# Questions tagged [decidability]

The decidability tag has no usage guidance.

155
questions

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### Can one compute the automorphism group of a curve of genus >1?

Given a sufficiently nice perfect field $k$ and a smooth projective curve $C$ of genus $g_C>1$ over $k$, can one compute the automorphism group ${\rm Aut}(C)$? It is known that ${\rm Aut}(C)$ is ...

14
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3
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### Guaranteed correct digits of elementary expressions

Let $E$ be the set of elementary expressions, defined as the smallest set of mathematical expressions such that $1 \in E$, and if $a,b \in E$ then $a + b$, $a - b$, $ab$, $a/b$, $\exp(a)$, $\log(a)$ ...

8
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1
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### Why don't Zeilberger and Gosper's algorithms contradict Richardson's theorem?

Richardson's theorem proves that whether an expression A is equal to zero is undecidable. A is in this case an expression, constructed from $x,e^x,\sin(x)$ and the constant function $\pi$ and $\ln(2)$ ...

5
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2
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### MIP*=RE theorem and its impact on logic and proof theory

In the monumental paper MIP*=RE five authors, Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, managed to show that two complexity classes: RE and MIP* do in fact coincide. ...

53
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7
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### Are there any undecidability results that are not known to have a diagonal argument proof?

Is there a problem which is known to be undecidable (in the algorithmic sense), but for which the only known proofs of undecidability do not use some form of the Cantor diagonal argument in any ...

3
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0
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### Maximal number of aperiodic Wang tiles

I was wondering whether there is an analogue result to the minimality of Wang tiling, in the direction of maximality.
I think that the paper by Jeandel and Rao, shows that the minimal number of Wang ...

3
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1
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779
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### Language equivalence between deterministic and non-deterministic counter net

One-Counter Nets (OCNs) are finite-state machines equipped with an integer counter that
cannot decrease below zero and cannot be explicitly tested for zero.
An OCN $A$ over alphabet $\sum$ accepts a ...

18
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1
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725
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### Is solvability semi-decidable?

Let $G = \langle A \mid R \rangle$ be a finitely presented group, given by a finite presentation. If $G$ is abelian, then we can verify this fact: simply verify the fact that $[a, b] = 1$ for all ...

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### Counter net decidability [closed]

Let one Deterministic Counter Net ($\mathrm{1DCN}$), which is a finite-state automata where every state is complete means all states has transition of all input symbols and their respective weight ...

7
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239
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### Decidability of completing Penrose tilings

Is the following problem known to be un/decidable? Problem: Given a finite configuration of Penrose tiles in the plane, determine if there is an extension of the configuration tiling the whole plane.

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### How can Kőnig's Lemma be expressed in Monadic Second-Order Logic of 2 Successors?

I read the following on Wikipedia's page on Monadic Second-Order Logic of Two Successors (MS2S):
Weak S2S (WS2S) requires all sets to be finite (note that finiteness
is expressible in S2S using Kőnig'...

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1
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### Are 100% of statements undecidable, in Gödel's numbering? [duplicate]

Gödel's incompleteness theorem shows that there are undecidable statements, i.e., formal logical claims which neither have proofs nor disproofs.
In doing so, Gödel famously enumerated all well-formed ...

5
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1
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301
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### Parity of number of solutions to Diophantine equations

By $MRDP$ resolution of Hilbert's tenth, we infer, counting number of solutions to Diophantine equations is undecidable.
Is parity of number of solutions to Diophantine equations undecidable?

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2
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### A variation of domino tiling problem with fusions

I know several specific variations of the domino tiling problem has been determined to be decidable or undecidable, such as the seed domino problem. I have a variation which I have not been able to ...

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### What theories are larger than the real closed field but still decidable?

It's well known that sentences about the real closed field can be decided by algorithm and the complexity of this is about $d^{2^{O(n)}}$ where $d$ is the product of the degrees of polynomials in the ...

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### Computability of fillability of unit cube in $\mathbb{R}^n$ by $k$ $\varepsilon$-balls

Let $\mathbb{N}$ denote the set of positive integers. We define a relation $R \subseteq \mathbb{N}^4$ in the following way:
$(p,q,n,s)\in R$ if and only if there is $S\subseteq [0,1]^n$ with $|S| = s$...

