instruction stringlengths 0 29.8k ⌀ |
|---|
A function in mathematics is a map between input and output values. Fixed points are input values (for a function) which map to output values satisfying equality with the input.
For the equality function $f(x) = x$ the set of input value equals to the set of fixed points of the function. For a function $f(x) = x^2$ ... |
A function in mathematics is a map between input and output values. Fixed points are input values (for a function) which map to output values satisfying equality with the input.
For the equality function $f(x) = x$ the set of input value equals to the set of fixed points of the function. For a function $f(x) = x^2$ ... |
A function in mathematics is a map between input and output values. Fixed points are input values (for a function) which map to output values satisfying equality with the input.
For the equality function $f(x) = x$ the set of input value equals to the set of fixed points of the function. For a function $f(x) = x^2$ ... |
The problem is:
$$∃x∀y(x ≥ y)$$
With a domain of all real positive integers.
The negation is:
$$∀x∃y(x < y)$$
so, if $y = x + 1$, the negation is true.
That means the negation of the negation (or, the original problem) is false.
My question is, that if the original problem is $∃x∀y(x ≥ y)$, why can... |
null |
The problem is:
$$\exists x \forall y (x \ge y)$$
With a domain of all real positive integers.
The negation is:
$$\forall x \exists y (x < y)$$
so, if $y = x + 1$, the negation is true.
That means the negation of the negation (i.e. the original problem) is false.
My question is, that if the origina... |
Negation of nested quantifiers |
|logic|first-order-logic| |
null |
Let me elaborate a bit on meisterluk's answer: Imagine we are trying to define the factorial function: remember the definition of the factorial function:
fact 0 = 1
fact (n+1) = n*(fact n)
Now in some PL frameworks (namely the [$\lambda$-calculus][1], it isn't immediately obvious how to define such a... |
What is does it mean when an argument to a function is called a **dummy argument**? I have not encountered this term outside Fortran, is it a general term in computer science? What would be examples of arguments passed as dummy and not passed as dummy?
I found [this definition][1], which I am not sure that I underst... |
What does "dummy argument" mean? |
|terminology|functional-programming| |
In Big Theta notation used for defining the running time of algorithm complexity, are the constants **c1 and c2 different** for every value of n??.
Definition: Theta (g(n)) = { f(n): there exist positive constants c1,c2 and n0 such that 0<= c1g(n) <= f(n) <= c2g(n) for all n >= n0 }
|
Value of constants in Big Theta notation |
|algorithms|asymptotics| |
|terminology|programming-languages|functional-programming| |
|asymptotics|mathematical-analysis|landau-notation| |
## "Dummy" as "unused"
In general (not in Fortran) a dummy argument is an argument that will not be used by the body of the function.
For example, in the following function, `z` is a dummy argument but `x` and `y` are not.
int f(int x, int y, int z) {
return (x + y);
}
They can be used for s... |
Garey and Johnson are referring to the fact that any encoding scheme for some instance $I$ of a problem $\Pi$ will only differ in length (i.e. number of bits) by a polynomial amount. For example, consider two possible ways to encode a graph: adjacency matrix, and adjacency list.
It is not possible to obtain a super-... |
In Big Theta notation used for defining the running time of an algorithm, are the constants $c_1$ and $c_2$ *different* for every value of $n$?
Definition:
$$\Theta (g(n)) =\\ \{ f(n): \text{there exist constants $c_1,c_2,n_0>0$ s.t. $c_1g(n) \leq f(n) \leq c_2g(n)$ for all $n \geq n_0$} \}.$$
|
No, the constants $c_1,c_2$ are the same for all values of $n$. If they could depend on $n$ then the definition would be meaningless since $\Theta(g(n))$ would contain all positive functions. Indeed, suppose $f,g$ are two positive functions. Then for all $n$, $$c_1g(n) \leq f(n) \leq c_2g(n) \text{ for } c_1 = c_2 = \f... |
null |
In Big Theta notation used for defining the running time of an algorithm, are the constants $c_1$ and $c_2$ *different* for every value of $n$?
Definition:
$\qquad \displaystyle \Theta (g(n)) = \{ f(n): \exists\, c_1,c_2,n_0>0. \forall n \geq n_0.\ \ c_1g(n) \leq f(n) \leq c_2g(n) \}.$
|
|terminology|asymptotics|mathematical-analysis|landau-notation| |
I'm no logician, but your base predicate states that there exists some real number that's larger than or equal to all other real numbers (i.e that *there is a highest real number*).
The inverse states that every real number has a number larger than it (i.e. that *there is no highest real number*).
Your test cases... |
This question was answered by M. Alaggan at cs.stackexchange. Below is the original answer:
"I'll start with your last question (in the comments); namely "Why doesn't x = y satisfy the initial problem".
The answer is in the quantifiers. Read from left to right. It starts with "there exists" X. So pick an X in you... |
The problem is:
∃x∀y(x ≥ y)
With a domain of all real positive integers.
