.
Optimize design parameters Initial value (mm) Upper limit (mm) lower limit (mm)
Width 21 23.1 18.9
Thickness 7 7.7 6.3
Length 20 22 18
Table 1. Design of optimize design parameters
2.2 Sensitivity analysis of optimization parameters
Through the sensitivity analysis, the influence degree of the design parameters on the system or model can be determined, and the design variables that are actually applied to various iterative
methods to achieve the optimization requirements can be obtained, and the parameter sensitivity analysis can be completed. The program analyzes the sensitivity by calculating the measured values of
various parameters, so as to determine the degree of influence of the design variables on the target variables, which helps to eliminate design variables that have little influence in the
optimization process, narrow the research scope of parameters, and reduce the number of parameters in the optimization process.
Figure 4 is a histogram of sensitivity analysis.
From Figure 4, it can be seen that the influence of three design variables on the maximum strain: the sensitivity coefficient of width is -0.229 31, the sensitivity coefficient of thickness is -0.165
29, and the sensitivity coefficient of length is 0.071 898. The influence of the maximum strain is greater; the influence of 3 design variables on the maximum stress: the sensitivity coefficient of
width is -0.229 32, the sensitivity coefficient of thickness is -0.165 18, and the sensitivity coefficient of length is 0.071802. The influence of the maximum stress is greater, and the negative
value of the sensitivity coefficient indicates that the design variable is inversely proportional to the target variable. Therefore, in the optimization design, the width and thickness of the
structure should be mainly considered, and the influence of the length of the structure should be properly considered.
2.3 Optimization parameter evaluation
Through sensitivity analysis, two design variables with greater influence were extracted from the three optimized design variables, namely the width and thickness of the elastomer. According to the
input parameters, ANSYS Workbench will select enough measurement points to combine within each design parameter range to obtain design points. Each design point represents a structural dimension
design scheme of an elastomer. After the calculation is completed, each input point is fitted with the output point to form a response surface, so as to determine the region of the optimal solution.
Figures 5 are the output response surfaces.
Table 2 shows the three groups of optimization result parameters. The parameter design variable value of group A is relatively small, the obtained strain is the largest, and the maximum stress is
also less than the allowable stress. Therefore, the candidate data of the - - group is selected. In order to facilitate processing and manufacturing, the structure size is rounded, and the width of
the elastomer is determined to be 19 mm, the thickness is 6.5 mm, and the length is 18 mm. Using this data as a verification, the maximum strain of the elastomer is calculated to be 1.3752x10-^3, the
maximum stress is 283.29 MPa, and the patch position strain in the middle of the elastomer is between 4.848x 10-^4~ 5.056x 10-^4, on average strain is 4. 952x10-^4. The stress at the patch position
in the middle of the elastomer is between 98.91~105. 69 MPa, and the average stress is 102.3 MPa. Compared with the initial structural design, the maximum strain of the elastomer increases by 11.81%,
and the maximum stress increases by about 11.805%, the average strain at the patch position in the middle of the elastomer increases by 9.31%, and the average stress increases by 9.06%.
Items A B C
Width/mm 18.921 19.446 22.071
Thickness/mm 6.307 6.9292 6.4626
Length/mm 18.02 21.22 20.42
Maximum stress 277.32 265.68 261.14
Maximum strain 1.3456 1.2896 1.2675
Table 2. Optimize design results
3. Conclusion
1. On the basis of the force analysis of the elastomer, it is verified that the material and structural size of the elastomer have an important influence on the strain variable of the torque sensor
when the torque sensor is subjected to the time-time.
2. The static analysis of the elastomer was carried out by ANSYS Workbench, and the stress and strain cloud diagrams were obtained.
3. Through the sensitivity analysis, it is known that the width and thickness of the elastomer have a great influence on its static characteristics. The optimization analysis function of Design
Exploration is used to obtain the reasonable optimized size of the elastomer. Under the condition of satisfying the strength, The strain amount of the elastomer is increased, and the sensitivity
of the coupling torque sensor is improved from the aspect of mechanical structure.
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How To Output Spelling Of Numbers In C++ - SpellingNumbers.com
How To Output Spelling Of Numbers In C++
How To Output Spelling Of Numbers In C++ – Learning how to spell numbers can be a challenge. However, learning to spell may be made much simpler with the correct resources. There are numerous sources
that will aid you in improving your spelling abilities, regardless of whether you are at school or working. These consist of advice and tricks, workbooks, and even games online.
The Associated Press format
If you are writing for a newspaper, or any other printed media, you need to be able to write numbers using the AP style. To make your writing more concise, the AP style offers guidelines for writing
numbers and other things.
Since its first appearance in 1953, The Associated Press Stylebook has seen hundreds of changes. The stylebook is now entering its 55th year. The stylebook is employed by the majority of American
newspapers, periodicals as well as news websites and websites that provide news.
Journalism is usually governed by AP Style. These guidelines include punctuation and the use of language. AP Style is a set of the best practices, which includes the use of capitalization, the use of
dates and times and references.
Regular numbers
An ordinal is a number that uniquely indicates a specific place in a list or sequence. These numbers are used frequently to indicate size, significance, or time passing. These figures can also show
the order in which things occur.
Depending on the circumstance depending on the situation, numbers are expressed verbally or numerically. The unique suffix is what makes the distinction between them.
To make it ordinal, add a ‘th’ to the top of any number. The ordinal number 31 may be represented as 31.
You can use ordinals to serve a variety of purposes, such as dates and names. It is equally crucial to differentiate between the ordinal and the cardinal.
Millions and trillions
Numerology can be used in numerous situations, including the stock exchange as well as geology, history and politics. Millions of dollars and billions are just two instances. A million is a natural
number prior to 1000,001, whereas trillions are after 999.999.999.
Millions is the annual revenue of a company. They are also used to determine the value the value of a fund, stock or other money item is worth. Additionally, billions serve as a measure of the value
of a company’s capital stock. You can check the accuracy of your estimates by using a calculator for unit conversion to convert billions into millions.
