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Questions tagged [undecidability]

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

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11
votes
3answers
615 views

Is it possible to decide if a given algorithm is asymptotically optimal?

Is there an algorithm for the following problem: Given a Turing machine $M_1$ that decides a language $L$, Is there a Turing machine $M_2$ deciding $L$ such that $t_2(n) = o(t_1(n))$? The ...
3
votes
1answer
191 views

Why does $A_\text{TM} \le_m \text{HALTING} \le_m \text{HALTING}^\varepsilon$?

I have a book that proves the halting problem with this simple statement: $$ A_\text{TM} \le_m \text{HALTING} \le_m \text{HALTING}^\varepsilon $$ It states that halting problem reduces to the ...
5
votes
2answers
233 views

Language comprising of Turing machine encodings

Let $A$ be the language $\{\langle M\rangle\mid M\text{ is a Turing machine that accepts only one string}\}$ According to my understanding, if a Turing machine is able to decide if another Turing ...
3
votes
1answer
1k views

Deciding Recursive/Recursively Enumerable when given Turning machine encoding as a language

Let $L_{1}$ and $L_{2}$ be two languages defined as follows : $L_1 = \{ \langle M\rangle \mid L(M) \neq \emptyset \}$ $L_2 = \{ \langle M\rangle \mid L(M) = \emptyset \}$ where $\langle M\...
5
votes
2answers
446 views

Is the undecidable function $UC$ well-defined for proving the undecidability of Halting Problem?

I am new to Computability Theory and find it is both amazing and confusing. Specifically, it is difficult for me to get through the undecidability of the well-known Halting Problem. Halting ...
5
votes
1answer
2k views

Decision problem and algorithm

I was reading about decision problem. I understand that decision problem tell yes/no answer for an input. The decision is based on a decision procedure also called an algorithm. The wikipedia says ...
10
votes
4answers
5k views

Is there an undecidable finite language of finite words?

Is there a need for $L\subseteq \Sigma^*$ to be infinite to be undecidable? I mean what if we choose a language $L'$ be a bounded finite version of $L\subseteq \Sigma^*$, that is $|L'|\leq N$, ($N \...
1
vote
1answer
520 views

Is the intersection of two regular languages regular?

Trivially decidable problem is one in which the problem is a known property of the language/grammar. So intersection of two regular languages is regular should be trivially decidable? But it is given ...
8
votes
1answer
147 views

Question related to Hilbert's 10th problem

Given $n \in \mathbb{N}$ and $p,q \in \mathbb{N}[x_1,\ldots,x_n]$ one can define the following formula in the language of formal arithmetics $$\varphi(n,p,q) = \forall x_1 \cdots \forall x_n : \neg ...
23
votes
2answers
1k views

Is there a "natural" undecidable language?

Is there any "natural" language which is undecidable? by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the ...
2
votes
3answers
2k views

Undecidability of the following language

So we can prove that the language say $A = \{ \langle M,w \rangle \mid \text{M is TM that accepts } w^R \text{ whenever it accepts } w \}$ is undecidable by assuming it is decidable and use that to ...
2
votes
1answer
871 views

Solvability of Turing Machines

I'm preparing for an exam, and on a sample one provided (without solutions), we have this question: Is the following solvable or non-solvable: Given a turing machine $T$, does it accept a word of even ...
6
votes
1answer
220 views

Showing the function=? is impossible

Here's a lab from a first-year computer science course, taught in Scheme: https://www.student.cs.uwaterloo.ca/~cs135/assns/a07/a07.pdf At the end of the lab, it basically presents the halting problem,...
2
votes
1answer
273 views

Reduction of A_LBA to E_LBA

I have a rather interesting one to ponder and would love if I could get an answer for it. We were discussing the topic of mapping reduction today in my Computing theory course and I was wondering why ...
4
votes
2answers
293 views

First-order logic arity defines decidability?