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### Is parquetability decidable?

Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is parquettable by $T$ if there is a partition $\frak P$ of $\mathbb{Z}...

2
votes

1
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### Axiomatization of S2S

What is a reasonable axiomatization of S2S?
S2S is the monadic second order theory with two successors (Wikipedia link). It has finite binary strings, operations $s→s0$ and $s→s1$ on strings, and ...

3
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0
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### Decidability of theory of modules over a ring of finite representation type

I have read from Mike Prest's model theory for modules (London lecture note series) chapter 17 that a Ring of finite representation type has a decidable theory of modules. Here decidability was ...

7
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### What is known about first order logic of $\mathbb{N}$ with + and a unary predicate?

In "Weak Second-Order Arithmetic and Finite Automata", Büchi claims that the first order theory of $\mathbb{N}$ with + and a predicate for recognizing powers of 2 ($Pw_2$) is expressively ...

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### Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable?

According to this source (p. 10), determining whether a simplicial complex is a simplicial sphere (the sphere recognition problem) is undecidable.
According to this source, determining whether a ...

1
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1
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182
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### Possible weaker version of the Domino/Wang tiling problem

This may be a dumb question, but I was wondering whether the question of 'periodically tiling the plane from a finite set of tiles' is the same as the domino tiling problem or a weaker version. I ...

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2
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### Congruences of binomial sums

Let $a_n$ is a binomial sum, for example
$$
a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...

8
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1
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350
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### Decidable theories with arbitrary complexity

Are there complete finitely axiomatizable first order theories (with equality) with arbitrarily high computational complexity?
Here, arbitrarily high (computational) complexity means that for every ...

6
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1
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544
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### How constructive is Matiyasevich's theorem?

A famous corollary of Matiyasevich's theorem is that there exists a Diophantine equation such that it is undecidable (under some recursively axiomatizable theory $T$, such as ZFC) whether that ...

2
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0
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### Decidability of the solvability of quadratic systems

Let a finite collection of (complex) unknowns $\{x_1,\ldots,x_n\}$ be given, as well as an affine system $AX=B$ in the quadratic variables $X:=[x_i x_j : i\leq j]$, with entries in a computable ...

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0
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### Decidability of a polynomial-exponential equation in two variables

My question is with regards to the following (algorithmic) problem:
Problem. Given $f\in \mathbb{Z}[x,y], a,b\in \mathbb{Q}, r\in \mathbb{Z}$, do there exist positive integers $m,n$ such that $f(m,n) =...

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2
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### Tarski's original proof of quantifier elimination in algebraically closed fields

I am currently helping teach a course about foundations of mathematics, which has thus far focused mostly on propositional and first-order logic. As part of the course, the students are each required ...

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271
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### Decidability of completeness in propositional logic

Propositional logic can be presented as in Mendelson’s book, with the sole inference rule of modus ponens, and with the following three axioms:
$$B \Rightarrow (C \Rightarrow B)$$
$$(B \Rightarrow (C \...

5
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1
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### Given a quasi-convex subgroup $H$ of hyperbolic $G$, can we decide if two elements $x,y \in G$ lie in the same double coset of $H$?

I've come across the following question in my research, which seems elusive but is almost surely decidable.
Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish ...

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0
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### Game with Turing machines

Introduction
The following game is quite nice: Alice has, in secret, constructed a polynomial $P \in \mathbb{Z}[x]$.
On day $n=1,2,3,...$, she secretly writes down $P(n)$ on a piece of paper.
Each day,...

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2
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### Is irreducibility of polynomials $\in \mathbb{Z} [X]$ over $\mathbb{Q}$ an undecidable problem?

There are a number of criteria for determining whether a polynomial $\in \mathbb{Z} [X]$ is irreducible over $\mathbb{Q}$ (the traditional ones being Eisenstein criterion and irreducibility over a ...

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### Why is the proof of decidability of arithmetic (Theorem 2.1) in Hamkins & Lewis (2000) enough?