In the English language, fractions are used to denote specific items or parts of numbers. The denominator as well as the numerator are split into two distinct pieces. The denominator shows the number
of pieces of similar size were taken. While the numerator shows how many portions were broken down into.
Fractions can be expressed in mathematical terms or written in words. When writing fractions in words, you must be careful about spelling them out. This might be challenging since you may need to use
several hyphens, in particular when dealing with bigger fractions.
You can adhere to the following basic guidelines in writing fractions in words. You can start sentences by writing the numbers complete. Another option is writing fractions using decimal forms.
There is a good chance that you will use years in spelling numbers, whether you’re creating a research paper, a thesis, or an email. You may avoid typing out the same number again and maintain proper
formatting by employing a few tricks and tactics.
The numbers should be written in formal language. There are many style guides that provide different guidelines. The Chicago Manual of Style recommends using numbers between 100 and 1. However, it’s
not advised to use numbers that are larger than 401.
Of course, there are exceptions. One example is the American Psychological Association style guide (APA). Although it is not a specialized one, is widely used in scientific writing.
Time and date
The Associated Press style book provides some general guidelines regarding how to style numbers. Numbers 10 through higher are able to be styled using the numerals system. Numerology is also utilized
in different contexts. The most common rule for the first 5 numbers on your document is to write “n-mandated”. There are some exceptions.
The Chicago Manual of Technique as along with the AP stylebook recommend that you use plenty of numbers. This doesn’t mean that a version without numbers is not possible, of course. However, I am
able to confirm that there is a difference since I am an AP graduate.
A stylebook must always be consulted to find out which ones you are omitting. For instance, it’s crucial to remember to include a “t” for instance “time”.
Gallery of How To Output Spelling Of Numbers In C++
C Program For Complex Numbers Using Class C Programming Tutorial
C Write A Program That Will Receive Numbers From The User Until The
132 Decimal Number As Double Or Float In C Hindi YouTube | {"url":"https://www.spellingnumbers.com/how-to-output-spelling-of-numbers-in-c/","timestamp":"2024-11-02T08:06:59Z","content_type":"text/html","content_length":"61375","record_id":"<urn:uuid:2507c7a2-5465-4d3f-a37c-816a53667d56>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00348.warc.gz"} |
This presentation is based on a part of an academic course on Enterprise Risk Management (ERM) titled ‘Correlation, co-dependency and risk aggregation’ and covers topics such as: the Central Limit
Theorem (CLT), risk modelling using factor structures and copula based dependency structures
1Session 7: Correlation, co-dependency and risk aggregation 2Session 7: Correlation, co-dependency and risk aggregation 3Introduction 4Consider first multivariate Normal, i.e. Gaussian, case 5MVaR in
Gaussian Case 6E.g. outcomes uncorrelated, equal weights 7Session 7: Correlation, co-dependency and risk aggregation 8Central Limit Theorem 9CLT potentially applicable at two levels 10CLT can break
down in the following ways: 11Mathematical axioms and No arbitrage principle 12Session 7: Correlation, co-dependency and risk aggregation 13Dependency (aka co-dependency/co-movement) 14Factor
structure - notation 15Factor structure - handling idiosyncratic risk 16Advantages of introducing a factor structure 17Identifying factor structures - 3 main model types 18Identifying factor
structures in practice (1) 19Identifying factor structures in practice (2) 20Session 7: Correlation, co-dependency and risk aggregation 21Illustrative distribution (two risk factors) (1) 22
Illustrative distribution (two risk factors) (2) 23E.g. bivariate copula (1) 24E.g. bivariate copula (2) 25Copula and copula density 26Copulas 27Copulas and Sklar’s theorem 28Example Copulas 29Tail
dependence 30Interpretation of tail index 31Gaussian and Independence copula 32Simulating r.v.s linked by Gaussian copula 33Simulations with non-Gaussian copulas 34Fitting copulas empirically 35Risk
aggregation 36Risk aggregation using copulas 37Important Information
NAVIGATION LINKS
Contents | Next | ERM Lecture Series | {"url":"http://www.nematrian.com/ERM_7s","timestamp":"2024-11-11T17:15:27Z","content_type":"text/html","content_length":"32341","record_id":"<urn:uuid:aaa2fa0e-91fb-42d6-a01f-a3e9a5c6b9d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00142.warc.gz"} |
What caused the stock market crash of 1893?
The strike began at the Pullman Company in Chicago after Pullman refused to either lower rent in the company town or raise wages for its workers due to increased economic pressure from the Panic of
What caused the panic of 1890?
The crisis was precipitated by the near insolvency of Barings Bank in London. Barings, led by Edward Baring, 1st Baron Revelstoke, faced bankruptcy in November 1890 due mainly to excessive
risk-taking on poor investments in Argentina.
What was the Panic of 1893 and how did it impact farmers?
Between 1889 and 1893, more than eleven thousand Kansas farms went into foreclosure. Western farmers were being evicted from (thrown out of) their homes and farms; many were homeless.
What was the primary cause of the capitalist crash of 1873?
The discovery of large quantities of silver in the United States and several European colonies caused the panic of 1873 and thus a decline in the value of silver relative to gold, devaluing India’s
standard currency. This event was known as “the fall of the rupee”.
What ended the Panic of 1893?
Local police arrested Coxey and the march’s other leaders. The rest of the marchers quickly dispersed. The government refused to intervene. Fortunately for the United States populace, the Panic of
1893 ended by the end of 1897.
How did JP Morgan help the Panic of 1893?
The Federal Treasury was quickly running out of gold reserves, where President Cleveland was forced to turn to J.P. Morgan to bail out the U.S. government from economic failure. Morgan loaned the
treasury $65 million in gold in order to preserve the gold standard and preventing economic collapse.
What ended the Panic of 1873?
1873 – 1879Panic of 1873 / Period
Did Rockefeller bailout the US?
The Panic of 1907 from the U.S. Treasury and millions from John Pierpont (J.P.) Morgan, J.D. Rockefeller, and other bankers. Sum: $73 million (over $1.9 billion in 2019 dollars) from the U.S.
Treasury and millions from John Pierpont (J.P.) Morgan, J.D. Rockefeller, and other bankers.
