Questions tagged [decidability]

Use this tag for questions about the existence of an algorithm that can and will return a correct true or false value to a decision problem.

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Is deciding termination easier than deciding rationality?

Irrationality proofs are usually extremely hard. Unknown cases include the Euler-Mascheroni-constant $\gamma$ , $e+\pi$ and $e\cdot \pi$ Is it easier to decide whether a given number has a ...
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30 views

Do the Robinson and/or Peano arithmetics suffice to prove that the Halting problem is undecidable?

Robinson and Peano are both rather weak systems, but they suffice to capture the rules of the natural numbers sufficiently to talk about the Gödel numbering etc., and Kleene's T predicate indicates ...
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Diagonalization for Etm

$E_{TM} = \{\langle M\rangle\mid M$ is a TM and $L(M) = \emptyset\}$ We want a proof by diagonalization to show that $E_{TM}$ is undecidable. But the form of inputs are like $<M>$ and the Table ...
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Decidability Results for $\mathcal{L}_{\omega_1, \omega}$?

Recently I was giving a talk on decidability results in logic and I wanted to emphasize that there is more to life than than just First Order Logic. In particular, I made sure to include results such ...
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Degree 2, 3 variable homogenous instance for Hilbert's Tenth Problem

I know that Hilbert's Tenth problem is uncomputable as shown by Matiyasevich's theorem, but for a special case, is Hilbert's tenth problem decidable for degree two homogenous $3$ variable instance of ...
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137 views

Given a Turing Machine $M$ and a word $w$, can we tell if the Turing Machine will move its head at every step?

The exercise I'm trying to solve is: Is the language $L = \{<M>,w\ |\ M \text{ moves its head on input } $w$ \text{ at every step}\}$ decidable or undecidable. We work with a version of the TM ...
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Prove that the problem [input is program Q and the question is does Q calculates the function f(y)=y^2] is undecidable

I need to prove the following problem is undecidable: Input for the problem is: A program called Q; The question is: does the program Q calculates the function f(y)=y^2; My solution is: Reduction from ...
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1answer
80 views

Peano Arithmetic - Decidability

I need to show that if Peano Arithmetic does not decide a sentence $\varphi$ then the standard model of Peano Arithmetic satisfies the negation of $\varphi$. I know this partly has to do with Godel's ...
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117 views

What to do with undecidable problems?

Several fundamental problems in math are undecidable; see the sphere recognition problem for example. To make it worse, quantum computers will not help. Therefore, I wonder what we can do about ...
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60 views

Is it true that every closed formula is decidable? Why?

I was wondering if every closed formula is decidable (in a complete system). Clearly a non closed formula is not decidable, since it can take some values for which it is true, and other for which it ...
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46 views

Let $f$ be a computable and injective function. Is $f^{-1}$ computable and injective?

So I just started learning about computability, undecidability and Turing machines. And I wonder if: Given a computable and injective function $f$, is $f^{-1}$ also computable and injective? I don't ...
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Why is it enough to test only the words up to length n ( number of state) in the given algorithm for a decidability problem

My question is related to the problem below, basically it's a decidability problem and the algorithm prooves it's decidable. My question : After reading the step 2 of the algorithm below, why is it ...
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1answer
38 views

How to show that this language is Turing recognizable

So this question has two questions and i have to use the answer from 1 to answer question 2. Assuming that my answer for 1 is good. I need help with 2. ( Correct me if wrong please.) Question 1 : Show ...
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44 views

If L1 is decidable and if L2 is included in L1, is L2 also decidable?

Is this statement true or false and why ? If $L_{1}$ is decidable and if $L_{2}$$\subseteq$$L_{1}$ then $L_{2}$ is also decidable. I would be tempted to say yes, but i am really not sure. I am also ...
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82 views

How to show that this language is decidable

The question : The language is $L=$ { $< G,w > $ : $G$ is a grammar in normal form of Chomsky and $w$ is a word on the terminal alphabet that can be derivated in $G$ by at least 2 differents ...
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153 views

NP, NP-complete and Reducibility

Suppose there is polynomial time reduce from problem $A$ to $B$. fact $1)$ if problem $B$ is $NP$-hard then Problem $A$ is $NP$-complete. fact $2)$ if problem $A$ is $NP$-complete then Problem $B$ is $...
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1answer
81 views

Show that the following language is decidable by finding the algorithm for the finite automaton

Given the language $K$ = $\{<M>: M$ is a finite automaton on the alphabet {0,1}) and $L(M)$ contains at least one word of the form $0^k1^l$ with $k,l\geq 0$}. In other words, describe an ...
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Show that this language is undecidable

