Questions tagged [undecidability]

Questions about problems which cannot be solved by any Turing machine.

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Decidability of a language and inclusion between two other languages

I have this assignement that asks to say if the following statement is true or false, and possibly justifying the answer: "Let L₁, L₂ be decidable languages. For every language L s.t. L₁ ⊆ L ⊆ L₂, L ...
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Check if language is decidable

I would like to determine if the following language is decidable or not. L = { w $\in$ $\Sigma^*$ | $T(M_w)$ is recognized by a Turing machine with at most 42 states}. I know that every finite ...
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1answer
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Decide if a language has a word of a given size

Suppose that $L$ is some language over the alphabet $\Sigma$. I was asked to show that the following languages is decidable: $$L' = \{w \in \Sigma^* | \text{ there exists a word } w'\in L \text{ ...
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Many-one reductions between the set of true sentences and a particular arithmetical set

Never used this site before so not sure the best way to get help. However, I've been looking at many-one reductions in relations to sentences in logic. Let TH(N) = {ϕ : ϕ is a first order sentence ...
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1answer
199 views

Decidability of Turing machines that never move their heads past any input string

$L_1 = \{ \langle M, w\rangle : M \text{ is a TM that never moves its head past the input string } w \}$ $L_2 = \{ \langle M\rangle : M\text{ is a TM that never moves its head past any input string} ...
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41 views

Decidability of equality, and soundness of expressions involving elementary arithmetic and exponentials

Let's have expressions that are composed of elements of $\mathbb N$ and a limited set of binary operations {$+,\times,-,/$} and functions {$\exp, \ln$}. The expressions are always well-formed and form ...
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Is the undecidability of a given problem undecidable?

Given an input problem P, can you construct an algorithm A to compute whether or not P is decidable or undecidable? In other words, is the undecidabiliy of a problem undecidable? My initial guess is ...
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Halting problem theory vs. practice

It is often asserted that the halting problem is undecidable. And proving it is indeed trivial. But that only applies to an arbitrary program. Has there been any study regarding classes of programs ...
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About computable sets

Let TOT be the set of all Turing Machines that halt on all inputs. Find a computable set B of ordered triples such that: TOT = {e : ($\forall$x)($\exists$y)[(e, x, y) $\in$ B] This definition means ...
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How could you “solve” the halting problem if, hypothetically, the busy beaver numbers were “small”?

I read that if BB(n) did not grow faster than all computable sequences of integers, you could solve the halting problem and contradict Turing's theorem. I'm trying to figure out how you could ...
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What does “Every CFL is decidable” exactly mean?

I am trying to prove the fact that every CFL is decidable, however I can't come to terms with what the statement exactly means. I know that generation of a particular string by a given CFG is a ...
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Mapping reduction from $A_{TM}$ to $INFINITE_{TM}$ same as to $ALL_{TM}$?

I was trying to solve a problem with a mapping reduction from $A_{TM}$ to $INFINITE_{TM}$, and came across a solution that was 100% identical to another solution I saw for $A_{TM} \leq_M ALL_{TM}$. ...
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Turing machine on input w tries to move its head past the left end of the tape

Consider the language $$ L = \{ \langle M,w \rangle \mid \text{$M$ on input $w$ tries to move its head past the left end of the tape}\}. $$ Prove whether L is decidable or not. I tried to prove ...
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1answer
55 views

Turing machines moving left at least once

Is the following language decidable? $$ L = \{ \langle M,w \rangle \mid \text{$M$ moves its head left at least once when run on $w$}\}. $$ I feel like this is a decidable language. But I don't know ...
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Is there a language that cannot be polynomially reduced to?

Is there a language A that cannot be polynomially reduced to by some language B? Or is it always possible to reduce a language B to A?
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Difference between regular grammar and CFG in generating computation histories and $\Sigma^*$

I would like to ask for intuition to understand the difference between a CFG generating $\Sigma^*$ and a regular grammar generating $\Sigma^*$.. I got the examples here from Sipser. Let $ALL_{CFG}$ ...
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Is it semidecidable to test whether a Turing decidable language is empty?

I'm not sure how to go about solving this. I tried this: Suppose L is a Turing decidable language. Turing Machine M1 is a decider of L and M2 is a decider of the complement L We construct a TM U ...
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Is it decidable for a NPDA to halt?

