Questions tagged [computability]

Questions related to computability theory, a.k.a. recursion theory

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9 views

How to Plot Logistic Regression using Seaborn? [closed]

I am trying to plot logistic regression model but seaborn regplot does not worked. What should I do?
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1answer
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Decidability of languages with dfa/turing-machines

For any alphabet and any natural number k, a language of strings at least k is decidable. So my question is having some alphabet (let's say (0,1)) and some number let's say k=5 then my language has ...
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Question regarding notation of a language decidability

$1.\: A_{DFA} = \{\langle B, w \rangle \mid B \text{ is a } DFA \text{ that accepts input string } w \}$ $2.\:A_{DFA} = \{\langle B \rangle \mid B \text{ is a } DFA \text{ that accepts input string } ...
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1answer
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Are the sets in $\Sigma_{p}$ (or $\Pi_{p}$) totally ordered?

Are the sets in each stage of the arithmetic hierarchy well-ordered, with respect to : $T-$reductions, or $m-$reductions? It is something which I have been unclear with for a while (from a ...
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1answer
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Strings of length $n$ enumerated by Turing machine

If $E$ is an enumerator and $n$ is a natural number, we define $\langle E,n \rangle$ to be all strings of length $n$ enumerated by $E$. Is $\langle E,n \rangle$ recognizable? Is it decidable?
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Converse of (two sets are RE => their union is RE) doesn't hold?

E.g. union of an RE set and its complement (where complement set is not RE) is Sigma star, wich is definitely RE. But what if you first attach to the beginning of elements in S with a #, and elements ...
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Infinite jump of a language

Let $A(n)$ be the $n$-th jump of a set $A$ for any natural number $n$. Define the infinite iteration of the jump, $A(\omega)$, as follows: $$ A(\omega) = \{ \langle m,n \rangle : m \in A(n) \}. $$ ...
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If $A$ is c.e.-complete then the join of $A$ and $A^c$ isn't c.e

Let $A\subseteq\{0, 1\}^∗$, and let $A^c = \{0, 1\}^∗ \setminus A$ be the complement of $A$. Prove: If A is $\le_m$-complete for CE, then $A$ join $A^c$ is neither c.e. nor co-c.e. c.e --> ...
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Classification of language of all words whose length is a perfect cube

L = {x belongs to {a, b} : |x| is a perfect cube } What is the smallest class of that the following language belongs to, between decidable and recognizable? $$ \{ x \in \{a,b\}^* : |x| \text{ is a ...
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1answer
56 views

TM decidability using pigeonhole principle

I think since the input word is delimited by special symbols, which the machine cannot move past, the language accepted by such a device should be finite. We know that all finite languages are regular,...
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Show that a set is reducible to another set

Consider an operator $+$ defined on $P(\mathbb{N})$ as follows $$A + B = \{2x:\ x\in A\}\cup\{2x + 1:\ x\in B\}$$ Show that $A$ is $m$-reducible to $A+B$ and $B$ is $m$-reducible to $A+B$ As per the ...
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What can be proven regarding the differences in power between unary ECMAScript regex functions and primitive recursive functions?

In 2014, inspired by Regex Golf, I started exploring, along with a mathematician going by the name teukon, what could be done in the unary domain in ECMAScript regex that went significantly beyond ...
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1answer
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Prove/Disprove: NP is closed under “mixed” complexity

Let $\displaystyle S_{1} ,S_{2} \subseteq \{0,1\}^{*}$, we say $\displaystyle x\in S_{1}°S_{2}$ if it's of the form $\displaystyle x=x_{1} x_{2} ...x_{n}$, for $\displaystyle n$ even, such that: $\...
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34 views

Determining whether a language $L_{a}$ is recursively enumerable

I'm trying to determine whether a language $L_{a}$ is recursively enumerable, but first I'm having trouble deciphering the definition $L_{a}$ given the following question: Given an recursively ...
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2answers
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If $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ is $L_1=L_2$

If the same $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ does it imply that $L_1=L_2$? My intuition says no, but I couldn't find a counterexample.
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Can the intersection two non-recursive sets be recursive? Prove it

I am still really new to compsci theory and some of the topics are really hard to understand. For this Problem I would think that say we have two sets A and B and they are both non-recursive. If we ...
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1answer
39 views

Vertex cover of minimal graph

I'm looking for algorithm that, for given undirected graph $G=(V,E)$, find graph $G'=(V,E')$ with minimal amount of edges that have same vertex cover as G. I mean, vertices $U$ are vertex cover of $G$ ...
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For every Non Deterministic polynomial Turing Machine $M$ exists $L(\overline{M})\in P \Leftrightarrow P=NP$