The negation is:
∀x∃y(x < y)
so, if y = x + 1
the negation is true.
That means the negation of the negation (or, the original problem) is false.
My question is, that if the original problem is ∃x∀y(x ≥ y), why can't x = y a... |
Predicate Logic Paradox |
|logic|discrete-mathematics| |
Your base predicate states that there exists some real number that's larger than or equal to all other real numbers (i.e that *there is a highest real number*).
The inverse states that every real number has a number larger than it (i.e. that *there is no highest real number*).
Your test cases don't produce a para... |
null |
I'm reading through `"Computers and Intractability: A guide to the Theory of NP-Completeness" by Michael R. Garey and David S. Johnson, p. 20` and I came across this concept of a function that is polynomially related to input lengths obtained using some encoding scheme. Let $$Len:D_{\Pi}\rightarrow \mathbb Z^+$$ be a f... |
null |
null |
A function is a well defined as: [relation between a set of inputs and a set of permissible outputs](http://en.wikipedia.org/wiki/Function_(mathematics)).
A dummy variable would be any input that had no effect on the output.
In "C" the time function takes a pointer to a time_t object as input. As this pointer is o... |
Actually the answer lies already in your question. You write:
> ... are the constants $c_1$ and $c_2$ different for every value of $n$?
A quantity $c$ is a *constant* only if it's value is unaffected by changes to the asymptotic variable (i.e. $n$). Therefore, $c$ must have the same value for all values of $n$. |
A fixed point of a function $f: A \to A$ is an element $x$ for which $f(x)$ is equal to $x$. For example, the function $x^2$ has two fixed points, which are the values $0$ and $1$, and the function $x^3$ has three fixed points. Mathematically that is the definition.
Now, depending on the mathematical structure you ... |
Actually the answer lies already in your question. You write:
> ... are the constants $c_1$ and $c_2$ different for every value of $n$?
A quantity $c$ is a *constant* only if its value is unaffected by changes to the asymptotic variable (i.e. $n$). Therefore, $c$ must have the same value for all values of $n$. |
I wonder whether there is a simple algorithm to do this work:
Say I have a collection of objects $C$, and a binary relation $R: C\times C$ that is auto-reflective ($\forall c\in C: c R c$) , and symmetric ($\forall b,c\in C: b R c \implies c R b$). The relation is not necessarily transitive.
I want to find ... |
Algorthm that hashes a colection of sets following their disjointness relation |
|algorithms|hash| |
In computer science, the arguably most prominent use of fixed-points is in [lattice theory](https://en.wikipedia.org/wiki/Lattice_(order))¹. A lattice is a partially ordered set $(S, \leq)$ with the additional property that given any two elements $x,y \in S$, the set $\{x,y\}$ has both a supremum and infimum (in $S$).
... |
> **Possible Duplicate:**
> [Negation of nested quantifiers](http://cs.stackexchange.com/questions/3463/negation-of-nested-quantifiers)
<!-- End of automatically inserted text -->
The problem is:
∃x∀y(x ≥ y)
With a domain of all real positive integers.
The negation is:
∀x∃y(x < y)
so, if y = x + 1
... |
{"OriginalQuestionIds":[3463],"Voters":[{"Id":29,"DisplayName":"svick"},{"Id":19,"DisplayName":"Nicholas Mancuso"},{"Id":98,"DisplayName":"Raphael"}]} |
null |
I wonder whether there is a simple algorithm to do this work:
Say I have a collection of objects $C$, and a binary relation $R: C\times C$ that is auto-reflective ($\forall c\in C: c R c$) , and symmetric ($\forall b,c\in C: b R c \implies c R b$). The relation is not necessarily transitive.
I want to find ... |
null |
Algorithm that hashes a collection of sets following their disjointness relation |
It's certainly not cheating. Think in calculus how substitution may be used to solve a tricky integral. The substitution makes the equation more manageable for manipulation. Additionally, substitution may transform somewhat complex recurrences into familiar ones.
This is exactly what occurred in your example. We def... |
Currently, I am self-studying Intro to Algorithms (CLRS) and there is one particular method they outline in the book to solve recurrence relations.
The following method can be illustrated with this example. Suppose we have the recurrence
$$T(n) = 2T(\sqrt n) + \log n$$
Initially they make the substitution m =... |
Changing variables in recurrence relations |
|algorithms|recurrence-relation| |
|asymptotics|recurrence-relation|mathematical-analysis|landau-notation| |
null |
I have a function to count upper bits of a 32 bit value. So if a number is 11100011111..., the result is 3 as there are 3 ones in the most significant place before a 0 is hit.
I need to convert the function to use only bitwise operations (no if statements or while loops) and the total number of operations should be ... |
Converting function to bitwise only? |
|logic| |
I've done some searching and it seems that fuzzy logic is used in (electronic) controllers. I'm interested in seeing how well it fares as a tool for reasoning. Are there any robots or programs that do clever things with it? |
Successful applications of fuzzy logic? |
|artificial-intelligence|reasoning| |
Given a directed graph G=(V,E) and a node r∈V, I need to grow a tree T rooted at r that has a minimum weight and spans all reachable nodes in G.
