instruction stringlengths 0 29.8k ⌀ |
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Polynomial time reductions |
|time-complexity|reductions| |
The best case running time and the worst case running time on inputs of length $n$ can each be expressed in this way (as can any function of $n$), however, the two measures do not need to coincide. Details follow.
The running time of an algorithm is not a function only of input size $n$ but a function of the input a... |
I'm having a very hard time understanding what's what.
$$L_{1}\leq_{p}L_{2}$$
If $L_2$ is stated to be in $\textbf{NP}$, is it necessarily true that $L_1$ is $\textbf{NP}$-Complete? I need to show the following for an assignment, but I'm having a dispute with a fellow student because he claims that I can't claim ... |
You asked two questions. What you are missing is that those two questions are unrelated.
Question 1:
> $$L_{1}\leq_{p}L_{2}$$
>
> If $L_2$ is stated to be in $\textbf{NP}$, is it necessarily true that $L_1$ is $\textbf{NP}$-Complete?
No. Consider the case where <i>L</i><sub>1</sub> and <i>L</i><sub>2</sub>... |
You asked two questions. What you are missing is that those two questions are unrelated.
Question 1:
> $$L_{1}\leq_{p}L_{2}$$
>
> If $L_2$ is stated to be in $\textbf{NP}$, is it necessarily true that $L_1$ is $\textbf{NP}$-Complete?
No. Consider the case where <i>L</i><sub>1</sub> and <i>L</i><sub>2</sub>... |
null |
I have a question about how you would find a example of a non Turing-recognizable language from the symmetric difference of two Turing-recognizable languages. I believe this is possible, but I am having problems coming up with a example in my head. Does anyone have any hints, or ideas about how to think about this? |
Non Turing-Recognizable from symmetric difference |
|turing-machines| |
|terminology|lambda-calculus|computation-models| |
|complexity-theory|terminology|np-complete|reductions| |
|computability| |
I'm really trying to wrap my head around vectorization but I can't seem to understand it. I don't know if I don't understand how to vectorize, or if I don't understand the array notation that's being used. An example of a loop I'm working on with school is below:
for (M=0; M< number_of_iterations/2; M++){
... |
|reference-request|np-complete|optimization|heuristics| |
|formal-languages|computability|formal-grammars|ambiguity|tree-grammars| |
|complexity-theory|terminology|reductions| |
null |
How do you say "δ:Q×Σ→Q" in English? Describing what "×" and "→" mean would also help. |
How does "δ:Q×Σ→Q" read in the definition of a DFA (deterministic finite acceptor)? |
|formal-languages|finite-automata|nondeterminism|language-design| |
Delta is a function which takes current state and letter from language alphabet as arguments and produces next state.
It is definition of transition function which is used for example in turing machines.
'x' is cartesian product
→ means "produces" |
The title of your talk contains an unwarranted assumption, namely that programming languages are "based on foundations of mathematics". This is generally not the case, although the two areas do have important relationships. A more accurate statement would be that (some) programming languages were designed using foundat... |
The title of your talk contains an unwarranted assumption, namely that programming languages are "based on foundations of mathematics". This is generally not the case, although the two areas do have important relationships. A more accurate statement would be that (some) programming languages were designed using foundat... |
Vor's answer gives the standard definition. Let me try to explain the difference a bit more intuitively.
Let $M$ be a bounded error probabilistic polynomial-time algorithm for a language $L$ that answers correctly with probability at least $p\geq\frac{1}{2}+\delta$. Let $x$ be the input and $n$ the size of the inp... |
Vor's answer gives the standard definition. Let me try to explain the difference a bit more intuitively.
Let $M$ be a bounded error probabilistic polynomial-time algorithm for a language $L$ that answers correctly with probability at least $p\geq\frac{1}{2}+\delta$. Let $x$ be the input and $n$ the size of the inp... |
How do you say $\delta\colon Q \times \Sigma \to Q$ in English? Describing what $\times$" and $\to$ mean would also help. |
|formal-languages|finite-automata| |
The statement $\delta\colon Q \times \Sigma \to Q$ translates to: The function $\delta$ assigns every pair formed by an element of $Q$ and an element of $\Sigma$, an element of $Q$. Instead of "assigns" you could also say "maps to". I assume that $\delta$ is the transition function for a finite automaton. Then you can ... |
I am asking for help to explain some crucial points of the central lemma and it's proof of famous paper [NP is as easy as detecting unique solutions by V.Vazirani and L.Valiant](http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf).
The proof can be found in the section **2. The proof** of... |
Proof of SAT is randomly reducible to UNIQUE-SAT |
|complexity-theory|reductions|proof-techniques| |
null |
$\delta$ is a function which takes a current state and a letter from the alphabet as arguments and produces next state.