Did J.P. Morgan hurt the economy?
Morgan was instrumental in helping to create the modern American economy. After the Panic of 1893, he reorganized many bankrupt railroads and industrial companies. He assembled U.S. Steel, the
world’s first billion-dollar corporation, and helped establish International Harvester and General Electric.
What was Grant’s response to the Panic of 1873?
The more he thought about it, however, the more he came to view the bill as an inflationary threat to the nation’s long-term credit. Grant vetoed the bill on April 22. In his veto message, Grant
feared that passage of the bill would lead to future efforts to print even more inflationary greenbacks.
Who was president during the Panic of 1873?
President Rutherford B. Hayes was forced to send federal troops to more than a half dozen states to stop the strikes. In the end, the fighting between strikers and troops left more than 100 people
dead and many more injured. Southern blacks suffered greatly during the depression.
What was happening in the USA in 1893?
May 1 – The 1893 World’s Fair, also known as the World’s Columbian Exposition, opens to the public in Chicago, Illinois. The first U.S. commemorative postage stamps and Coins are issued for the
Exposition. May 5 – Panic of 1893: A crash on the New York Stock Exchange starts a depression.
How did J.P. Morgan help the Panic of 1893?
What substance was Rockefeller throwing away because they couldn’t find a use for it?
Rockefeller, whose Standard Oil monopoly depended on widespread automotive consumption of gasoline, saw the possibility of ethanol-powered vehicles as enough of a threat to his business to warrant a
ban on ethanol under the guise of the temperance movement and Prohibition.
Who stopped the Panic of 1907?
Morgan’s deal-making finally stopped the Wall Street panic. Much economic damage, however, had al- ready spread across the country. The resulting depression of 1907–08 was severe, but probably would
have been greater if the bank panic had continued.
Who was to blame for the Panic of 1837?
Martin Van Buren
Martin Van Buren, who became president in March 1837, was largely blamed for the panic even though his inauguration had preceded the panic by only five weeks.
What happened to the Chicago area in 1893?
During the summer of 1893 commercial, industrial and manufacturing depression accompanied financial panic. Businesses failed and several major railroads, with Chicago as their transportation hub,
went into receivership, and control of ‘unprecedented mileage’ was handed over to the state and federal courts in bankruptcy.
How did the Panic of 1893 affect the economy?
The Panic of 1893 was an economic depression in the United States that began in 1893 and ended in 1897. It deeply affected every sector of the economy, and produced political upheaval that led to the
political realignment of 1896 and the presidency of William McKinley .
What caused the Great Depression of 1893-1894?
The economic misery was exacerbated by an extraordinarily harsh winter in 1893, Coxey’s army of unemployed marched to Washington, D.C. in 1894, and in April of 1894 more than 40,000 workers were
reported to be involved in over thirty national strikes.
What happened to the railroads in 1894?
The bad omen of investors switching from potentially volatile stocks to more stable bonds in 1894 was mirrored in railroads by slower acquisition of rolling stock. Railroad expansion rose again in
1895, but was arrested in 1897 by another economic trough. In 1893, the total railroad mileage in the U.S. was 176,803.6 miles. | {"url":"https://vidque.com/what-caused-the-stock-market-crash-of-1893/","timestamp":"2024-11-01T21:01:32Z","content_type":"text/html","content_length":"54920","record_id":"<urn:uuid:189ed3d5-475e-4626-b152-5b80f07394e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00783.warc.gz"} |
Introduction to bayNorm
bayNorm has been submitted to Bioconductor, once it is accepted, it can be installed via:
Currently bayNorm can be installed via github:
Quick start: for either one or multi groups of cells
The main function is bayNorm which is a wrapper function of prior parameters estimation and normalized array or matrix generation.
Essential parameters for running bayNorm are:
• Data: a SummarizedExperiment object or matrix (rows: genes, columns: cells).
• BETA_vec: a vector of probabilities which is of length equal to the number of cells.
• Conditions: If Conditions is provided, prior parameters will be estimated within each group of cells (we name this kind of procedure as “LL” procedure where “LL” stands for estimating both \(\mu
\) and \(\phi\) locally). Otherwise, bayNorm applied “GG” procedure for estimating prior parameters (estimating both \(\mu\) and \(\phi\) globally).
• Prior_type: Even if you have specified the Conditions, you can still choose to estimate prior parameters across all the cells by setting Prior_type="GG".
The input parameters BETA_vec, Conditions (if specified), UMI_sffl (if specified), Prior_type, FIX_MU, BB_SIZE and GR are stored in the list input_params which should be output from bayNorm. Objects
PRIORS together with input_params output from bayNorm should be input in bayNorm_sup for transforming between 3D array, mode or mean version’s output of normalized count matricies.
Estimation of capture efficiencies
Apart from the raw data, another parameter which user may need to provide is the mean capture efficiency \(<\beta>\) and hence \(\beta\) can be further calculated using scaling factors estimated from
any other methods. The default \(\beta\) is calculated to be total counts normalized to 6%. Or you can use BetaFun in bayNorm to estimate \(\beta\):
#Suppose the input data is a SummarizedExperiment object:
#Here we just used 30 cells for illustration
rse <- SummarizedExperiment::SummarizedExperiment(assays=SimpleList(counts=EXAMPLE_DATA_list$inputdata[,seq(1,30)]))
#SingleCellExperiment object can also be input in bayNorm:
#rse <- SingleCellExperiment::SingleCellExperiment(assays=list(counts=EXAMPLE_DATA_list$inputdata))
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.02299 0.03859 0.05255 0.06000 0.07473 0.11168
The function BetaFun selects a subset of genes such that outlier genes and genes with high proportion of zeros across cells are excluded. Then the total counts of the subset of genes in each cell are
normalized to \(<\beta>\). Another way is to normalize scaling factors estimated from the R package scran to \(<\beta>\).
Run bayNorm
BETA_vec can be set to be NULL, then the \(\beta\) are estimated to be total counts normalized to 6%.