I know that it is decidable for an NPDA to accept a string $w$, i.e. a TM can receive as input the description of an NPDA along with a string and test if the NPDA accepts the string. But are there ...
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Is it decidable to know the number of positions used by a Turing machine for a fixed input?

I'm having trouble proving if the following language is recursive, recursively enumerable, or not r.e. at all: the set of all encodings of Turing machines $M$ such that the number of positions in the ...
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How does a PDA compare two configurations of accepting histories?

In Michael Sipser's book, they prove that ALL_CFG is undecidable using accepting computation histories and PDAs. My question is how exactly (with details of implementation) a PDA goes on to compare ...
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A Turing machine for which it is impossible to predict whether it halts or not on a fixed input

The halting problem is undecidable, i.e. $\not \exists$ $M$ Turing machine s.t. for every $(M_0,w_0)$ input where $M$ is the description of a Turing machine and $w_0$ is an input word, the output of $...
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Halting problem for fixed Turing machine and fixed input

It is known that the halting problem is undecidable even when we fix either the Turing machine $M$ or the input $w$. What if we fixed both the machine and the input? I.e., is it decidable for every ...
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Find decidable sets such that $A$ reduces to $B$ but not vice versa

I am stuck in this problem, so any help is appreciated. The problem asks to show that there exists decidable sets $A$ and $B$ such that $A \leq_{m}^{p} B$ but $B \not \leq_{m}^{p} A$, and that $A$, $B$...
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EVEN-CFL Decidable / Undecidable - Rice-Theorem

Let EVEN-CFL $=\left\{w | M_{w} \text { is a } \mathrm{TM}, \text { such that } L\left(M_{w} \right) \\ \text{ has only words with even length and is context free.}\right .\}$ Question : Is EVEN-CFL ...
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Design a DFA recognising the following language

Design a DFA over alphabet (a,b) such that for all it's string no prefix contain two more a's than b's and two more b's than a's and the number of a's is equal to b's. Is it possible to design a DFA ...
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show that this decidable set $C$ exists

I came across this problem which says that given disjoint sets $A$ and $B$ s.t $\bar{A}$ and $\bar{B}$ are both computably enumerable (c.e.), there exists a decidable set $C$ s.t. $A \subseteq C$ and $...
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2answers
245 views

show that in every infinite computably enumerable set, there exists an infinite decidable set

I came across this problem: Show that in every infinite computably enumerable set, there exists an infinite decidable set. As an attempt to solve the problem, I could only think of a proof by ...
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2answers
344 views

Why doesn't the recursion theorem prove there is an undecidable finite set?

I created something similar to Sipser's proof for the undecidability of $A_{TM}$ (theorem 6.5), "proving" the undecidability of a set that must be finite. Presumably, it's wrong, but I can't figure ...
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Is this correct : whether or not a type 3 grammar generates $\Sigma^*$ is not c.e

An example from Sipser's book, Introduction to the Theory of Computation, shows that it is not decidable for a $TM$ to recognize whether a $CFG$ (or a type 2 grammar) generates $\Sigma^*$, where $\...
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Prove {<M> | TM M on input 3 at some point writes symbol “3” on the third cell of its tape} is recursively enumerable but not recursive

Question: Let $$S = \{\langle M\rangle\mid \text{TM }M\text{ on input 3 at some point writes symbol “3” on the third cell of its tape} \}.$$ Show that $S$ is r.e. (Turing acceptable) but not recursive ...
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Is L decidable or not

Let $L = \{\lt M\gt | M$ is a $TM, L(M) = \{1^n0^n | n\ge0\}\}$. Create a reduction from $A_{TM}$ (acceptance problem) to $L$. Is $L$ not decidable? But isn't $L$ decidable since it is possible to ...
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Can you apply Rice's Theorem on the following languages? Are they decidable?

Can you apply Rice's Theorem on the following languages? Are they decidable? $$L_1:=\{v\mid v \text{ is the Code of a TM } M_v \text{ and } M_v \text{ has an even number of states.}\}$$ $$L_2:=\...
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1answer
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Union of halting-like problem and non-halting-like problem

I came across the following problem: Define languages $L_0$ and $L_1$ as follows : $L_0=\{⟨M,w,0⟩∣M\text{ halts on }w\}$ $L_1=\{⟨M,w,1⟩∣M\text{ does not halt on }w\}$ Here $⟨M,w,i⟩$ is ...
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how is the set of undecidable programs related to the set of non-halting programs?