The $\Leftarrow$ direction is straightforward. On the other hand for $\Rightarrow$ direction I have an idea of the prove but I don't sure about it. For NTM, Non Deterministic Turing Machine, $M$, for ...
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1answer
26 views

Is there a non-deterministic polynomial by time Turing machine such that: $L(M)\in NPC$ and $L(\overline{M})\in P$

When $\overline{M}$ is a non-deterministic polynomial by time Turing machine that final states switched: accept to reject and vice versa. I'm thinking that this equal to $P=NP$, but I saw a solution (...
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For what reason this function is recursive primitive?

Let $\forall n \in \mathbb{N}\ \ P(n)$ a primitive recursive predicate such that $\neg P(n)$ for a finite number of values of $n$. Why this function: $$ f(x) = \begin{cases} 1 & \text{ if there ...
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1answer
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Why is $\forall x \in \mathbb{N}\ \Phi(x,x)$ unary but $\Phi$ is binary?

May anyone explain me why $\forall x \in \mathbb{N}\ \Phi(x,x)$ is unary but $\Phi$ is binary? In my latest exam I wrote that the universal function $\Phi^n(x_1, x_2, ..., x_n,y) = \psi^n_y(x_1,x_2,......
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2answers
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Turing machine that checks whether a given string is an output of a given machine and input

Is there a Turing machine such that, given a description $\langle M \rangle$ of a Turing machine $M$, an input $x$ and a string $y$, computes whether or not $y$ is the output of $M$ input $x$? My ...
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1answer
59 views

Minimum absolute value of subset sums of integer values

$f(x_1,...,x_m)=\min_{\emptyset\subset I\subseteq[m] }\left|\sum_{i\in I}x_i\right|, x_i\in \mathbb{Z}\setminus\{0\}$ How to prove $f\in \mathbf{POLY} \Leftrightarrow \mathbf{P}=\mathbf{NP}$? When $\...
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2answers
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Why can't we compute the lexicographically-least word of a given length on which a given TM halts?

I had this question in my exam. but my answer is wrong(I didn't receive explanations why...) $$f(\langle M\rangle,1^n)=\left \{ \texttt{the lexicographically smallest } x\in\left \{ 0,1 \right \}^n \...
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1answer
35 views

Why every finite language is polynomial?

I understand that it's possible to build TM that check all the finite number of cases, so it's definitely in $R$, but I'm not sure why it's in $P$
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11answers
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Can a computer determine whether a mathematical statement is true or not?

I was reading Introduction to the Theory of Computation by Michael Sipser and I found the following paragraph quite interesting: During the first half of the twentieth century, mathematicians such as ...
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1answer
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Proof of existence of $L\in R\setminus P$

I saw some proof but I didn't understood it, any simple one?
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1answer
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$\overline{SAT}$ vs. $UNSAT$, Is it the same?

I know this question may look stupid, but still.. Is the meaning of both "have no satisfiable assignment"?
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1answer
35 views

Deteremine if Language is in $R$ or $RE$

$$L =\left \{ \langle M \rangle \mid \exists x\in \Sigma^* \left(\left | x \right |\leq 10000 \wedge H(M, x\right) \right \}$$ Where $H(M, x)$ denotes whether Turing machine $M$ halts on input $x$. My ...
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1answer
34 views

$L_{\Sigma^*}=\{\langle M\rangle|L(M)=\Sigma^*\}\notin coRE$

I'm trying to understand why: $$L_{\Sigma^*}=\{\langle M\rangle|L(M)=\Sigma^*\}\notin coRE$$ As I see it TM, $\langle M\rangle$, should accept all the inputs, and if one of the inputs rejected it's ...
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1answer
77 views

How can primitive recursion be a special case of minimization?

In several posts on StackExchange and elsewhere, I have seen claims that you only need to show you can construct constants, successor, projection, composition, and minimization to prove a language ...
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1answer
36 views

Computability of function composition

I have some problems to understand computability and hope you can help me. In the lecture we had following problem: Consider the three partial functions $f,g,h\colon N \to N$, where $f$ is computable ...
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1answer
26 views

Condition to prove $f$ is a reduction

A theorem says if $f$ is a computable function and we can prove $x \in A \Leftrightarrow f(x) \in B$, then we can use reduction so $A \leq_m B$. But i'm confused if should I prove if : $(x \in A \...
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1answer
51 views

