Questions tagged [undecidability]

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

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Undecidable: $w$ on which a TM M $M$ halts after $\leq w$ steps

The detailed question is: Is there a word $w$ on which a TM M $M$ halts after a maximum of $|w|$ (word length) steps? I highly assume, that this problem is not decidable because in the worst case ...
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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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1answer
59 views

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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Does undecidability violate Turing completeness? Shouldn't “complete” include “decidability”? [closed]

Does undecidability violate Turing completeness? Shouldn't "complete" include "decidability"? That is, if one has a language that's Turing complete, but expresses infinite computation (i.e. may not ...
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27 views

PCP decidable if all words have the same length?

If $X=(x_1,...,x_n) , Y=(y_1,...,y_n)$ with $|x_i|=|y_i|=|x_{i+1}|=|y_{i+1}|=...$ for all $i \leq n$ Is PCP now still undecidable ? I would say it's now decidable because we just have to find some ...
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Is decidability closed under the mapping f where f(a)=f(b)=0 and f(c)=1?

Consider the function $f$ that maps strings over $\{a, b, c\}$ to strings over $\{0, 1\}$ by replacing each $a$ by 0, each $b$ by 0, and each $c$ by 1. For example $f(cabbc) = 10001$. The function $f$ ...
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1answer
17 views

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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2answers
235 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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65 views

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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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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2answers
114 views

Can we enumerate finite sequences which have no halting continuation?

Note: this question has been cross-posted to Math.SE, after about a week here. I am trying to deepen my understanding of the relationship between the Halting Problem and Godel's Completeness Theorem (...
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76 views

Decidable questions of undecidable problems

Even if there is no general algorithm to decide if any program will halt, but there could be properties or meta-questions about the programs that is decidable. For example, given program $A$ and a ...
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1answer
211 views

A language which is neither r.e. nor co-r.e

First, consider $$L_\exists=\{\langle M\rangle \mid M \text{ is a Turing machine and accepts some input}\}$$ is RE. I tried to construct a Turing machine: ...
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2answers
320 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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33 views

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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1answer
104 views

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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2answers
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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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1answer
604 views

Is chess game movement TM decidable?

If we have a finite chess board and two figures x and y. Is it possible to get y from x by following chess rules and when white is y and white starts from y placement. Is this decidable? My ...
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2answers
700 views

Recursive language subtracted from recursively enumerable language

This is a homework problem but I am awfully confused. The problem reads as follows: If $L_1$ is recursively enumerable but not recursive, and $L_2$ is recursive, then which of the following is the ...
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230 views

Determining the classification of languages

$L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ halts on $w$}\}$ $L_1 = \{ \langle M, w, 1 \rangle \mid \text{$M$ does not halt on $w$}\}$ $L = L_0 \cup L_1$ I need to determine where in ...
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Prove or disprove if $L_{1}$ is undecidable and $L_{2}$ is finite language then $L_{1} \cup L_{2}$ is undecidable

I tried to prove by contradiction. $L_{1}$ is undecidable and $L_{2}$ is finite language then $\overline{L_{1}}\cap \overline{L_{2}}$ is decidable. $$L_{1} = \overline{HALT_{TM}} = \big\{ \langle M, ...
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Does showing a problem and its complement are not Turing-decidable means that the language & its complement are not Turing-recognizable?

I was reading the Sipser's book on the Theory of Computation, 3rd edition and came up with a question. "Does showing a problem and its complement are not Turing-decidable means that the language & ...
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4answers
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What approaches are most useful when proving uncomputability of a given function?

I'd like to understand what approaches should one adopt when deciding/proving that a given function F is uncomputable, by any Turing Machine (TM). The ones I've tried so far are as follows: Reduction,...
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1answer
34 views

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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1answer
51 views

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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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
38 views

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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1answer
27 views

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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1answer
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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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1answer
168 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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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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1answer
31 views

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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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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2answers
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Halting problem vs Universal Language

Wikipedia defines halting set as follows: $H = \{(i, x) |$ program $i$ halts when run on input $x\}$ Ullman defines universal language as follows $U = \{(M, w) |$ Turing machine $M$ accepts $w\}...
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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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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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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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340 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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1answer
346 views

Is it decidable “Given a TM M, whether M ever writes a non blank symbol when started on the empty tape.”

I came across below problem in this pdf: Given a TM M, whether M ever writes a non blank symbol when started on the empty tape. Solution given is as follows: Let the machine only writes blank ...
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2answers
128 views

Can I apply Rice's theorem to decide decidability status of these languages?

I came across these languages: A Turing machine prints a specific letter. A Turing machine computes the products of two numbers I was guessing whether I can apply Rice's theorem to decide upon above ...
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2answers
64 views

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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1answer
65 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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1answer
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Is checking if the length of a C program that can generate a string is less than a given number decidable?

I was given this question: Komplexity(S) is the length of the smallest C program that generates the string S as an output. Is the question "Komplexity(S) < K" decidable? With respect to ...
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32 views

Reducing universal language to language of palindromes

I am trying to understand proof for proving language of all palindromes is undecidable from these slides. It tried to reduce universal language to language of all palindromes on alphabet. The two ...
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1answer
38 views

Is the problem of deciding whether two programs have the same semantics decidable?

If I have program and I want to check whether other programs have the exact same semantics or not, could I always build a machine that could make that decision? This is a question relevant to ...
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19 views

Correct Turing machine representation for Rice Theorem proof

Consider the language L1. From Rice Theorem I know L1 is not decidable (i.e. undecidable). L1 = { R(M) | R(M) is a TM and 1011 ∈ L(M)} For example if I want to represent by diagram a TM $M_1$ ...

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