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Questions tagged [computability]

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

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#P=NP: All satisfying solutions are valid answers by programs for NP

In fact, the implication looks like an equivalence too. All satisfying solutions to a boolean formula are valid answers by programs for NP is equivalent to saying the class #P equals the class NP. ...
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Decidablity of time complexity

Let $t:\mathbb{N}\rightarrow\mathbb{N}$ be a time constructible function with $t(n)\geq n + 100$. Show that there is no TM $T$ that given the gödel number of another TM $M$, decides wether or not M is ...
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Undecidability of the “Single-Halting Problem”

I have to show for a turing machine S that is taking another TM T and a word x as input and only halt for one specific T and x, that it is not decidable. The idea is now to reduce it to the halting ...
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1answer
723 views

What is the complement of a language?

If given any language L, how do I find the complement of said language? I lack the basic understanding required to determine if a language is co-recognizable. I understand that a language, $L$, is co-...
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Is the language $L$ of coded CFG's Turing decidable?

Consider the following language $L$ = {$<G><w>$ | $G$ is a CFG and $w\in L(G)$} Now, I wish to prove that $L$ is Turing decidable. My gut tells me to construct a Turing machine that ...
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2answers
45 views

Recognizer for decidable language and words it doesn't halt on

Suppose we have a decidable language B (there exists some TM that decides it). Suppose we have another TM M which only ...
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1answer
22 views

Is the set of surjective recursive functions in RE/coRE?

Let L be a set of recursive funtions with $L = \{i\in \mathbb{N}|f_i\space is\space surjective\}$ where $i$ is a gödel number of f.Is $L\in RE,\space coRE$? I can't think of a way to show either of ...
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2answers
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Are the capabilities of programming languages the same?

Is the capability of every programming language the same since it is eventually translated into machine code. Python, Java etc. are all together instructions the CPU is going to process. So, you could ...
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How strong is a FSM with long enough(but not infinite) tape?

Like turing machine, but your tape is finite. To make a program valid it should have a limit result when the length of tape tends to infinity. Whether the tape has two ends or is cyclic doesn't ...
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36 views

Mathematical resource material accompanying TAPL

I'm currently reading Types and Programming Languages by Benjamin C. Pierce and just arrived at chapter 21 Metatheory of Recursive Types. Prior to this chapter I found the book challenging but ...
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1answer
51 views

Is Rice-Shapiro theorem bidirectional?

Rice-Shapiro theorem states that version A Let $\Gamma$ be a set of computably enumerable sets, and $I = \{e : W_e \in \Gamma\}$ its index set in some admissible enumeration of c.e sets. If $I$...
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1answer
24 views

Can the complement of an unrecognizable language be a recognizable language?

I know that complement of a language that is recursively enumerable, but not recursive, is definitely not recursively enumerable (or unrecognizable). So my question is what can be said about the ...
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1answer
31 views

How to prove that a problem is undecidable by using the Halting problem?

I cannot understand how to reduce the halting problem to a property to show that is undecidable. For example, I have this property of a Turing Machine and I have to prove if it's recursive or not: "...
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3answers
60 views

Proof Idea: There are irrational numbers whose decimal expansion cannot be computed

The online lecture I am watching stated a proof idea: The set of all possible programs is countably infinite, yet the set of irrational numbers is uncountably infinite. I don't think this is ...
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1answer
52 views

Halting problem with extra input

Can there be a function HALT(f, y) so that: There are some x such that f(x) halts iff there ...
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1answer
22 views

Is a set $B = \{y, \exists x \in A, f(x)=y\}$ recursive if A is a recursive set and f is a $N->N$ total computable function?

Obviously, B would be recursive if for every TCF f, there was an inverse fuction that would return all possible values, as we could just take these and then check if any of them is in A. However I ...
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1answer
29 views

Are physical laws uncomputable in any type of computation (according to this article)?

It seems that this article (https://arxiv.org/pdf/1312.4456.pdf) proposes that laws of physics are uncomputable (i.e., they could not be reproduced on a computer), but I'm not sure about it. In some ...
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Is ${M :|L(M)| \leq 330}$ Recursively enumerable? [duplicate]

M is a Turing machine description, L(M) is the language recognized by M and |L(M)| is the size of this language.
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1answer
81 views

Which one of these two sets is computably enumerable?