#Return 3D array normalzied data, draw 20 samples from posterior distribution:
BETA_vec = NULL,
mean_version = FALSE,S=20
,verbose =FALSE,
parallel = TRUE)
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#Return 2D matrix normalized data (MAP of posterior):
#Simply set mode_version=TRUE, but keep mean_version=FALSE
#Return 2D matrix normalized data (mean of posterior):
#Simply set mean_version=TRUE, but keep mode_version=FALSE
Non-UMI scRNAseq dataset
bayNorm’s mathematical model is suitable for UMI dataset. However it can be also applied on non-UMI dataset. In bayNorm, you need to specify the following parameter: * UMI_sffl: bayNorm can also be
applied on the non-UMI dataset. However, user need to provide a scaled number. Raw data will be divided by the scaled number and bayNorm will be applied on the rounded scaled data. By doing so, the
Dropout vs Mean expression plots will be similar to that of UMI dataset.
Generate 3D array or 2D matrix with existing estimated prior parameters.
If you have run bayNorm on a dataset before but want to output another kind of data (3D array or 2D matrix), you can use the function bayNorm_sup. It is important to input the existing estimated
parameters by specifying the following parameter in bayNorm_sup:
• BETA_vec: If Conditions has been specified previously, then input unlist(bayNorm_output$BETA)
• PRIORS: input bayNorm_output$PRIORS_LIST
• Conditions: make sure to specify the same Conditions as before. You can find these two objects from the previous output of bayNorm function, which is a list.
#Now if you want to generate 2D matrix (MAP) using the same prior
#estimates as generated before:
mean_version = FALSE,
verbose =FALSE)
#Or you may want to generate 2D matrix
#(mean of posterior) using the same prior
#estimates as generated before:
mean_version = TRUE,
verbose =FALSE)
Other functions
1. Prior_fun: This is a wrapper function of EstPrior, BB_Fun and AdjustSIZE_fun:
□ EstPrior: Estimating priors using MME methods.
□ BB_Fun: Estimating priors by maximizing marginal likelihood function with respect to either \(\phi\) or both \(\mu\) and \(\phi\).
□ AdjustSIZE_fun: Adjusting \(\phi\) estimated from BB_Fun.
2. noisy_gene_detection: This is a higher wrapper function over bayNorm, bayNorm_sup and SyntheticControl. For details about the rationale behind this function, see the Methods part in the paper 2.
3. SyntheticControl: Given a real scRNA-seq data with estimated \(\beta\), it will estimate \(\mu\) and \(\phi\) from it and simulate control data using Poisson distribution. See the Methods part in
the paper 2 for more details.
#Downstream analysis: DE genes detection DE function used in bayNorm paper, which was kindly provided by the author of SCnorm.
SCnorm_runMAST3 <- function(Data, NumCells) {
resultss<-SCnorm_runMAST(Data, NumCells)
} else if(length(dim(Data))==3){
resultss<- foreach(sampleind=1:dim(Data)[3],.combine=cbind)%do%{
qq<-SCnorm_runMAST(Data[,,sampleind], NumCells)
SCnorm_runMAST <- function(Data, NumCells) {
Data = as.matrix(log2(Data+1))
qq_temp<- rowMeans(G2)-rowMeans(G1)
numGenes = dim(Data)[1]
datalong = melt(Data)
Cond = c(rep("c1", NumCells[1]*numGenes), rep("c2", NumCells[2]*numGenes))
dataL = cbind(datalong, Cond)
colnames(dataL) = c("gene","cell","value","Cond")
dataL$gene = factor(dataL$gene)
dataL$cell = factor(dataL$cell)
vdata = FromFlatDF(dataframe = dataL, idvars = "cell", primerid = "gene", measurement = "value", id = numeric(0), cellvars = "Cond", featurevars = NULL, phenovars = NULL)
zlm.output = zlm(~ Cond, vdata, method='glm', ebayes=TRUE)
zlm.lr = lrTest(zlm.output, 'Cond')
gpval = zlm.lr[,,'Pr(>Chisq)']
adjpval = p.adjust(gpval[,1],"BH") ## Use only pvalues from the continuous part.
adjpval = adjpval[rownames(Data)]
return(list(adjpval=adjpval, logFC_re=qq_temp))
#Now, detect DE genes between two groups of cells with 15 cells in each group respectively
#For 3D array
DE_out<-SCnorm_runMAST3(Data=bayNorm_3D$Bay_out, NumCells=c(15,15))
#DE genes are called with threshold 0.05:
#For 2D array
DE_out<-SCnorm_runMAST3(Data=bayNorm_2D$Bay_out, NumCells=c(15,15))
Rationale of bayNorm
A scRNA-seq dataset is typically represented in a matrix of dimension \(P\times Q\), where P denotes the total number of genes observed and Q denotes the total number of cells studied. The element \
(x_{ij}\) (\(i \in \{1,2,\ldots, P\}\) and \(j \in \{1,2,\ldots, Q\}\)) in the matrix represents the number of transcripts reported for the \(i^{\text{th}}\) gene in the \(j^{\text{th}}\) cell. This
is equal to the total number of sequencing reads mapping to that gene in that cell for a non-UMI protocol. For UMI based protocols this is equal to the number of individual UMIs mapping to each gene.