Given the language $K$ $=\{<M> $ where $M$ is a turing machine ( that is on the alphabet {0,1}) and $L(M)$ contains at least one word of form $0^k1^l$ with $k,l\geq 0\}$ I would like to know if ...
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1answer
38 views

Best approach for Undecidability proof

Context: Hi, my professor sent me this challenge and I got stuck. I thought using Rice's Theorem for this question, since $M$ is non-trivial, but he told me to use a reduction. Is he right? Should I ...
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56 views

show this language is undecidable

$L = \{ \langle G \rangle \mid G$ is context-free, $L(G)$ contains a palindrome $\}$ Reduce the post correspondence problem to $L$. So I have to show that if I could decide L then I could decide PCP. ...
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Boolos’s proof of the first incompleteness theorem. Predicate $C(x,y)$ and assumption of completeness.

I am trying really hard to understand how Boolos’s proof works, but I keep having doubts about it. Can you spot any logical flaws in my reasoning? This question arose from a previous question of mine, ...
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36 views

Show that $G_{u,v}$ is not decidable and not recognizable (nor is its complement)

Let $\Sigma$ be an alphabet, $u,v \in \Sigma^*, u \neq v$. $G_{u,v} = \{ \langle M \rangle \mid M$ is turing machine, $u \in L(M) \Leftrightarrow v \in L(M)\}$. I understand that if it is not ...
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Are the following languages decidable?

$L_2 = \{ \langle M \rangle \mid M $is a turing machine, there is a different turing machine M' with $L(M')=L(M) \}$ $L_4 = \{ \langle M \rangle \mid M(\varepsilon) $writes three consecutive 1s at ...
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$\mathsf{DSPACE}(n) \subsetneq \mathbb{E}$: there is a decidable language no LBA can handle..

Use the following lemma and diagonalisation: Lemma: Let $M$ be a linear bounded automaton. Then, there is a lba $M'$, that accepts the same language and halts on every input. ($\mathsf{DSPACE}(n)$ is ...
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1answer
29 views

Show that the following language is decidable

$E_{cfg}=\{ \langle G \rangle \mid G \text{ is context-free grammar}, \ L(G)=\emptyset \}$ I think the following algorithm should work: Test whether or not the input $w$ codes a cfg. If not, reject. ...
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106 views

Has Any Currently Open Problem in Mathematics Definitively Been Shown to be Decidable?

There is a fairly extensive list of problems in various fields that have been shown to be undecidable. For example, see https://en.wikipedia.org/wiki/List_of_undecidable_problems And certainly, an ...
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Decidability of convergence of real series given an oracle for positive series

I've been reading for the last hour a few posts on this site about series that no one knows if they converge or not. I was quite surprised that they were almost all consisting of positive terms. ...
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1answer
287 views

Boolos’s proof of Gödel’s first incompleteness theorem. What am I getting wrong?

Sorry in advance, the following is quite messy, I didn’t find a way to express myself more clearly and rigorously. I would appreciate suggestions on how to make the following better and try to act ...
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1answer
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Gödel's first incompleteness theorem. What did I get wrong? [closed]

I'd like to point out that obviously I don't claim the following to be right, and I now recognise that the way my question was phrased before editing could have been interpreted as very arrogant, and ...
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1answer
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Open problems in the first order theory of the real numbers

It is a well-known result that the first-order theory of the real numbers is decidable. However the decision algorithm for this language is in double exponential time, so such an algorithm cannot be ...
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1answer
40 views

Prove decidability. Sets A and B. B is finite (have inite number of elements), A is undecidable. Can $A$ $cup$ $B$ be decidable?

Sets $A$ and $B$. $B$ is finite (have inite number of elements), $A$ is undecidable. Can $A$ $\cup$ $B$ be decidable? I have and idea, but i'm not sure that it's correct. What if $A \cup B$ is ...
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1answer
43 views

Prove the decidability of an enumerable set. A and B are enumerable, C is decidable.

Sets A and B are enumerable, C is decidable. $A$ $\subseteq$ $C$ $\subseteq$ $A$ $\cup$ $B$. $A$ $\cap$ $B$ = $\emptyset $.Prove that A is decidable too
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1answer
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Is the statement $A \leq_{\operatorname{mo}}^{\operatorname{poly}}B \land A\in \operatorname{REC}\implies B\in\operatorname{REC}$ true?

