Questions tagged [decidability]

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12
votes
1answer
271 views

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 ...
4
votes
1answer
170 views

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 ...
3
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0answers
96 views

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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1answer
520 views

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 ...
6
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0answers
66 views

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
votes
1answer
502 views

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 ...
5
votes
2answers
248 views

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 $\...
0
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0answers
109 views

Genus $0$ algebraic curves integral points decidable?

It is known there is an explicit algebraic variety in $\mathbb Z[x_1,\dots,x_t]$ a bounded $t>2$ whose integral zero-set is non-empty is undecidable. If the variety has genus $0$ is there anything ...
4
votes
1answer
86 views

Monadic second order theory of “backwards tree” — is it decidable?

A famous result by Rabin states that the monadic second order theory of the binary tree is decidable. By the binary tree, understand the free monoid $\{0,1\}^*$ of words, with operations $S_0(w)=w0$ ...
11
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2answers
420 views

Decidability of a first-order theory of hyperreals

The theory of real closed fields is decidable. The hyperreals satisfy that theory, so we can interpret statements in the theory of real closed fields as being about hyperreals. If we add a unary ...
0
votes
1answer
104 views

Linear programming with exponential inequalities and rational variables

If we are given a set of real linear inequalities then using elimination theory or just linear programming we can decide. If the program also has inequalities of form $2^x\leq g$ in addition to linear ...
8
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0answers
134 views

Membership problem in general linear group

This is surely a very well known problem, but I could not find an answer on MO or on Google, so here I am. Given some finitely generated free subgroup $H$ of $\operatorname{GL}_n(\mathbb{Z}[t,t^{-1}])...
3
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0answers
88 views

Post correspondence problem: Busy beaver variant

The Post correspondence problem (Wikipedia link) is to decide for $k$ pairs of strings $$(a_1,b_1), (a_2, b_2), ..., (a_k,b_k),$$ if there exists a finite sequence of numbers $c_j, 0\le j\le j_\max $ ...
9
votes
2answers
316 views

Reference request: Recent progress on the conjugacy problem for torsion-free one-relator groups?

I am aware that the Spelling Theorem of B. B. Newman implies that one-relator groups with torsion are hyperbolic, and thus have a solvable conjugacy problem. My understanding is that for one-relator ...
-2
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1answer
107 views

Undecidable definition of mathematical expressions?

I am arguing a bit on Facebook regarding the definition of a mathematical expression. Some argue that equations are not expressions (and there are a few possibly dubious online sources which states ...
1
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1answer
309 views

Which Hilbert's 10th polynomials are known to have solutions?

The Diophantine equation $$x^3 + y^3 + z^3 = 42$$ was recently solved by Booker and Sutherland: Sum of three cubes for 42 finally solved. Is there a clean partition of the form of those polynomial ...
6
votes
1answer
163 views

Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?

It is a well-known fact that the Henselization of the function field $\mathbb{F}_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}_{p}(t)^{alg} \cap \mathbb{F}_{p}((t))$, so of course $\...
12
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1answer
631 views

Is there a ring for which the reducibility of a polynomial is undecidable?

Let $R$ be a ring such that all of its elements have a finite number of divisors, ie $\forall r\in R\, |\{x\in R: x|r\}|<\infty$. Then we can decide whether a polynomial in $R[t]$ is reducible ...
3
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0answers
250 views

Are there any references in the literature relating to work on finding a Diophantine equation representing abc

The Davis-Putnam-Robinson-Matiyasevich theorem is: Diophantine is equivalent to listable This result has some known applications: (1) Prime-producing polynomials. (2) Diophantine statement of the ...
6
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3answers
439 views

How much of the Cantor-Schröder-Bernstein theorem is constructively recoverable if the injections have retractions and decidable images?

This is cross-posted from MSE at the suggestion of a comment after receiving no answers over a few weeks. Suppose we have $f : A \to B$ and $g : B \to A$, as well as left inverses $f_r : B \to A$ of $...
16
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3answers
1k views

Is Multilinear Hilbert's tenth problem version undecidable?

