Questions tagged [decidability]

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98
votes
15answers
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A simple decision problem whose decidability is not known

I am preparing for a talk aimed at undergraduate math majors, and as part of it, I am considering discussing the concept of decidability. I want to give an example of a problem that we do not ...
34
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3answers
3k views

Turing Machine restrictions that render halting decidable

If one restricts Turing Machines to a finite tape (i.e., to use bounded space $S$), then the halting problem is decidable, essentially because after a number of steps (which can be calculated from the ...
25
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2answers
2k views

Is it decidable to determine if a given shape can tile the plane?

I know that it is undecidable to determine if a set of tiles can tile the plane, a result of Berger using Wang tiles. My question is whether it is also known to be undecidable to determine if a single ...
21
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2answers
1k views

Decidability of diophantine equations over {=, +, gcd}

It is well-known that polynomial diophantine equations are undecidable (Hilbert's 10th problem): that is, given a quantifier-free formula over the language $\{=, +, \cdot, 1\}$ (of equality, addition, ...
20
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5answers
730 views

What notable automaton models have polynomially-decidable containment?

I'm trying to solve a particular problem, and I thought I might be able to solve it using automata theory. I'm wondering, what models of automata have containment decidable in polynomial time? i.e. if ...
19
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4answers
918 views

Checking if all products of a set of matrices eventually equal zero

I am interested in the following problem: given integer matrices $A_1,A_2, \ldots, A_k$ decide if every infinite product of these matrices eventually equals the zero matrix. This means exactly what ...
19
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1answer
945 views

Is equivalence of unambiguous context-free languages decidable?

It is well known that the equivalence problem is undecidable for general context-free languages. However, all proofs of this fact that I am aware of seem to involve some ambiguous context-free ...
18
votes
2answers
652 views

Is it possible to decide $\beta$-equivalence within System F (or another normalizing typed λ-calculus)?

I know that's impossible to decide $\beta$-equivalence for untyped lambda calculus. Quoting Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984).: If A ...
18
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2answers
695 views

Deciding whether a unary context-sensitive language is regular

It is a well-known result that the question Does a context-free grammar generate a regular language? is undecidable. However, it becomes decidable on a unary alphabet, simply because in this case,...
15
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1answer
635 views

Computability and continuity

Say $L_1$ and $L_2$ are computable languages. Let $f$ be a function $L_1 \rightarrow L_2$. Let $C$ be the statement, "if $l \subseteq L_2$ and $l$ is a computable language, the preimage $f^{-1}(...
12
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1answer
718 views

What is the simplest computational model for which the emptiness problem is undecidable?

What is the simplest computational model for which the emptiness problem is undecidable? Emptiness problem for a computational model (e.g. finite state automaton, alternating pushdown automaton, ...
12
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1answer
140 views

Decidable theory of asymptotic growth

What are the known limits of the decidability of the comparison of growth rate of functions from $\mathbb{N} \to \mathbb{N}$? I am here thinking of the decidability of questions like "Is $x^x \sim 2^{\...
11
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3answers
1k views

Does P contain incomprehensible languages? (TCS community wiki)

Answer: not known Many thanks to all who helped refine this question and the definitions associated to it. The definitions of this wiki provided the starting point for the more recent TCS wiki "...
11
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2answers
256 views

Vector Addition Systems with finite “obstacles”

A Vector Addition System (VAS) is a finite set of actions $A \subset \mathbb{Z}^d$. $\mathbb{N}^d$ is the set of markings. A run is a non-empty word of markings $m_0 m_1\dots m_n$ s.t. $\forall i \in \...
9
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1answer
1k views

Does every Turing-recognizable undecidable language have a NP-complete subset?

Does every Turing-recognizable undecidable language have a NP-complete subset? The question could be seen as a stronger version of the fact that every infinite Turing-recognizable language has an ...
9
votes
1answer
394 views

Is there an algorithm that finds the forbidden minors?

The Robertson–Seymour theorem says that any minor-closed family $\mathcal G$ of graphs can be characterized by finitely many forbidden minors. Is there an algorithm that for an input $\mathcal G$ ...
9
votes
3answers
849 views

Is it possible to compute whether two functions are extensional equal?

If you have two functions implementing a different sorting algorithm, is it then possible to infer by source code that they both have the same external properties? Meaning that they both will have a ...
9
votes
2answers
437 views

Is meta-undecidability possible?