M is a turing machine description, L(M) is recognized by M, |L(M)| is the size of this language. {M : |L(M)| <= 330} {M : |L(M)| >= 330} I don't quite understand what this question is asking. ...
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1answer
34 views

Reduce ATM to REGULAR_TM

Consider $\mathsf{REGULAR_{TM}} = \{\langle M \rangle \mid \text{$M$ is a TM and $L(M)$ is a regular language}\}$. Let $S$ be the following algorithm, which solves $\mathsf{A_{TM}}$: “On input $\...
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1answer
131 views

Language that fulfills pumping lemma but is not in RE

I am supposed to find a language $$L\subseteq \Sigma ^*, \Sigma \subseteq \mathbb{N}$$ that fullfills the pumping lemma and is not in RE and not in coRE. I've never constructed a language with a given ...
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1answer
29 views

Are non-regular languages decidable?

Given a language L, I've shown that L is not regular. Can I conclude that it is not decidable or are there non-regular languages that are decidable?
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2answers
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Quotient in LOOP program [closed]

I want to construct a LOOP-computable program for the integer division (quotient): x = a DIV b The LOOP specification can be seen here: https://en.wikipedia.org/wiki/LOOP_(programming_language) I ...
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0answers
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What is the relationship between “model of computation” and “algorithm”?

Traditionally, the usual definition you find for model of computation is "an abstract description of how an output is computed given an input" (Wikipedia and my TCS course are my sources, but the ...
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2answers
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Relation between Undecidable problems and NP-Hard

I drew these pictures to check whether I comprehended the ideas of P, NP, NP Complete and NP Hard correctly. And then, I realized that it is not certain where undecidable problems should be placed. ...
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1answer
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Decidability of factoring algebraic equations

Given an arbitrary algebraic equation, say for example the likelihood of the bernoulli distribution: $$\prod_{i}^{n}\theta^{x_i}(1-\theta)^{1-x_i}$$ And some arbitrary factorization constraints, say:...
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2answers
46 views

Can a RE language be reduced to a non-RE language?

In our lecture notes about many-one reduction we showed that the following statements hold: $$ L, L' \subseteq \mathbb{N}\space and \space L\leq L'$$ $$(I)\space L' \in RE \implies L\in RE$$ $$(II)\...
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Witnessing for partial recursive functions

For all $f\colon \mathbb{N}^2 \rightarrow \mathbb{N}$ partial recursive there exists partial recursive $g\colon \mathbb{N} \rightarrow \mathbb{N}$ such that a) $x \in \operatorname{Dom}(g) \...
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0answers
17 views

Whether language of all turing machines is decidable or undecidable or semi-decidable?

I recently came across this language: $L=\{<TM>| \text{TM accepts recursively enumerable languages}\}$ It was asked in the question to find out whether language L is decidable or undecidable. ...
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0answers
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Can a CFG generate an accepting configuration? - or is there a turing-recognizable CFG language that is not decidable

I could not think of a way to concisely write down my question clearly, but I'd like to ask, from Sipser's book, $ALLCFG$ is an undecidable language (where $ALLCFG$ means that $G$ is a $CFG$ that ...
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1answer
63 views

Trying to prove semidecidability of an undecidable language

I have been having a hard time understanding whether the set $S = \{ M \mid |L(M)| = 5 \}$ is semidecidable or not, where $M$ is a generic Turing Machine and $L(M)$ the language accepted by such TM, ...
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Understanding the computational power of neural networks

It is known that a recurrent neural network with rational weights is computationally equivalent to a Turing Machine (a proof can be found in this paper). I don't understand how is it possible, it ...
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0answers
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Power of Turing machines that are not allowed to overwrite the input string [duplicate]

The question asks what kind of languages (regular, context free) can a Turing machine accept if you are not allowed to overwrite the input string. The initial configuration of the machine is start ...
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2answers
97 views

Is there any other computation theory besides the one in automata theory?

I'm a bit confused at a fundamental level. In Computer Science, maybe some of us mostly use discrete mathematics since our computer is digital (like during studying operating system, algorithms, ...
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1answer
26 views

In which environment we use NFA(Non Deterministic Finite Automata)?