It might be easier to understand when you realize that it is just a function that takes a pair, for instance $(q_0, a)$ where $q_0 \in Q$ is a state and $a \in \Sigma$ is a letter in the alphabet, ... |
Basically I am aware of three foundations for math
1. Set theory
2. Type theory
3. Category theory
So in what ways are programming languages and foundations of mathematics related?
EDIT
The original question was "Programming languages based on foundations of math"
with the added paragarph of
And im... |
How are programming languages and foundations of mathematics related? |
Basically I am aware of three foundations for math
1. Set theory
2. Type theory
3. Category theory
So in what ways are programming languages and foundations of mathematics related?
EDIT
The original question was "Programming languages based on foundations of math"
with the added paragarph of
And im... |
{"Voters":null} |
{"Voters":null} |
Technically, you could say that a database can be represented as a huge array. The benefit of a DB, however, is the ability to represent this huge array very succinctly.
More theoretically, the benefit of a DB can be compared to the topological fact that cylinder sets can be represented succinctly, as opposed to mo... |
In terms of comparing datastructures, is a relational database just a >1 dimensional array? I'm just asking because I don't know much about databases, but I know a bit about datastructures. I am just trying to make this clear in my mind. |
So is a relational database just a >1 dimensional array? |
|data-structures| |
Along the same thinking as this statement by [Andrej Bauer][1] in this [answer][2]
> The Haskell community has developed a number of techniques inspired by
> category theory, of which [monads][3] are best known but should not be
> confused with [monads][4].
What is the relation between [functors][5] in SML and ... |
What is the relation between functors in SML and Category theory? |
|programming-languages|category-theory| |
The statement $\delta: Q\times\Sigma\mapsto Q$ can be read as follows: given a state and a symbol the automaton moves to another state (possibly the state it was in). The $\times$ is like "and" in English meaning that you can pick any element from the first set $Q$ and any element from the second set $\Sigma$. More pre... |
null |
No, it isn't. A data structure can be described in two ways:
1. How does it store the data.
2. Which operations are supported, their performance (usually given as asymptotic time- and/or space-complexity), and what laws do they obey.
Obviously, a database doesn't store all its data as a single array.
Arrays ... |
The Halting problem is a natural undecidable language which is complete for the set of recursivly enumrable sets. I am interested in undecidable non Turing-complete language such that we can not reduce the Halting problem to it.
What is the most **natural** undecidable RE problem which is not Turing-complete?
|
Natural RE undecidable problems but not Turing-complete |
|computability|turing-completeness| |
I am asking for help to explain some crucial points of the central lemma and it's proof of famous paper [NP is as easy as detecting unique solutions by L.Valiant and V.Vazirani](http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf).
The proof can be found in the section **2. The proof** of... |
Is any language with finite number of strings regular? Is the language of all strings regular?
Iam new to all this topics and got confused.Can any one please help me in letting me know the answer |
Is any language with finite number of strings regular? |
|regular-languages|finite-automata| |
1. Is any language with a finite number of strings regular?
2. Is the language of all strings regular?
I am new to this topic and got confused. Can any one please help me with this? |
1. Yes. Assume the language is $L=\{w_1,w_2,\ldots, w_n\}$. then you can form the [regular expression](http://en.wikipedia.org/wiki/Regular_expression) $w_1 + w_2 + \cdots + w_n$, which describes $L$.
You can also argue with finite automata or regular grammars.
2. Yes, if your alphabet is $\Sigma=(a_1,\ldots,a... |
null |
1. Is every language with a finite number of strings regular?
1. Is the language of all strings regular?
I am new to this topic and got confused. Can any one please help me with this? |
Is every language with a finite number of strings regular? |
For a regular language which of the following are true?
1)R U S = S U R
2) (R U S) . T = RT U ST
3) R*.S* = (R U S )* |
Regular language and automata theory |
|regular-languages| |
For a regular language which of the following are true?
1. $R \cup S = S \cup R$
1. $(R \cup S) \cdot T = RT \cup ST $
1. $R^* \cdot S^* = (R \cup S)^*$ |
There indeed are sets that are undecidable but not Turing complete. However, I'm not sure there is such a thing as a _natural_ undecidable not Turing complete language.
Emil Post set out on a search for exactly this (See [Post's problem][2]) and his research resulted in the [Simple sets][1]. To quote from the Wik... |
For regular languages $R, S$ and $T$, which of the following are true?
1. $R \cup S = S \cup R$
2. $(R \cup S) \cdot T = RT \cup ST $
3. $R^* \cdot S^* = (R \cup S)^*$ |
Regular language properties |
null |
1. True
2. True
3. False
To find the proofs, proceed as follows:
1. Show that any string $x \in R \cup S$ belongs to $S \cup R$, and vice versa.
2. Ditto 1.
3. Find a string $x$ that is in $(R \cup S)^*$ but not the LHS. *Hint: what about $rsr$ where $r \in R, s \in S$?* |
>Given a graph G: is there an algorithm, bounded by a polynomial of order k, that can determine if G is 3-colorable-or-non-3-colorable and provide a certificate?