Is there a non-halting program for every undecidable program? is undecidable the "same thing" as non-halting? Thanks!
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Undecidability of language involving two TMs

I am currently browsing the lecture notes on computability/decidability and I have encountered the following exercise I am unable to solve. Given $M_1$, $M_2$ Turing machines, is it true that for ...
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How can I apply Rice's theorem?

I am learning for my computability and complexity exam in which there is always an exercise to decide whether some problem is decidable or not. In one of the past exams, there was the following ...
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Is it decidable whether Turing Machine never scans any tape cell more than once when started with given string

The problem: Is it decidable that the set of pairs $(M,w)$ such that TM $M$, started with input $w$, never scans any tape cell more than once. How can I easily prove above to be decidable. I found ...
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Is it decidable if there exists some input such that the TM makes at least five moves?

I am reading this excerpt from Ullman's book: I have following doubts: (related to red underline) TM can make 5 left moves or 5 right moves. So it will need at max 11 cells. Then how it says 9? (...
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196 views

Undecidability of two Turing machines acting the same way on an input

So I need to find a reduction to the (undecidable) problem of deciding if two Turing machines $M_1$ and $M_2$ behave the same way on an input $x$. "Behaving the same way" is defined like this: $M_1$ ...
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Proving sets of regular expressions and context free grammars are decidable [duplicate]

Consider below languages: $L_1=\{<M>|M$ is a regular expression which generates at least one string containing an odd number of 1's$\}$ $L_2=\{<G>|G$ is context free grammar which ...
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In the reduction from HALT to ALLHALT, why does the constructed Turing machine loop indefinitely when the inputted Turing machine rejects?

Let HALT be the language $\{\langle M, w\rangle : M\text{ is a TM that halts on }w \}$. Let ALLHALT be the language $\{\langle M\rangle : M\text{ is a TM that halts on all inputs}\}$. Use a reduction ...
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reduction: L1’s decidability is unknown and L2 is undecidable

About this question: Reductions can be tricky to get the hang of, and you want to avoid “going the wrong way” with them. In which of these scenarios does L1 ≤m L2 provide useful information (and ...
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Proving Problems are Undecidable/ Semi decidable? E.g. Halting Problem, Membership Problem? [duplicate]

I am having issues finding similarities in different cases where a problem such as the Halting Problem or the Accept-Λ problem is reduced to the Membership problem to prove that it is semi-decidable ...
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TM1 accepts w1 vs TM1 halts on w1

What is difference between following two problems, their decidability and recognizability status: Given Turing Machine TM "accepts" given string w. Given Turing Machine TM "halts on" given string w. ...
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is the empty language L = ∅ a subset of every languages?

I need to show false the following claim Every language L which is a subset of $A_{TM}$ ($L \subseteq A_{TM}$) is undecidable. For this, I wish to use the empty language L = ∅ (I know is decidable)...
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if A is decidable then B is decidable too

Assume that a language A is reducible to language B. The claim is true? if A is decidable then B is decidable too. The correct answer is: This claim is wrong. If A is e.g. the empty language (...
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368 views

Prove that the class of CFG languages that are closed under reversal is undecidable

Note The wording of the title may be a bit vague, but I'm not asking if CFLs are closed under reversal. Please see below. Problem Description Given a word $w$, define $w^{r}$ to be its reversal. ...
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A special case of subset sum

I came across the following problem in my complexity-theory course: Given a set of numbers $A := \{a_1, \dots, a_n\} \subset_{\mathrm{finite}} \mathbb{N}$ and a number $b$ also in $\mathbb{N}$ such ...
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Is it possible to write down every Prolog program+query as the sequent in the sequent calculus?

Prolog program P is set of Horn (definite) clauses, effectively it is the conjunction of implicational formulas. I guess that every Prolog program P with some query Q can be written as ...
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74 views

Is it decidable whether a given Turing machine moves its head more than 481 cells away from the left-end marker, on input ε?

So, while reading some problems on decidability, I came across the following resource: https://www.isical.ac.in/~ansuman/flat2018/tm-more-undecidable.pdf Here, on page no 12, it is written that the ...

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