There are problems that are decidable, there are some that are undecidable, there is semidecidability, etc. In this case I wonder whether a problem can be meta-undecidable. This means (at least in my ...
9
votes
1answer
325 views

Membership problem for certain class of unrestricted grammars

Consider an arbitrary context-free grammar $G$ over the alphabet $\lbrace 0,1,\overline{0} ,\overline{1} \rbrace$. To the productions of this grammar, add two fixed non-context-free productions $P$: $...
9
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0answers
343 views

Deciding if a language induced by a Presburger formula is context-free

Is the following problem decidable? Given $n$ and a Presburger arithmetic formula $\phi(x_1,x_2,\dots,x_n)$, determine whether the language $\{a_1^{i_1} \dots a_n^{i_n}:\phi(i_1,i_2,\dots,i_n)\}$ ...
8
votes
1answer
989 views

Do natural generalizations of P versus NP exist?

Accepted Answer Scott Aaronson's answer has been "accepted" (mainly because it's the only answer!) One-sentence summary of answer  Plausibly natural generalizations of the P versus NP question ...
8
votes
1answer
198 views

Decidability of inductive invariant existence in Presburger arithmetic

Problem: Consider a finite number of control states (including an "initial" and a "bad" state), a finite number of integer variables, and for each ordered pair of states a transition relation ...
8
votes
0answers
332 views

Decidability of existential first-order theory of reals with exponential

The first-order theory over the reals as an ordered field with polynomials is decidable with doubly exponential complexity. However, if we additionally allow the exponential function, that is $e^x$ ...
8
votes
0answers
63 views

Oracle-Decidability of Algebraic Independence

Consider numbers $x_1,...,x_n\in \mathbb{R}$ given by TMs $M_1,...,M_n$ such that $M_i$ approximates $x_i$ to an arbitrary precision (by allowing it to run longer and longer). I am interested in the ...
7
votes
1answer
268 views

Uniform mortality problem for Turing Machines

Consider the following generalisation of the mortality problem for Turing Machines. Given a Turing Machine $M$. Is there a bound $k_M$ such that starting from any configuration $c$ machine $M$ ...
6
votes
3answers
238 views

Solving problems by deciding a logic

I am curious to know when open problems have been solved by expressing them in a specific logic, and then showing that this logic is decidable. I have two distinct cases in mind: The problem is ...
6
votes
1answer
633 views

What is known about $CFL \cap coCFL$?

CFL is the class of context-free languages; co-CFL the languages whose complements are context-free. So CFL $\neq$ co-CFL. Are there any nice characterizations or other basic facts about CFL $\cap$ ...
6
votes
1answer
251 views

Proof of decidability of type checking of calculus of (co)inductive constructions?

I often see it asserted that type checking is decidable for CIC, but I haven't seen it proven. Is there a good paper (or simple demonstration) of this?
6
votes
1answer
1k views

Show that minimal CFG is undecidable via mapping reduction

Actually the problem below is an exercise in Sipser's textbook (Problem 5.36). However, from my perspective, it is not so trivial so that I post this question on this site instead of CS.SE. The ...
6
votes
2answers
254 views

FSM transducer sequential composition decidability

this is a followup/ sequel to this recent question which was answered, this one presumably significantly harder. consider a deterministic FSM transducer $F$ and its mapping $F(x)$ of an input word $x$....
5
votes
1answer
188 views

Post correspondence problem for finite monoids

The Post correspondence problem has the following version for finite monoids: Input: a finite monoid $M$ and a finite list $(m_1,m_1'),\ldots, (m_n,m_n')$ of pairs of elements of $M$ Question: is ...
5
votes
1answer
167 views

What language $L \in NCM$ has $\overline{L} \not \in NCM$?

$NCM$, the class of non-deterministic reversal-bounded counter machines, has a lot of interesting dependability and closure properties. It's known that, unlike the deterministic version, NCM is not ...
5
votes
1answer
102 views

Decidability of rank-k polymorphism vs. System F

There's a paper by Kfoury from 1992, "Type Reconstruction in Finite Rank Fragments of the Second-Order $\lambda$-Calculus", that proves that type inference for Curry-style rank-$k$ polymorphic lambda ...
5
votes
1answer
390 views

Proof that the theory of rationals is convex

In Example 10.12 of the book The calculus of computation by Bradley and Manna, it is said The theory of rationals is convex, as it is convex in a geometric sense. How does the geometric sense of ...
5
votes
1answer
198 views

Is algebraic dependency decidable?