A multilinear polynomial $f\in\mathbb Z[x_1,\dots,x_t]$ has terms only of form $$b\prod_{i=1}^tx_i^{a_i}$$ where $a_i\in\{0,1\}$ and $b\in\mathbb Z$. Is there no general purpose algorithm for ...
5
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0answers
127 views

Example of applying real quantifier elimination algorithm for polynomials

Sorry if any of this is unclear, or doesn't make much sense, I'm still trying to figure it out, a practical example such as this would likely help me understand better than anything else. I have read ...
8
votes
1answer
266 views

Is equality of formulas with floor rounding or integer division decidable?

As far as I know, formulae involving rationals and basic arithmetic ($+$, $-$, $\cdot$ and $/$) have decidable equality. Is this still the case if we add floor rounding (or integer division)? Define ...
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0answers
66 views

Decidability of linear equation about Sine and Cosine

Given integers $n,d$, and rational numbers $a_i,b_i,l_{i,j},s_{i,j}$ for $1\leq i\leq d$, $1\leq j\leq n$, we are considering the following equation $$ \sum_{i} [a_i \sin(\sum l_{i,j}\theta_j)+b_i \...
0
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0answers
70 views

Multivariate polynomial with infinite but discrete roots on one variable

I want to know if there exists a polynomial $ P(z, x_1,x_2,\ldots,x_n)$ over the rationals such that the set $$ Z_P = \{z | \exists x_1,\ldots,x_n. P(z, x_1,x_2,\ldots,x_n) = 0 \} \subsetneq \mathbb Q ...
2
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0answers
131 views

Compare my software's representation of exponential numbers and 0?

Suppose I have a real number $$ x=\sum_{i=1}^n a_i e^{\lambda_i} $$ where $a_i,\lambda_i$s are complex algebraic numbers. Is there an algorithm to determine whether it is greater than 0 or less than ...
1
vote
1answer
122 views

decidability of regularity of a language depending on representation

It is well known that many decision problems for regular languages are decidable. However, the proofs seem to rely on a witness of the regularity of said language, be it an automaton, a grammar, a ...
53
votes
8answers
5k views

Two (probably) equal real numbers which are not proved to be equal?

Can someone give me a nice example of two computable real numbers which are believed but not proved to be equal? I never really understood the assertion that "the reals do not have decidable equality"...
2
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0answers
123 views

Can we “invert” Diophantine equations such as $x^3+y^3+z^3=k$ in to halting probabilities for some universal Turing machine?

Following Pooten [1], Davis[2], Chaitin [3], and Ord and Kieu [4]: Is it possible that there is a polynomial $P$ of degree $d\le 4$, along with a prefix-free universal Turing machine $T$, such that ...
8
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1answer
437 views

Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
1
vote
1answer
122 views

Decidability of S2S with real numbers

Is the theory of natural numbers and functions $ℕ → ℝ$ decidable under: - for natural numbers: $\mathrm{succ1}(n) = 2n+1$; $\mathrm{succ2}(n) = 2n+2$; equality - for functions: pointwise addition and ...
4
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0answers
133 views

Subgroup membership problem for Noetherian groups

I am interested in the status of the subgroup membership problem (MP) for finitely presented Noetherian groups. That is, given a finite presentation $\langle X,R\rangle$ for a Noetherian group, \begin{...
1
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0answers
50 views

Equality of combinations of exponentials and logarithms

Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...
2
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0answers
227 views

minimum size of undecidable quadratic diophantine problems

According to Matiyasevich, the existence of integer solutions of systems of polynomial equations with integer coefficients is undecidable. By introducing additional variables substituting factors of ...
2
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0answers
172 views

The elementary theory of finite commutative rings

I have wondered the decidability of elementary theory of finite commutative rings. Since we know that the elementary theory of finite fields is decidable shown by J.Ax (The Elementary Theory of Finite ...
1
vote
1answer
206 views

Is Calculus of Constructions type inhabitance semi-decideable?

