# Questions tagged [computability]

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

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### Do languages in $\mathsf{coRE} \setminus \mathsf{R}$ have Turing machines?

What can we say about languages in $\mathsf{coRE} \setminus \mathsf{R}$? Are there Turing machines for these languages? I know that $\overline{HP} \in \mathsf{coRE}$ doesn't have a Turing machine, and ...
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### Proof that languages are Turing-recognizable iff computably-enumerable

A very small question on this proof, which I found as Theorem 3.21 in Sipser's, and in my lecture notes. In the "only if" direction, we assume that a Turing machine $M$ recognizes some ...
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### Can you say anything interesting about a language knowing only that it is prefix-closed?

Suppose $L$ is an arbitrary formal language over a finite alphabet $A$, and suppose that $L$ is closed under prefixes (i.e. if $w \in L$, and $u$ is any prefix of $w$, then $u \in L$). Knowing only ...
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### algorithm for checking satisfiability

In order to prove that SAT is in NP, I need to come up with a polynomial time verfier (an algorithm). The Cooks Levin Theorem uses a non-deterministic Turing machine but that's not what I am looking ...
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### Church–Turing thesis and infinite Turing machines

What exactly is the definition of church turing thesis? It's really confusing. I want to prove the following statement: A Turing machine with infinitely many states is more powerful than a regular ...
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### Does the term “continuity” have a different meaning in maths and in CS?

I ask this question because of some statements in the question "What is the 'continuity' as a term in computable analysis?" making me suspicious. I'm engineer, not computer scientist, so I ...
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### What is the “continuity” as a term in computable analysis?

Background I once implemented a datatype representing arbitrary real numbers in Haskell. It labels every real numbers by having a Cauchy sequence converging to it. That will let $\mathbb{R}$ be in the ...
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### what is the relevance of computability when applying diagonallization?

When thinking about diagonalization, I've always glossed over whether or not the enumeration, or the diagonal is computable or not. When does it matter? Say for example, that have an enumeration of ...
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### running RAM on a given input

I understand how RAM commands work but I am unable to understand how we use a given input string and find the output. For instance, there's a Random Access Machine which has an input {0,1}*. The ...
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### change turing machine to RAM

How can we convert a given Turing Machine into a Random Access Machine? I understand that we can use the transition function to come up with a sort of algorithm but how can we translate all of it into ...
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### For an NFA, can we always find a RAM?

For an NFA, can we always find a RAM, which recognises the same language?
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### How do you write a python\pseudo code that generates all pair permutations?

What would be a good pseudo code or Python 3 code for the following permutations problem? Let us define a n-permutation as a bijective function $\pi: \{0,...,n-1\}\rightarrow \{0,...,n-1\}$ and ...
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### In an NFA, what if there are no transitions out of an accept state but there are symbols left in the string?

Let's say I have a string 0110 and after 011 I reach an accept state (let's call the accept state "q") in an NFA. However, there is no transition mentioned in the diagram from q for the ...
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### Inhabitation of STLC is in PSPACE

Urzyczyn: Inhabitation in Typed Lambda-Calculi (A syntactic approach) gives a proof that STLC inhabitation problem is in PSPACE (section 2, lemma 1). I don't understand certain aspects of the proof: ...
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### Undecidability of the language of PDAs that accept some ww

I'm trying to solve problem 5.33 from Sipser's Introduction to the Theory of Computation, "Consider the problem of determining whether a PDA accepts some string of the form $\{ww|w\in \{0,1\}^∗\}$...
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### If $A\in RE$ then $f(A)\in RE$

Let $A\in RE$, and define$f(A) = \{y |\ y= f(x),\ x\in A\}$ for some computable function $f$. Then $f(A)\in RE$. I can't figure out why this is true. Since $f$ is computable there is a Turing machine ...
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### Mathematical limits on lossless data compression

Let's say Bob wants to send a particular binary sequence to Alice. Imagine that Bob and Alice both have powerful machines but slow Internet connections. Bob could just send the sequence directly but ...
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### Decidability for $\exists w´, w´´\in L:$ so that |w´´| - |w´| is prime

I tried to decide wheter the given Language $L = \{ \langle M \rangle | M \space is \space TM \space and \space \exists \space w´,w´´\in L(M):|w´´|-|w´| \space is \space prime \}$ is recursive or ...
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### Enumerator for Word and Halting Problem

in theoretical computer science I learned for every recursive enumerable language there would be an enumerator and a grammar. So since word problem and halting problem are recursively enumerable, I ...
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### Are all Recursively Enumerable languages which are not Recursive also Undecidable?