The matrix can include data from different groups or batches of cells, representing different biological conditions. This can be represented as a vector of labels for the cell groups or conditions (\
(C_j\)). bayNorm generates for each gene (i) in each cell (j) a posterior distribution of original expression counts (\(x_{ij}^0\)), given the observed scRNA-seq read count for that gene (\(x_{ij}\)
A common approach for normalizing scRNA-seq data is based on the use of a global scaling factor (\(s_j\)), ignoring any gene specific biases. The normalized data \(\tilde{x}_{ij}\) is obtained by
dividing the raw data for each cell \(j\) by the its global scaling factor \(s_j\):
\[$$\label{equ::scaling} \tilde{x}_{ij} = \frac{x_{ij}}{s_j}$$\]
In bayNorm, we implement global scaling using a Bayesian approach. We assume given the original number of transcripts in the cell (\(x_{ij}^0\)), the number of transcripts observed (\(x_{ij}\))
follows a Binomial model with probability \(\beta_j\), which we refer to as capture effeiciency and it represents the probability of original transcripts in the cell to be observed. In addition, we
assume that the original number or true count of the \(i^{\text{th}}\) gene in the \(j^{\text{th}}\) cell (\(x_{ij}^0\)) follows Negative Binomial distribution with parameters mean (\(\mu\)), size
(or dispersion parameter, \(\phi\)), such that: \[\Pr(x^0_{ij}=n|\phi_i,\mu_i)=\frac{\Gamma(n+\phi_i)}{\Gamma(\phi_i)n!}(\frac{\phi_i}{\mu_i+\phi_i})^{\phi_i}(\frac{\mu_i}{\mu_i+\phi_i})^{n}\]
So, overall we have the following model:
\[$$\label{model:bay} \begin{split} &x_{ij}\sim \text{Binom}(x_{ij}^0,\text{prob}=\beta_j)\\ &x_{ij}^0 \sim \text{NB}(\text{mean}=\mu_i,\text{size}=\phi_i) \end{split}$$\]
Using the Bayes rule, we have the following posterior distribution of original number of mRNAs for each gene in each cell:
\[$$\underbrace{\Pr(x_{ij}^0|x_{ij},\beta_j,\mu_i,\phi_i)}_\text{Posterior} = \dfrac{ \overbrace{\Pr(x_{ij}|x_{ij}^0,\beta_j)}^\text{Likelihood} \times \overbrace{\Pr(x_{ij}^0|\mu_i,\phi_i)}^\text
{Prior}}{\underbrace{\Pr(x_{ij}|\mu_i,\phi_i,\beta_j)}_\text{Marginal likelihood}}$$\]
The prior parameters \(\mu\) and \(\phi\) of each gene were estimated using an empirical Bayesian method as discussed in detail in Supplementary Information of 1.
For more details about the rationale of bayNorm, please see 1.
Estimation of capture efficiencies
Cell specific capture efficiency \(\beta_j\) and global scaling factor (\(s_j\)) are closely related. We can transform scaling factors estimated by different methods (see below) into \(\beta_j\)
values with the following formula:
\[$$\beta_j = (s_j/\bar{s})\bar{\beta}$$\]
\(\bar{\beta}\) or \(<\beta>\), a scalar, is an estimate of global mean capture efficiency across all cells, which ranges between 0 and 1.
There are two different methods for estimating \(\bar{\beta}\) and \(\beta_j\):
1. If spike-ins or smFISH data are available they can be used to estimate capture efficiencies. We can either divide the total number of observed spik-ins in each cell by the total number of input
spike-ins, or we can fit a linear regression to estimate the cell specific \(\beta_j\). If smFISH data is available, we can fit a linear regression between the mean expression of raw data
(response variable) and the mean expression of the smFISH data (explanatory variable). The coefficient of the explanatory variable can be used as \(\bar{\beta}\).
2. The raw data itself can be directly used for estimation of cell specific global scaling factors (\(s_j\)). Then an estimate of \(\bar{\beta}\) can be used to estimate \(\beta_j\). There are
different methods available for estimation of global scaling factors. Some were developed for bulk RNA-seq data and some are specific to scRNA-seq data. The value of \(\bar{\beta}\) depends on
the protocol used and can be batch dependent. For example, for Droplet based protocol, it is about 0.06 (inDrop) or 0.12 (Drop-seq). \(\bar{\beta}\) can also be estimated by spike-ins or smFISH
data as explained above.
We finally note, that estimates of capture efficiency discussed above will assume cells have simular original transcript content. Therefore, the bayNorm outputs estimates of original transcript
counts for a typical cell, which is corrected for variation in cell size and transcript content. This is usually desirable for down-stream analysis such as DE detection. However, if one is interested
in absolute origianl count and has additional information such as cell size or total transcirpt content per cell, the capture efficiencies can be approporiatly rescaled for this purpose.
Estimation of prior parameters
Maximisation of marginal distribution
Using an emperical bayes approach, one can use the maximisation of marginal likelihood distribution of the observed counts across cells to estimate prior parameters. Let \(M_i\) denotes the marginal
likelihood function for the \(i^\text{th}\) gene across cells. Assuming independence between cells, the log-marginal distribution for the \(i^\text{th}\) gene is
\[$$\log M_i = \sum_{j=1}^Q \log \Pr(x_{ij}|\mu_i,\phi_i,\beta_j),$$\]
where \(\Pr(x_{ij}|\mu_i,\phi_i,\beta_j)\) is the Negative Binomial (see Methods). Maximizing of this equation yields the pair \((\mu_i,\phi_i)\).
The above optimization needs to be done for each of the P genes. We refer to the \(\phi\) and/or \(\mu\) estimated by maximizing marginal distribution as BB estimates for convenience, because bayNorm
utilizes spectral projected gradient method (spg) from the R package named ``BB’’. Optimizing the marginal distribution with respect to both \(\mu\) and \(\phi\) (2D optimization) is computationally
intensive. If we had a good estimate \(\mu\), then we could optimize the marginal distribution with respect to \(\phi\) alone, which would be much more efficient.
Method of Moments
A heuristic way of estimating \(\mu_i\) and \(\phi_i\) is through a variant of the Method of Moments. The first step is to do a simple normalization of the raw data, to scale expressions given the
cell specific capture efficiencies (\(\beta_j\)). The simple normalized count \(x_{ij}^s\) is calculated as following:
where the numerator of the scaling factor of \(x_{ij}\) is obtained by taking the average of scaled total counts across cells.
Based on simple normalized data, we are able to estimate prior parameters \(\mu\) and \(\phi\) of the Negative Binomial distribution using the Method of Moments Estimation (MME), which simply equates
the theoretical and empirical moments. This estimation method is fast and simulations suggests it provides good estimates of \(\mu\) but the drawback is that the estimation of \(\phi\) show a
systematic bias (see Supplementary Figure S24 a-b in 1).