Complementary for $SAT$

I have tried to find a definition of complementary language to $SAT$, I mean $\overline{SAT}$. But I still confused, in case of $L\in \overline{SAT}$ is it mean: if $\varphi\in L$ then all ...
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2answers
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Recursively enumerable notation $RE$ vs. $RE\setminus R$

I know that it's a bit stupid question.. , but still, Is there any difference between $RE$ and $RE\setminus R$ notations? I'm asking because I saw that in some places using both of the notations, for ...
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1answer
57 views

Cook–Levin theorem and reduction as injective function

I saw that the injectivity "derives directly from the theorem", but i can't see how it's happen, any explanation?
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1answer
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Characterization of computationally universal functions

Is it correct to state that $u$ is a universal function if and only if $$ \forall f : \text{RE} \quad \exists g : \text{R} \quad \exists h : \text{R} \quad f = h \circ u \circ g $$ where RE is the set ...
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1answer
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For s set $S\subseteq RE$, so call feature of language $S=\emptyset$ vs. $S=\{\emptyset\}$

I'm trying to understand what's the difference between $S=\emptyset$ and $S=\{\emptyset\}$ The diffenition that I found for $L_S=\{\langle M\rangle\ | L(M)\in S \}$ I understood that $S=\emptyset$ and ...
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1answer
30 views

A language is decidable iff it is Turing-recognizable and co-Turing-recognizable (WHY?)

I am trying to understand the proof for this theorem (theorem 4.22 of the book 'An introduction to the theory of computation'): ...
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1answer
37 views

How does CNN deal with rotation invariant pictures?

I am trying to make a CNN model . Training the image . Want to know that When we apply kernel on image and take out the features of images. That features are rotation invariant or we have to apply ...
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Is my assumption about non trivial propery correct?

"make sure you understand why for a non trivial property $S$, $\bar{S}$ is also non trivial" My assumption is: $S$ is non trivial property: There are L1,L2 such that $L_{1},L_{2}\in RE$ and ...
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1answer
48 views

Is a language recursive? 2 wrong ways of solving

Let's define: $Disagree(M_1,M_2) = \{x| $The result of $M_1$ on $x$ different from the result of $M_2$ on $x\}$ that means: if $M_1$ accept, $M_2$ reject and vice versa $NPA=\{L|\exists M_1,M_2$ ...
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495 views

(Un)computability of a restricted Halting Problem

Before I start with my question, I want to state some notation I am using. I fix some arbitrary but fixed enumeration of Turing Machines (TMs) and denote with $\Phi_i : \mathbb{N}\to\mathbb{N}$ the ...
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For every non-trivial language $A$ and every finite strict subset $B \subsetneq A$, it's holds that $A \le_m A \setminus B$

It's claim 1 from Bader Abu Radi's solution to this question. My solution (have no idea how wrong it is): $B$ finite $\Rightarrow$ $B\in R \Rightarrow$ exists TM $\langle M_B\rangle$ that halts $B$. *...
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2answers
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State whether the language is in $R$, $RE$, etc. The intuition for the solution

I saw the solution but can't understand the intuition of the following question: Let's define $$L^{\ge k} = \{w\in L : |w| \ge k\}$$ and $$L=\{\langle M\rangle | \exists k:L(M)^{\ge k} = \overline{HP}^...
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1answer
37 views

Reduction from $A$ to $B$ as execution of Turing machines

As explained in answers to this question, reduction from $A \le B$ can be represented in the following way. But in this example: from here At least as I understand it: The reduction is from $\...
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2answers
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Is it valid to make an admission of a topological space by a “partial quotient map”?

It is well-known that the Sierpiński space, $\{F,T\}$ endowed with topology $\{\emptyset, \{F\},\{F,T\}\}$, is admissible. I tried to implement it in Haskell. First I implement $\mathbb{N}$ (including ...
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How can I simulate nested WHILE loops in a theoretical programming language to show Turing completeness?

PRE-WORK-POST is a theoretical programming language with the following structure, where P,Q and R are LOOP program: $$\text{PRE} \ P \ \text{WORK} \ Q \ \text{POST} \ R \ \text{END}$$ First $P$ is ...
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1answer
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Is there a connection between the Undecidability Theorem and “software complexity”?

I was reading Complexity: The Emerging Science at the Edge of Order and Chaos and a certain passage got me really intrigued. When discussing Chris Langton's explorations of artificial life algorithms,...
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Explain the difference between Turing Complete and Turing Equivalence

I'm not sure if I understand the difference between Turing Complete and Turing Equivalent programming languages. A computational system that can compute every Turing-computable function is called ...

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