A set of numbers $S=\{x_1,...,x_n\}$ is said to be algebraically dependent if there exists a (multivariate) polynomial $p$ with coefficients in $\mathbb Q$ whose roots contain $x_1,...,x_n$ (or a ...
5
votes
1answer
253 views

Random walk returning probability

Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum_{i=1}^4 p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ ...
5
votes
0answers
91 views

Which computational models support bigotous programs?

A bigotous program is a program which decides if its input is semantically equivalent to itself. Of course, this is impossible in a Turing complete language due to Rice's theorem. In fact, its pretty ...
4
votes
2answers
1k views

Enumerating decidable languages

[The assumption in this question is wrong. It is possible to enumerate exactly the decidable languages with semideciders.] Lets say we have a TM $M_E$ enumerator that writes out codes of TM's on a ...
4
votes
2answers
480 views

About the decidability of sets enumerated in non decreasing order

It is well known that a set of numbers enumerable in nondecreasing order is decidable. However, the typical proof, by cases on the finiteness of the enumerated set, is not constructive. In general, it ...
4
votes
1answer
130 views

Deciding whether a 2nfa halts on every input on every branch

A 2nfa is a nondeterministic finite automaton that can move its head left or right on the input tape, or not. Is the following language known to be decidable? $$ \textit{H}_{\mathsf{2nfa}} = \{ \...
3
votes
2answers
211 views

Dependence of decidability on the encoding of Turing machines

Let $f : \{0, 1\}^* \to \{0, 1\}^*$ be a computable function. Given any encoding $\left<M\right>$ of Turing machines over binary (i.e., a function from the set of Turing machines to the set of ...
3
votes
1answer
295 views

Can the halting problem be solved probabilistically? [closed]

Let $H$ be the halting oracle, meaning that $H$ is a function on pairs of strings such that $H(P,X) = 1$ iff $P$ halts on $X$. A probabilistic program is a program that has (oracle) access to a random ...
3
votes
1answer
223 views

Decidability of first-order theory of real closed fields with functions

By a famous theorem of Tarski, the first-order theory of real closed fields is decidable, as it admits quantifier elimination. Can this result be extended so that propositions can be quantified over ...
3
votes
1answer
124 views

How to solve an unification problem on $\mathbb{N}$?

Usually, the unification problem for two given terms $t$ and $s$ is to find a substitution $\theta$ such that $\theta t = \theta s$, which is equal to finding the certain $\langle x_1 , \cdots , x_n \...
3
votes
0answers
64 views

Deciding whether an arbitrary context-free grammar generates a deterministic push-down automata?

I know that it's undecidable whether an arbitrary context-free grammar is ambiguous, but is it decidable whether that grammar is deterministic? I can't find the answer to this question anywhere on the ...
2
votes
1answer
3k views

Deciding whether a context-free language is regular [closed]

Does anyone know whether the following decision problem is decidable: Given a context-free language $L$, is $L$ regular? Here $L$ can be expressed, e.g., using a context-free grammar. Does anyone ...
2
votes
2answers
178 views

How can you state how abstract interpretation gets around Rice's Theorem succinctly?

At this question, abstract interpretation has a nice in-depth look. However, I'd like someone to clearly and very precisely state how abstract interpretation takes the result of Rice's Theorem over ...
2
votes
1answer
153 views

Petri net termination

Termination is the following problem. Input: a Petri Net with initial marking Output: "yes" iff there exists an infinite firing sequence. The naive algorithm in the case of bounded nets for example ...
2
votes
1answer
112 views

Is it decidable whether the langauge accepted by a reversal-bounded counter machine is deterministic?

I'm wondering if anyone can point me to either an algorithm or an undecidability proof for the following question: Given a non-deterministic reversal-bounded multicounter machine $M$, is there some ...
2
votes
0answers
84 views

Tenth Hilbert problem on interval arithmetic

What is the decidability status of Hilbert's Tenth problem interpreted over interval arithmetic? In details, let $p\in\mathbb{Q}[x_1,\ldots,x_n]$ be a Diophantine polynomial. The problem is that of ...