The weight function assigns a non-negative weight to a node and depends on node's antecedents in T. More precisely, I have some sets of nodes S1,S2,...,Sk and my weight fun... |
Minimum vertex-weight directed spanning tree where the weight function depends on the tree |
|algorithms|graphs|optimization| |
I have an question relating to the elementary foundations of Turing Machine theory. I would like to have a clarification of the status of a function $\phi$ (a mapping between TM indexes) I shall introduce in the formal question after the preamble. First to the preamble.
Let us assume that we have a fixed Turing Mach... |
Using a step-counting function in a Turing Machine construction |
|turing-machines| |
null |
I have an question relating to the elementary foundations of Turing Machine theory. I would like to have a clarification of the status of a function $\phi$ (a mapping between TM indexes) I shall introduce in the formal question after the preamble. First to the preamble.
Let us assume that we have a fixed Turing Mach... |
The answer is yes.
A possible quick approach: UTM2($TM_i$,c,x) $\neq 0$ only on $|x| \leq c$ so you can build $TM_{\phi(i,c)}$ simply simulating $TM_i$ on all inputs and hard encoding its output on the finite input strings $|x| < c$, the index of the resulting TM is the value of $\phi(i,c)$
|
I am unable to understand why there is a requirement of 3m+1 generals overall given m traitors. As per my understanding, the steps are as follows:
1. Each general decides a strategy and forwards their strategy to all the other (3m) generals.
2. Each general forwards a vector containing the strategies (of all the othe... |
Impossibility condition in the Byzantine Generals problem |
|complexity-theory|distributed-systems|complexity-classes| |
**Note:** Please excuse the unlinked links; I cannot yet link more than two.
The Mizar system is a huge repository of math proofs. See the [wikipedia page][1] and the [official website][2].
> of all of the known mathematical proofs that can be expressed using English statements
From [wikipedia/Mizar_system#Miz... |
The answer is yes.
A possible quick approach: UTM2($TM_i$,c,x) $\neq 0$ only on $|x| \leq c$ so you can build $TM_{\phi(i,c)}$ simply simulating $TM_i$ on all inputs and hard encoding its output on the finite input strings $|x| < c$, the index of the resulting TM is the value of $\phi(i,c)$
**EDIT**: if you want,... |
The Mizar system is a huge repository of math proofs. See the [wikipedia][1] page and the [official website][2].
> of all of the known mathematical proofs that can be expressed using English statements
From [wikipedia/Mizar_system#Mizar_language][3]:
> The distinctive feature of the Mizar language is its reada... |
null |
I am unable to understand why there is a requirement of 3m+1 generals overall given m traitors. As per my understanding, the steps are as follows:
1. Each general decides a strategy and forwards their strategy to all the other ($3m$) generals.
2. Each general forwards a vector containing the strategies (of all the ... |
|distributed-systems|fault-tolerance| |
Read the notation aloud. "There exist constants $c_1, c_2$ that are greater than $0$ such that for all $n$ greater than some $n_0$, $c_1g(n)$ is less than or equal to $f(n)$ and $f(n)$ is less than or equal to $c_2g(n)$.
Hence, $c_1$ and $c_2$ are constants that apply for all $n > n_0$. |
I have an question relating to the elementary foundations of Turing Machine theory. I would like to have a clarification of the status of a function $\phi$ (a mapping between TM indexes) I shall introduce in the formal question after the preamble. First to the preamble.
Let us assume that we have a fixed Turing Mach... |
I have an question relating to the elementary foundations of Turing Machine theory. I would like to have a clarification of the status of a function $\phi$ (a mapping between TM indexes) I shall introduce in the formal question after the preamble. First to the preamble.
Let us assume that we have a fixed Turing Mach... |
I think your miss-conception lies in the assumption
> m+1 loyal generals should vote the same way
Why should this hold?
Assuming you are talking about the problem as defined in LSP82:
For simplicity let's set $m=1$. Then with just $2m+1=3$ generals
you have one commanding general and two lieutenant generals.... |
> A nondeterministic machine trying to decide membership in a language
> is presented with a hint (called a "witness" or "certificate") which
> proves membership (no such witness is provided for elements outside
> the language; the definition is asymmetric).
So, if a non-deterministic algorithm can solve a proble... |
Length of the certificate in complexity classes |
|complexity-theory| |
I have an question relating to the elementary foundations of Turing Machine theory. I would like to have a clarification of the status of a function $\phi$ (a mapping between TM indexes) I shall introduce in the formal question after the preamble. First to the preamble.
Let us assume that we have a fixed Turing Mach... |
[{"Id":39,"DisplayName":"Gilles"}] |
[{"Id":39,"DisplayName":"Gilles"}] |
null |
Is it possible an algorithm complexity decreases by input size? Simply $O(1/n)$ possible? |
Complexity inversely propotional to $n$ |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.