We have two types of Automata. One is NFA and second is DFA. These are little bit different but thing is that in which environment we prefer NFA over the DFA?
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2answers
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Weaker, but similar conditions to Turing completeness?

A model of computation is called Turing complete if it can simulate any Turing machine. This rules out for example a combinational logic circuit. However, there is a sense in which combinational ...
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1answer
32 views

Would hypercomputation machines be capable of simulating/computing/programming everything?

If uncomputable numbers existed, could this hypercomputation machines compute them? Could hypercomputation compute all types of uncomputable things? Even truly inconsistent things? Even things that ...
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1answer
30 views

Prove that $L = \{a^i \;:\; (\exists x \in \mathrm{Lang}(M_i))\;[ xx \notin \mathrm{Lang}(M_i) ] \}$ not recursively enumerable [duplicate]

Past year paper question: Let $M_i$ denote the Turing machine with code $i$ using the alphabet $\Sigma=\{a,b\}$. Show that the following language is not recursively enumerable: $L = \{a^i \;:\; (\...
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0answers
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Non-Turing-recognizable Language [duplicate]

I have been stuck on this problem for a while: Show that $L=\{\langle M \rangle : L(M) \text{ contains an even number of strings} \}$ is not Turing-recognizable. I know that by Rice Theorem, this ...
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3answers
110 views

Is a Turing machine too strong of a model to model physical computation?

I've heard many times people debate the possibility of a real world computation that is impossible for a Turing machine, especially in the context of a human mind. Implying that the Church-Turing ...
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1answer
35 views

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference

Prove/disprove that the class of decidable (resp. partially decidable) languages is closed under symmetric difference. A symmetric difference of sets A and B is the set (A \ B) ∪ (B \ A). I know that ...
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2answers
85 views

Converse of halting problem

It is well known that if some computing apparatus is Turing-complete, then the halting problem is undecidable for that computing apparatus. However, is it true that if the halting problem is ...
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1answer
26 views

Why aren't recognizable languages and co-recognizable languages not reducible to each other?

While learning to prove undecidability of problems, I came across a statement that you can't reduce a recognizable language to a co-recognizable language and vice-versa to prove undecidability. Why is ...
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1answer
41 views

Decidability of Turing Machine accepting exactly 14 words

Would you say that the following problem is undecidable? $$L_1 = \{\langle T \rangle \mid T \text { accepts 14 words}\}$$ My intuition says that this must be undecidable, and I want to try to reduce ...
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1answer
59 views

Is the language of all TMs accepting all strings starting with 010 decidable?

I am trying to figure out if this language is decidable: $$ \{ \langle M \rangle \mid \text{$M$ accepts all strings starting with 010}\}. $$ My intuition is that it is. Whatever string $w$ starts ...
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2answers
53 views

Primitive recursive plus Ackermann

Let us consider the class $\cal F$ of functions that contains all constant functions all projections the successor function the Ackermann function as basic functions, and that is closed under ...
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0answers
28 views

Semidecidable properties of computable reals

By computable real I mean $x\in\mathbb{R}$ such that there is some computable total function $p_x$ that takes a natural number $n$ and returns a dyadic rational $r$ such that $|x-r|<2^{-n}$. I ...
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2answers
29 views

Error in proof of decidability

$L=\{\left<M\right> \ | \ M $ is a TM s.t. $M$ does not accept any string starting with a '1' $\}$. Assume the alphabet to be $\Sigma = \{0,1\}$. By Rice's theorem $L$ is undecidable. I ...
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1answer
58 views

Prove $L = \{M \mid L(M)\text{ is infinite}\}$ is not Turing-recognizable

I'm supposed to prove this through mapping reducibility. I think I'm supposed to show that $A_{\mathrm{TM}} \le_\mathrm{m}\overline{L}$, which means that $\overline{A_{\mathrm{TM}}}\le_\mathrm{m} L$ ...
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2answers
76 views

Does Halts reduce to all other undecidable languages?

In a CS theory class I'm taking, we showed Halts was undecidable via a diagonalization argument. All other undecidable problems we looked at we either got by reducing Halts to them, or some chain of ...