The combined method (default setting in bayNorm for estimating priors)
Based on simulation studies (Supplementary Figure S24 in 1), the most robust and efficient estimation of \(\mu\) and \(\phi\) can be obtained using the following combined approach, which is the
default setting in bayNorm:
1. Based on simple normalized data, we use the MME method for each gene to obtain MME estimated \(\mu\) and \(\phi\).
2. Although the BB estimated \(\phi\) is much closer to the true \(\phi\), many estimates are at the upper boundary of the search space (Supplementary Figures S24 c-d in 1). So, we find adjusting
the MME estimated \(\phi\) by a factor which can be estimated by fitting a linear regression between MME estimated \(\phi\) and BB estimated \(\phi\) works best (Supplementary Figures S24 c-d in
1). This adjusted MME estimated \(\phi\) together with the MME estimated \(\mu\) and estimates of \(\beta_j\) can be used in approximating posterior distribution for each gene in each cell.
Cells are grouped together for prior estimation, based on cell-specific attributes (\(C_j\)). Prior estimation can be done over all cells irrespective of the experimental condition. We refer to this
procedure as “global”. Alternatively, suppose that there are multiple groups of cells in the datasets and we have reasons to believe each group could behave differently. Then we can estimate the
prior parameters “\(\mu\) and \(\phi\)” within each group respectively (within groups with the same \(C_j\) value). We refer to this procedure as “local”. Estimating prior parameters across a certain
group of cells based on “global” procedure allow for removing potential batch effects. Multiple groups normalization based on “local” procedure allows for amplifying the inter-groups’ differences
while mitigating the intra-group’s variability, which is suitable for DE detection.
Session information
## R version 4.0.3 (2020-10-10)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.5 LTS
## Matrix products: default
## BLAS: /home/biocbuild/bbs-3.12-bioc/R/lib/libRblas.so
## LAPACK: /home/biocbuild/bbs-3.12-bioc/R/lib/libRlapack.so
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
## attached base packages:
## [1] parallel stats4 stats graphics grDevices utils datasets
## [8] methods base
## other attached packages:
## [1] SingleCellExperiment_1.12.0 SummarizedExperiment_1.20.0
## [3] Biobase_2.50.0 GenomicRanges_1.42.0
## [5] GenomeInfoDb_1.26.0 IRanges_2.24.0
## [7] S4Vectors_0.28.0 BiocGenerics_0.36.0
## [9] MatrixGenerics_1.2.0 matrixStats_0.57.0
## [11] bayNorm_1.8.0 knitr_1.30
## [13] BiocStyle_2.18.0
## loaded via a namespace (and not attached):
## [1] Rcpp_1.0.5 XVector_0.30.0 compiler_4.0.3
## [4] BiocManager_1.30.10 zlibbioc_1.36.0 bitops_1.0-6
## [7] iterators_1.0.13 tools_4.0.3 fitdistrplus_1.1-1
## [10] digest_0.6.27 evaluate_0.14 lattice_0.20-41
## [13] doSNOW_1.0.19 rlang_0.4.8 Matrix_1.2-18
## [16] foreach_1.5.1 DelayedArray_0.16.0 yaml_2.2.1
## [19] xfun_0.18 GenomeInfoDbData_1.2.4 stringr_1.4.0
## [22] locfit_1.5-9.4 grid_4.0.3 snow_0.4-3
## [25] survival_3.2-7 BiocParallel_1.24.0 rmarkdown_2.5
## [28] bookdown_0.21 magrittr_1.5 codetools_0.2-16
## [31] htmltools_0.5.0 MASS_7.3-53 splines_4.0.3
## [34] stringi_1.5.3 RCurl_1.98-1.2 | {"url":"https://bioconductor.statistik.tu-dortmund.de/packages/3.12/bioc/vignettes/bayNorm/inst/doc/bayNorm.html","timestamp":"2024-11-13T09:16:34Z","content_type":"text/html","content_length":"671483","record_id":"<urn:uuid:2eac7e8a-234b-4162-9875-d1751be9cac3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00011.warc.gz"} |
How I got the CS:GO bomb beep pattern
Posted on 21th of July 2020.
The formula
In short, here is the formula for the beep pattern of the CS:GO bomb:
Where f(t) gives the BPS (Beeps Per Second), and t (0 ≤ t ≤ 45) is the time in seconds (the in-game bomb explodes after 45 seconds).
For a more generalized formula:
Where g(t) gives the BPS (Beeps Per Second), and p (0.0 ≤ p ≤ 1.0) is the percentage of the time that has passed since the bomb has been armed. This is more useful when you don't want to be stuck
using the 45 second explosion time frame.
A plot of time v.s. beeps per second
The length of each beep is fixed at 125ms per beep. These formulas approximate the beep pattern of CS:GO's bomb very accurately.
For information about the note(s) of the bomb beeping, I suggest taking a look at this reddit post. Apparently the notes are different depending on if the bomb is planted on bombsite A or B,
fascinating stuff!
How I got the formula
The interval between the beeps are very clearly exponential, so I needed to find or generate a matching exponential formula. After some digging online I couldn't find any. At first, I tried
estimating the beeps per second from a YouTube video of the bomb sound roughly every 5-10 seconds, and filling this data into an Excel table. When I got this data, I tried to let excel guesstimate
the (exponential) formula for it. This was without any luck; the beep pattern still seemed a bit off. Looking back at it, it was pretty obvious seeing as I was just loosely guessing the beeps per
second of the original sound.
At this time I had the choice to go all-in and find the exact formula or to make do with the relatively bad formula I had figured out. Of course I chose to get the exact formula!
So, it was time to download an mp3 file of the bomb sound of the game. Luckily I found one pretty easily on YouTube, even without any background music. I imported the file into Audacity and started
to find the exact beeps per second for some timestamps, roughly 5 seconds apart and 2 seconds apart when the bomb was about to explode. To find the beeps per second, I selected the part between the
start of a beep and the start of the beep after that one. Audacity told me the length of the selection in milliseconds which is accurate enough for what I'm trying to do. Dividing 1 by the length of
the selection in seconds gave me the bps at that time.
After gathering 17 data points, it was time to figure out an exponential equation that fits the given data. After using a really old-school application called CADRE Regression, it figured out the
values for A0, A1 and A2 for a second degree exponential function:
I used these values in my in-real-life CS:GO (fake!) bomb project to match the in-game beep pattern.