I'm wondering if type inhabitance for the calculus of constructions is semi-decideable. I know the following: System F inhabitance and, correspondingly, second-order unification are semi-decideable ...
2
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0answers
88 views

Is the equational theory of the variety of ternary self-distributive algebras decidable?

A ternary self-distributive algebra is an algebra $(X,t)$ that satisfies the identity $$t(u,v,t(x,y,z))=t(t(u,v,x),t(u,v,y),t(u,v,z)).$$ Is the equational theory of the variety of ternary self-...
11
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3answers
1k views

Undecidable easy arithmetical statement

Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some ...
4
votes
0answers
102 views

Deciding equality in free models of a (generalized) Lawvere theory

Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be functor of Lawvere theories $\mathcal{C}, \mathcal{D}$ (i.e. cartesian categories where every object is isomorphic to some power of a chosen object) ...
6
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1answer
101 views

Decidability and Cluster algebras

Recall the definition of a cluster algebra, which can be seen as a (possibly infinite) graph, where each vertex is a tuple of a quiver and Laurent expressions at some of the vertices of the quiver. ...
5
votes
1answer
344 views

Undecidability of Diophantine equations with disjoint variables?

Consider a special case of the Hilbert's 10th problem: $f(\vec{x})=g(\vec{y})$, where $\vec{x}$ and $\vec{y}$ are disjoint ( i.e, the LHS and RHS do not have any common variables), moreover, $f$ and $...
6
votes
2answers
683 views

Are omega-consistent extensions of PA always consistent with each other?

The question is as in the title. In the edit history you can find my attempt to formalise the question, but that was a failure, for reasons stated clearly in the comments. Thus, my question is just: ...
5
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0answers
168 views

Rabin's proofs of emptiness and complementation problems for automata on infinite trees

I have originally asked this question on Math.SE, but I think it is more suitable here. I have been reading M. Rabin's 1969 article Decidability of Second-Order Theories and Automata on Infinite ...
2
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0answers
79 views

Matrix (geometric sum) orbit problem

Is the following algorithmic problem known to be decidable/undecidable? Input: an element $\mathbf{v} \in \mathbb{Z}^n$, a matrix $\mathbf{A} \in GL_n(\mathbb{Z})$, and a subgroup $H \leqslant \...
36
votes
3answers
2k views

Is it decidable to check if an element has finite order or not?

Suppose we have a finitely presented group $G$ with decidable word problem. Is it decidable to check whether a given element $x\in G$ has finite order or infinite?
10
votes
1answer
442 views

Are the terms of a linear recurrence integral?

Given rational numbers $a_1,\ldots, a_k$ and $u_0, \ldots, u_k$, let $(u_n)_{n \geq k}$ be the linear recurrence defined by $$u_n := a_1 u_{n-1} + \cdots + a_k u_{n-k}, \text{ for } n \geq k .$$ ...
8
votes
1answer
356 views

Can we algorithmically contract loops in a simply connected space?

It is well-known that the question whether a given connected simplicial complex (or simplicial set) is simply connected, is algorithmically undecidable as it can model the word problem. Assuming ...
2
votes
1answer
120 views

Understanding the paper: “Guarded Fixed Point Logic”

This question is specifically about the paper "Guarded Fixed Point Logic" by Gradel and Walukiewicz. Among other things they prove the decidability of the satisfiability problem for Fixpoint Loosely ...
3
votes
1answer
703 views

Provably undecidable problems within prime numbers context

A colleague of mine was stating there are no known undecidable statements that have explicit connection with prime numbers. What does this mean? I understand that it is unknown whether Goldbach ...
1
vote
0answers
86 views

determine the existence of positive semi-definite matrix

Given a $d\times d$ complex matrix subspace $S$, we are asking whether there is some finite integer $n$ such that there exists a non-zero positive semi-definite matrix is orthogonal to $S^{\otimes n}$....