I've read first-order logic is in general undecidable, and that could be decidable only when working with unary operators. (I think that's propositional logic, correct me if I am wrong) The question ...
5
votes
1answer
230 views

Hardness of ambiguity/non-ambiguity for context-free grammars

A grammar is ambiguous if at least one of the words in the language it defines can be parsed in more than one way. A simple example of an ambiguous grammar $$ E \rightarrow E+E \ |\ E*E \ |\ 0 \ |\ ...
8
votes
1answer
161 views

Can $f$ be not computable even if $L$ is decidable?

I am trying to teach myself computability theory with a textbook. According to my book, a function $f$ over an alphabet $A=\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, ...
5
votes
1answer
910 views

When are 2 decision/optimization problems equivalent?

Does anybody know a good definition of 2 decision / optimization problems being equivalent? I am asking since for example allowing polynomial time computations any 2 problems in NP could be ...
3
votes
1answer
1k views

Which properties of context sensitive languages are decidable?

There are two context-sensitive languages, $L_1$ and $L_2$. Which of the following statements about them are decidable respectively undecidable? $L_1 = \emptyset$ $L_1 = \Sigma^*$ $L_1 \cap L_2 = \...
20
votes
1answer
519 views

Ratio of decidable problems

Consider decision problems stated in some “reasonable” formal language. Let's say formulae in higher-order Peano arithmetic with one free variable as a frame of reference, but I'm equally interested ...
14
votes
1answer
385 views

For a Turing Machine $M_1$, how is the set of machines $M_2$ which are "shorter" than $M_1$ and which accept the same language decidable?

I wonder how come that the following language is in $\mathrm R$. $L_{M_1}=\Bigl\{\langle M_2\rangle \;\Big|\;\; M_2 \text{ is a TM, and } L(M_1)=L(M_2), \text{ and } |\langle M_1\rangle| > | \...
8
votes
1answer
235 views

How to show that the set of machines which accept languages in $\mathrm{NP}\smallsetminus\mathrm P$, is decidable only if $\mathrm P=\mathrm{NP}$?

I'd like your help with proving that the language $$L=\{\langle M \rangle \mathrel| L(M) \in \mathrm{NP}\smallsetminus \mathrm{P} \}$$ is decidable iff $\mathrm{P}=\mathrm{NP}$. If $\mathrm{P}=\...
5
votes
1answer
3k views

Examples of undecidable problems whose intersection is decidable

I know that given two problems are undecidable it does not follow that their intersection must be undecidable. For example, take a property of languages $P$ such that it is undecidable whether the ...
2
votes
1answer
882 views

Showing that the set of TMs which visit the starting state twice on the empty input is undecidable

I'm trying to prove that $L_1=\{\langle M\rangle \mid M \text{ is a Turing machine and visits } q_0 \text{ at least twice on } \varepsilon\} \notin R$. I'm not sure whether to reduce the halting ...
5
votes
2answers
203 views

Semi-decidable problems with linear bound

Take a semi-decidable problem and an algorithm that finds the positive answer in finite time. The run-time of the algorithm, restricted to inputs with a positive answer, cannot be bounded by a ...
4
votes
2answers
271 views

Why absence of surjection with the power set is not enough to prove the existence of an undecidable language?

From this statement As there is no surjection from $\mathbb{N}$ onto $\mathcal{P}(\mathbb{N})$, thus there must exist an undecidable language. I would like to understand why similar reasoning ...
11
votes
2answers
928 views

Can we show a language is not computably enumerable by showing there is no verifier for it?

One of the definitions of a computably enumerable (c.e., equivalent to recursively enumerable, equivalent to semidecidable) set is the following: $A \subseteq \Sigma^*$ is c.e. iff there is a ...
17
votes
4answers
970 views

Is this finite graph problem decidable? What factors make a problem decidable?

I want to know if the following problem is decidable and how to find out. Every problem I see I can say "yes" or "no" to it, so are most problems and algorithms decidable except a few (which is ...
11
votes
2answers
6k views

A Question relating to a Turing Machine with a useless state

OK, so here is a question from a past test in my Theory of Computation class: A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\...
18
votes
5answers
1k views

Is it possible to solve the halting problem if you have a constrained or a predictable input?

The halting problem cannot be solved in the general case. It is possible to come up with defined rules that restrict allowed inputs and can the halting problem be solved for that special case? For ...

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