Code snippet of the CS:GO bomb project which calculates beep time between beeps. Note that the A1 and A2 values are different because I adjusted the formula to accept time t in milliseconds instead
of seconds. | {"url":"https://gritter.me/posts/csgo-bomb-beep-pattern/","timestamp":"2024-11-06T19:50:08Z","content_type":"text/html","content_length":"7508","record_id":"<urn:uuid:63196c20-1f4a-45fc-9935-57b018e6eed4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00760.warc.gz"} |
Rolling Average - Futility Closet
In a standard 10-frame game of bowling, the lowest possible score is 0 (all gutterballs) and the highest is 300 (all strikes). An average player falls somewhere between these extremes. In 1985,
Central Missouri State University mathematicians Curtis Cooper and Robert Kennedy wondered what the game’s theoretical average score is — if you compiled the score sheets for every legally possible
game of bowling, what would be the arithmetic mean of the scores?
It turns out it’s pretty low. There are (66^9)(241) possible games, which is about 5.7 × 10^18. If we divide that into the total number of points scored in these games, we get
which is about 80 (79.7439 …).
This “might make you feel better about your average,” Cooper and Kennedy conclude. “The mean bowling score is indeed awful even if you are just an occasional bowler. Even though this information is
interesting, there are more difficult questions about the game of bowling that could be asked. For example, you might wish to determine the standard deviation of the set of bowling scores and hence
know more about the distribution of the set of all bowling scores. But the exact determination of the distibution of the set of scores is, in our opinion, a difficult problem. For example, given an
integer k between 0 and 300, how many different bowling games have the score k? This, we leave as an open problem.”
(Curtis N. Cooper and Robert E. Kennedy, “Is the Mean Bowling Score Awful?”, Journal of Recreational Mathematics 18:3, 1985-86.) | {"url":"https://www.futilitycloset.com/2014/11/13/rolling-average/","timestamp":"2024-11-07T10:59:33Z","content_type":"text/html","content_length":"58069","record_id":"<urn:uuid:8ad6efe5-c4df-4147-89f9-2d771589ea22>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00171.warc.gz"} |
linear regression u . nonlinear functions
with LINREG them is a useful application of the method of least squares are available - the description of a set of experimental data by a curve or a theoretical formula to obtain a linear or non -
linear relationship which best fits the data - when possible small errors .
LinReg consists of the parts of the program :
• linear regression
• Nonlinear functions - straight - equations
Evaluation of measured values and detecting the measured value characteristic
Calibration of measurement systems by linear regression
Detecting contexts within a series of studies
Representation of processes with linear or non- linear relation
Calculating a mathematical equation by y = ax + b or optionally for a non - linear relationship until 9 polynomial by:
Y = A0 + + a2x2 a1x + a3x3 ..... + amxm
Creating a formula from a value table
Examples : - Creation of characteristics for pumps
- Calibration curves for Photometry - polarimetry - AAS - etc.
- Preparation of performance curves for machines ,
graphical representation of curves with manual change possibility of coefficients
Creating a mathematical relation of graphic templates for acquisition
in computer programs
linear regression
- Mathematical relationship of value pairs as a function of x by:
f ( x) = ax + b for straight lines with and without intercept ( b = 0) possibility
calculating force with by 0/0
- Error for the slope and intercept .
- Error for each measured value in % and absolute.
- Correlation coefficient.
Graphical representation of the calculated function on the screen - at variable
Axis geometry
In order to optimize the calculation of additional pairs of values can be appended , deleted or changed .
non-linear functions
For the presentation of functions and technical processes with mathematical operations in a non-linear curve.
Creating a formula from a value table with nonlinear history
Detection of operations with non-linear character
- Creation of curves or absorption curves
- Performance curves for machines
- Forming Calculations n in a degree function.
For a number of measurements a mathematical function is calculated to 9 polynomial.
Graphical representation of the calculated function on the screen
Line equations
manual drawing straight in the coordinate system
Target of slope + intercept or 2 points
Calculation of intersection, intersection angles, axle breakdown
Representation of parabolas with X / Y symmetry and form Y = ax² + bx + c | {"url":"http://scient-service.de/index.php/en/software/linreg","timestamp":"2024-11-06T23:22:08Z","content_type":"text/html","content_length":"27811","record_id":"<urn:uuid:5feea3ce-6e76-4fd7-a422-28c438c131c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00795.warc.gz"} |
Code 4 Tomorrow
Comparison Operators
Comparison operators are ways to compare data
• == means "equal to"
Don't confuse == with =. = is for assignment while == is for checking equality
Example Code
# Comparison Operators
print(5 > 9) # False
print(10 >= 9) # True
print(5 < 9) # True
print(10 <= 9) # False
print(9 == 9) # True
print(10 == 9) # False
print(9 != 9) # False
print(10 != 9) # True
In Python syntax, what would you write to say you have at least $4.50?
Compare X
You start with x = 10. Compare x with another number using a comparison operator such that if you print x (comparison operator) (another number), it prints False.
Write a program that takes an integer as input and displays True if the integer is even. Otherwise, it displays False.
No Greater Than
Create a program that takes a POSITIVE integer as an input and checks if it is NO GREATER than 100. Print True if it is and False if it isn't. YOU MAY NOT USE GREATER THAN or LESS THAN OPERATORS (>,
<, >=, or <=). Find a way to do this problem using only the == operator and any math operators you want
Copyright © 2021 Code 4 Tomorrow. All rights reserved.