Knowing that all Recursive languanges are Decidable and All Not R.E. Languages are Undecidable (correct me if I am wrong), Are all languages which are R.E. but not Recursive also Undecidable? R.E. ==&...
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### What is minimization (μ-function) in layman tems?

In Computer Science μ-function is used to extend set of primitively recursive functions to generally recursive functions, and I can't understand what this function does. There is a lot of formulae, ...
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### What's wrong with this “proof” that $\mathbb{R}$ is enumerable?

The fake proof: We know that $\mathbb{R}$ is uncountable, hence we cannot enumerate over it. But what we do know is that $\mathbb{Q}$, the set of rationals, is countable, and even denumerable. We ...
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### Why is this language Turing recognizable and not not-Turing recognizable

I read that the following language is r.e. but not not-Turing recognizable $L$: On input $M$ (where $M$ is a Turing Machine), $M$ accepts at least 20 inputs I am not sure why it is not not-Turing ...
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### With Σ = {a,b}, give a dfa for L= w1aw2 : |w1 |≥ 3, |w2 |≤ 5}

I racked my brain,I saw other people's solutions and it don't make sense. I think my biggest problem is I don't know when one string ends,like for example w1 is >=3 and it can have how many ever b'...
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### Is there a logic-to-numeral mapping which preserves uniqueness (contrary to the Gödel coding)?

Given the two equivalent terms $A \vee B$ and $B \vee A$, Gödel numbering returns two various codes $2^{4}.3^{\overline{A}+1}.5^{\overline{B}+1}$ and $2^{4}.3^{\overline{B}+1}.5^{\overline{A}+1}$, ...
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### Is it true that if L* is recursive, L is also recursive?

Is it true that if $L^*$ is recursive, where $*$ is Kleene star, $L$ is also recursive? I know that the opposite direction is true: If $L$ is recursive, then $L^*$ is recursive. But I don't know how ...
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### Is it correct to say L is RE if I can map reduction from LH to L?

I seem to be not understanding correctly what reductions means for Languages. for example, Lets say there is a Language called LM. I want to see if LM is recursive or not, to do that lets say I find ...
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### Given A to C, and B to C with known complexities, what can be said about A to B?

Say I have two sets of values $A$ and $B$ and for each set I have a computable function from that set to a third set $C$. Now suppose that I want to construct a function from $A$ to $B$, such that if ...
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### Prove a language is not recursive enumerable

I need to prove $: L=\left\{<M>| M \text { is a } T M \text { and } L(M)=L\left((01)^{*}\right)\right\} \notin R e$ at first observation it looks like it's immediate from Rice's extended Thm, ...
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### how to prove that $NP \cap co - NP$ = { S | S such that there exist a Strong Deciding Algorithm for S}?

i need to prove that and i find it struggle: given: for deciding problem S: a non deterministic algorithm $A(x)$ is strong deciding algorithm if: $x \in S =>$ fo every run of $A(x)$ returns "Yes"...
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### Single-valued enumeration of all c.e. sets

Please help prove the following statement: There exists a single-valued computable enumeration of the family of computably enumerable sets. Definitions: 1) Let $S$ be nonempty countable set (...
### How to understand definition of $\Pi_k$ in arithmetical heirarchy
Am reading a text about computability theory, and according to the text, at each level $k$ of the arithmetical hierarchy, we have two sets, $\Sigma_k$ and $\Pi_k$, where $\Pi_k$ is defined as:  \...
I have a task to determine the upper bound of states in the Minimal Deterministic Finite Automata that recognizes the language: $L(A_1) \backslash L(A_2)$, where $A_1$ is a Deterministic Finite ...