Recently, I was reading the paper "Infinite Time Turing Machines" by Hamkins & Lewis. And one of the first theorems (Theorem 2.1) is about decidability of arithmetic.
The proof is quite ...

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273
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### Quantifier elimination in $S^1$

Does quantifier elimination (by cylindrical decomposition) work for systems of polynomial equations and inequalities where some or all of the variables are complex numbers of unit modulus, rather than ...

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0
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### Non-degenerate solutions in multiplicative subgroups of $\mathbb{Q}$ of finite ranks

I am trying to study whether of a set of first order sentences is decidable, and the key is to figure out an algorithm to compute all non-degenerated solutions. The setting is as follow. Let $q = (q_{...

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### Hilbert 10th problem for genus 2 equations

Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...

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### Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!

Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...

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### Undecidable infinite analogs of NP-complete problems?

In the paper Some undecidable problems involving edge-coloring of graphs, Burr proves that a certain k-coloring problems for certain infinite graphs (however, with finite descriptions - here "...

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0
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203
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### Integer points of rational function in 2 variables

Is there an algorithm that, given polynomials $P(x)$ and $Q(y)$ with integer coefficients, decides whether there exists integers $x$ and $y$ such that $\frac{P(x)}{Q(y)}$ is an integer?
This is a ...

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1
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### Decidability theory involving real parameters

In order to formally ask if a problem is decidable, one first needs to show how to encode each instance of said problem as a finite string of bits (or symbols over some other finite alphabet). For ...

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### Uniform word problem in finitely presented simple groups

The following question arose in the comments on this question, and it seems like a reasonable question to ask in its own right. I've added some additional details.
The word problem in any fixed ...

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1
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### Is "almost-solvability" of Diophantine equations decidable?

Say that a Diophantine equation is almost-satisfiable iff for each $n\in\mathbb{N}$ it has a solution mod $n$. Trivially genuine satisfiability over $\mathbb{N}$ implies almost-satisfiability, but the ...

6
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1
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### Given some recursive function, can we effectively associate it a polynomial as in the DPRM theorem?

I'm interested in the following assertion about the Davis-Putnam-Robinson-Matijasevich theorem
Given a recursive function $f:\mathbb{N}\rightarrow\mathbb{N}$, i.e. its index, we can effectively get ...

12
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1
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382
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### Commutator problem vs conjugacy/word problem

For a finitely presented group $G$, generated by a finite set $A$, the commutator problem is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a ...

7
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0
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271
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### Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?

Given a Diophantine equation it is not decidable if it has integer solution.
I. Is there a Diophantine set $\mathcal D_{unique}$ satisfying the properties
every member in $\mathcal D_{unique}$ is a ...

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### Variation in decidability of diophantine equations with field extension

Let $O_k$ be the ring of integers in a subfield $k$ of $\overline{\mathbb{Q}}$. Let's call an equation $p(x_1, \dots, x_n) = 0$ where $p$ is a polynomial in $n$-variables $x_1, \dots, x_n$ with ...

12
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1
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853
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### Testing whether $e^x+ax^2+bx+c$ has a zero

What is the simple test with exponential polynomials to determine whether
$$f(x)=e^x+ax^2+bx+c$$ has a positive zero?
This was prompted by the question about discriminants here. We have an ineffective ...

8
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0
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### The conjugacy problem for two-relator groups

Is the conjugacy problem for two-relator groups known to be undecidable?
The word problem for two-relator groups is a famous open problem (appearing e.g. as Question 9.29 in the Kourovka notebook), ...

6
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1
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### Do we have an algorithm for comparing $e^e$ with rationals?

Do we have an algorithm for comparing $e^e$ with rationals, with a known time to convergence?
In a non-constructive sense, there obviously is an algorithm.
If $e^e$ is some rational $q_0$, then we ...

4
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2
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274
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### Checking for finite fibers in hash functions

Let $\{0,1\}^{<\omega}$ denote the collection of finite binary sequences. By a hash function we mean a computable map $$h: \{0,1\}^{<\omega} \to \{0,1\}^n$$ for some fixed $n\in\omega$. Define $\...