Is the statement $A \leq_{\operatorname{mo}}^{\operatorname{poly}}B \land A\in \operatorname{REC}\implies B\in\operatorname{REC}$ true? I know, that $A \leq_{\operatorname{mo}}^{\operatorname{poly}}B$...
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1answer
73 views

How close is “$Y$ is connected” to a $\Sigma^0_1$, or $\Pi^0_1$ sentence?

I was giving some thought to this answer and I was curious about the sentence $Y$ is connected How close is $Y$ is connected to a $\Sigma^0_1$, or $\Pi^0_1$ sentence? For a little more context... ...
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How to disprove that a problem is undecidable

My question is similar a question already asked Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture But in general, given some well-known problems of mathematics for example the Riemann ...
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1answer
38 views

Given a TM T , is there an input string that causes T to enter every one of its nonhalting states?

Given a TM T , is there an input string that causes T to enter every one of its nonhalting states? I think it is undecideable and want to use reduction to prove it, but can't think of an undecideable ...
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1answer
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Prove that: For each language L that is accepted by a DTM in polynomial time, there is also a DTM M′ that decides L in polynomial time.

According to one of my professors, the statement "For each language L that is accepted by a DTM in polynomial time, there is also a DTM M′ that decides L in polynomial time." is true, but i ...
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1answer
50 views

Decidable Languages over an alphabet

How can we prove that exist as many decidable languages over {1, 0} as over the alphabet {0}. I understand that the number of turing machines over an alphabet is countable. But how can we formulate a ...
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Is there a “basis” or “subbasis” for semidecidability?

There seems to be an analogy between semidecidability and topology, but to me, the "topology" seems to be weak. The usual definition of a topology over a space $X$ is: Statement #1: $\...
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1answer
198 views

An undecidable problem and a non-semidecidable one

Prove that the decision problem "Does $f$ match this behaviour?" is undecidable (assume the behaviour is nontrivial) and that the problem "Is $h(x)$ undefined?" is not semi-...
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Decidability of $2^a \bmod a^3$?

From some limited testing, it seems to me like $2^a \bmod a^3$ has no repeat values except for some powers of $2$. More generally, it also seems that $x^a \bmod (a^3+k)$ often holds to this, although ...
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Is Hilbert's tenth problem decidable for degree $2$?

Hilbert's tenth problem is the problem to determine whether a given multivariate polyomial with integer coefficients has an integer solution. It is well known that this problem is undecidable and that ...
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1answer
32 views

Separation axiom implied by semidecidability of comparison

I am studying computable analysis. What I'm fascinated by is the analogy between computable analysis and general topology: a Wikipedia article Semidecidable sets are analogous to open sets. So I ...
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1answer
28 views

Prove decidability or undecidability of a language accepted by a turing machine.

consider this problem: "Given a TM $X$, determine if the language accepted by $X$ contains more than $100$ strings" Is this problem decidable or not? Opinion 1: (mine) I derived my proof by ...
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1answer
55 views

Prove that $L=\big\{\langle G\rangle \mid G\text{ is a CFG over }\Sigma=\{0,1\}\text{ and }1^* \cap L(G)\ne\varnothing\big\}$ is decidable.

How to prove that $L = \big\{\langle G\rangle \mid G \text { is a CFG over } \Sigma = \{0,1\} \text { and } 1^* \cap L(G) \ne \varnothing\big\}$ is decidable? I know I am supposed to prove that it is ...
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1answer
74 views

“Natural” example of an undecidable subset of $\Bbb N$

All the simple examples of undecidable problems that I know deal with symbolic computation or calculation. For example, the halting problem, whether Diophantine equations have solutions, the word ...
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What does “L being a CFL” as “a property of CFL's” mean?

In Ullman's Introduction to Automata, Languages and Computation (1979): 8.8 Use Theorem 8.14 to show that the following properties of CFL's are undecidable. a) L is a linear language. b) L is a CFL. ...
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Why is the blank-tape halting problem $H_0$ so relevant in literature?

After comparing some text books and lectures about computability theory I encountered not only the halting problem $$H = \{\langle M,x \rangle \mid \text{the Turing machine $M$ halts on input $x$}\}$$ ...
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Is there any example of a property which is true but undecidable?

Yesterday I thought I had found a paradox about undecidable statements. Let's say there is some property P about integers, which is formulated as follows: ∀ n ∈ ℕ, something(n). Where "something" is a ...
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Necessity of decidable type checking for formalizing mathematics

If a type theory such as Martin-Löf's dependent type theory (MLTT) is to be used as a foundation for mathematics, decidable type checking is certainly nice to have: it guarantees that for every proof ...