I think what you meant here was: given a graph $G$, find whether there is a certificate or short proof for whether it is 3-colorable. What you're missing h... |
1. $x\in R\cup S \iff x\in R \vee x \in S \iff x\in S \vee x \in R \iff x\in S\cup R$
2. $x\in(R\cup S)T \iff \exists u, v \mid (x=uv)\wedge (u \in R\cup S)\wedge (v\in T) \iff \exists u, v \mid (x=uv)\wedge (u \in R \vee u \in S)\wedge (v\in T) \iff \exists u, v \mid (x=uv)\wedge (u\in R \wedge v\in T) \vee (u\in... |
I am asking for help to explain some crucial points of the central lemma and it's proof of famous paper [NP is as easy as detecting unique solutions by L.Valiant and V.Vazirani](http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf).
The proof can be found in the section **2. The proof** of... |
Reading this question "[Natural RE undecidable problems but not Turing-complete][1]" it came to my mind the following language:
if $\Sigma(n)$ is the busy beaver function (maximum attainable score among all halting 2-symbol n-state Turing machines of the above-described type, when started on a blank tape), define th... |
A variant of the busy beaver function |
|turing-machines| |
Reading this question "[Natural RE undecidable problems but not Turing-complete][1]" it came to my mind the following language:
if $\Sigma(n)$ is the busy beaver function (maximum attainable score among all halting 2-symbol n-state Turing machines of the above-described type, when started on a blank tape), define th... |
|graphs|combinatorics|coding-theory|expanders| |
Can one have a condition like this in semidefinite programming? |
null |
Reading this question "[Natural RE undecidable problems but not Turing-complete][1]" the following language came to my mind:
If $\Sigma(n)$ is the busy beaver function (maximum attainable score among all halting 2-symbol n-state Turing machines of the above-described type, when started on a blank tape), define the f... |
In the book [Introduction to Languages and the Theory of Computation](http://www.amazon.com/Introduction-Languages-Theory-Computation-Martin/dp/0072322004), I'm reading section 2.6 on how to minimize the number of states in an FA.
I'm having trouble understanding a notation defined as $L_q$. Here's what the book say... |
How to find $L_q = \phi$, a state that is not reachable for a given string? |
|automata|finite-automata| |
You're right. There is no such state with $L_q = \emptyset$
The first step in the minimization algorithm is: "delete all nonreachable states". |
In the book [Introduction to Languages and the Theory of Computation](http://www.amazon.com/Introduction-Languages-Theory-Computation-Martin/dp/0072322004), I'm reading section 2.6 on how to minimize the number of states in an FA.
I'm having trouble understanding a notation defined as $L_q$. Here's what the book say... |
How to find $L_q = \emptyset$, a state that is not reachable for any given string? |
It is known that $Parity \notin AC^0$ (nonuniform), but the proof is rather involved and combinatorial. Are there simpler, but weaker lower bounds, say for $NP \not \subseteq AC^0$ or $NEXP \not \subseteq AC^0$?
For example, can nontrivial simplifications be obtained in the proof of $NEXP \not \subseteq ACC^0$ to de... |
Simple lower bounds against AC0 |
|lower-bounds|circuits| |
In database literature this idea is widely known as [universal relation][1].
[1]: http://en.wikipedia.org/wiki/Universal_relation_assumption |
|complexity-theory|lower-bounds|circuits| |
One area that you could consider is **Fault Tolerant Parallel Computation**. Expander codes and graphs could be efficiently used in them like:
[Highly fault-tolerant parallel computation][1], by [Daniel A. Spielman ][2]
[On word-level parallelism in fault-tolerant computing][3]
[1]: http://ieeexplore.ieee.... |
I guess it depends on your point of view, but the proof via approximating polynomials (along the lines of Razborov-Smolensky) that Parity isn't in AC0 is not so involved...
The natural way in which one would modify the proof that "NEXP is not in ACC0" to yield "NEXP not in AC0" would be to give a SAT algorithm for A... |
null |
There is, to the best of my knowledge, no formal relation between *functors* in category theory and *functors* in ML (SML or OCaml, they're close enough for our purpose here).
In category theory, functors are functions that operate on objects. They are one level above morphisms, which are often functions that operat... |
I have a simple problem of making a DFA which accepts all inputs starting with double letters (aa, bb) or ending with double letters (aa, bb), given $\Sigma =\{a, b\}$ is the alphabet set of the given language.
I tried to solve it in a roundabout way by:
1. Generating a regular expression
2. Making its correspon... |
Not able to convert from NFA to DFA |
|automata|finite-automata|assignment-problem| |
Categories form a (large) category whose objects are (small) categories and morphisms are functors. In this sense functors in category theory are "higher size morphisms".
ML functors are not functors in the categorical sense of the word. But they are "higher size functions" in a type-theoretic sense.
Think of con... |
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