The code in this course is licensed under the
MIT License
If you would like to use content from any of our courses, you must obtain our explicit written permission and provide credit. Please contact classes@code4tomorrow.org for inquiries. | {"url":"https://www.code4tomorrow.org/courses/python/beginner/ch.-3-operators/3.3-comparison-operators","timestamp":"2024-11-02T17:39:10Z","content_type":"text/html","content_length":"98330","record_id":"<urn:uuid:2612df26-0d33-41a6-9862-059de0840d33>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00151.warc.gz"} |
Blog: Logs for the MRL Meeting Held on 2019-11-18
Logs for the MRL Meeting Held on 2019-11-18
November 18, 2019
<sarang> First, GREETINGS
<sarang> Hello
<hyc> heylo
<mikerah> hello
<sarang> For today's ROUNDTABLE, I have a few things of interest to share
<sarang> Updates to the RingCT 3.0 analysis and code reflect its two provably-sound versions: https://github.com/SarangNoether/skunkworks/blob/sublinear/rct3.md
<sarang> One version is the authors' padded-input version that implies some restrictions on the signer count
<sarang> The other version is a backport I did of the exploit fix from their newer version to the original one, with corresponding changes to security proofs
<sarang> Triptych analysis now reflects an optimized multi-signer version: https://github.com/SarangNoether/skunkworks/blob/sublinear/triptych.md
<sarang> as does its code and the draft writeup
<sarang> I have shared the writeup with a few additional researchers as well, in the hope of getting extra eyes on the soundness proof
<sarang> The CLSAG paper was unfortunately rejected for the Financial Cryptography 2020 conference (which only had a 22% acceptance rate, according to the committee)
<sarang> Here are the reviewer comments: https://gist.github.com/SarangNoether/e39db743c3260448c1d67c3622b43f4b
<sarang> Reviewer A, who recommended rejection, made some detailed points about the security model: particularly how ambiguity and linkability are treated relative to some other papers
<sarang> Once suraeNoether returns, I want to discuss the particulars of this
<sarang> I'm not convinced that the more complete linkability treatment (which ties in ambiguity as well) is needed, given the use case
<sarang> However, the recommendation for a more robust treatment of signer ambiguity is fair
<sarang> The notes about the k-OMDL assumption being less common are also fair, but that's the best hardness assumption that was found for the proofs in question
<sarang> Related to CLSAG: Derek at OSTIF tells me that JP Aumasson, who quoted $7200 to review the paper, is available presently to do so, and it's not clear at the moment when he would be
unavailable again
<sarang> I didn't sense a lot of broad support for this earlier
<sarang> Given that the FC2020 review comments just came back, the paper should be updated to reflect the notes before being sent off to anyone for additional review
<sarang> So the timeline on this seems unclear right now
<sarang> Does anyone else wish to share interesting news or research?
<sarang> Righto
<sarang> My ACTION items are to address CLSAG reviewer comments to update the preprint, finalize single-signer Triptych analysis and the associated preprint, continue working with others on whether
multi-signer soundness is provable using known assumptions, and examine the current state of suraeNoether's graph-matching work
<sarang> What a quiet meeting =p
<MalMen> normaly this meetings are you talking with suraeNoether :P (at least the lasts I watch)
<MalMen> I am reading the links you posted, thank you for the heads up
<sarang> selsta also posted this link elsewhere, which I found very interesting: https://medium.com/dragonfly-research/breaking-mimblewimble-privacy-model-84bcd67bfe52
<sarang> Looking at a practical attack on Grin using network observation
<MalMen> I did start to understud better how MW work with that research
<MalMen> there is not much they can do about this kinda of attack unless users kinda of "coinjoin" the transactions with know peers before releasing them to the all network
<MalMen> for what I did understund about it
<sarang> It definitely highlights the importance of the network and propagation layer in transactions
<kico> seems like using 8 peers only by default is kinda dangerous
<sarang> vtnerd has been looking into the tricky interactions involved with integrating Dandelion++, I2P, Tor, and the like
<kico> I mean for dandelion to work "properly"
<MalMen> It should be possible to get some farly good level of privacy with MW, but the way the transactions are propagated would need to became more complex in order to ofuscate the way they are
<sarang> How so? Dandelion++ provides restrictions on stem neighbor selection for a given time epoch
<MalMen> kinda of what Bitcoin think they can do on layer2 to obuscate the linkability on layer1
<kico> I guess if one connects to more peers it increases the chance that it aggregates transactions before they're peered to the network for what I understood from that paper
<sarang> A good lesson on how it's tricky to assume things about transactions before they reach the chain, I suppose
<sarang> Well, anything else of interest to discuss before closing the meeting?
<sarang> If anyone has thoughts or comments relating to the CLSAG reviewer notes, I'd be glad to hear them
<MalMen> just one quick question from someone that dont know much abouth CLSAG
<sarang> sure
<MalMen> CLSAG will not make possible second layer networkds (missing the word here) ?
<sarang> No, it doesn't introduce any new functionality toward that
<sarang> Its only purpose is to make signatures smaller and a bit faster
<sarang> Off-chain solutions are limited by a lack of scripting and ambiguity around useful things like non-interactive refunds, etc.
<MalMen> there is something that can possibility second layer in the future for monero yet or we still far from it ?
<sarang> Introducing such things is tricky (DLSAG is one attempt, but suffers from a linking problem)
<MalMen> ok, thank you for the heads up
<sarang> If you are willing to work through the technical language and definitions, the DLSAG paper has some very clever constructions: https://eprint.iacr.org/2019/595
<sarang> (disclaimer: I am a co-author on the paper, but did not come up with the original idea)
<MalMen> Ok, I will take a look
<sarang> It highlights how tricky it can be to enable swaps, payment channels, and the like, given the protocol and indistinguishability restrictions
<sarang> Oh, I should also add that the DLSAG paper _was_ accepted to the FC2020 conferences
<MalMen> oh, that is good
<MalMen> I believe that it will be almost impossible to make any "Non-Interactive Refund Transactions" (just took it from the paper) seemless as the other transactions
<sarang> DLSAG comes very close, but the linking problem is troublesome
<MalMen> peraphs we would come to a place where we need to choose if we want to acept anything like DLSAG knowing its less private but allow us to have hotswaps and some kind of lightning network
<MalMen> or just keep with onechain transactions
<sarang> I would not expect broad support for such a tradeoff, but who knows
<sarang> And perhaps a new idea that _does_ solve the problem will arise at some point
<MalMen> we hope so
<sarang> Anyway, are there other topics of interest to bring up before closing the meeting?
<MalMen> well, thats all from me, thanks for your time sarang *
<sarang> no problem
<sarang> OK, the meeting is over! Thanks to everyone for attending
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