# Questions tagged [decidability]

The decidability tag has no usage guidance.

100
questions

**8**

votes

**0**answers

119 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**

votes

**0**answers

68 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 $ ...

**7**

votes

**2**answers

252 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**

votes

**1**answer

104 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**

vote

**1**answer

291 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

**1**answer

132 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 $\...

**10**

votes

**1**answer

486 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**

votes

**0**answers

239 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**

votes

**3**answers

375 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**

votes

**3**answers

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**

votes

**0**answers

122 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

**1**answer

241 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 ...

**1**

vote

**0**answers

56 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**

votes

**0**answers

61 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**

votes

**0**answers

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

**1**answer

82 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 ...

**52**

votes

**8**answers

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**

votes

**0**answers

115 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**

votes

**1**answer

385 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

**1**answer

105 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**

votes

**0**answers

125 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**

vote

**0**answers

49 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**

votes

**0**answers

207 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**

votes

**0**answers

158 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

**1**answer

183 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**

votes

**0**answers

87 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**

votes

**3**answers

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

**0**answers

99 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**

votes

**1**answer

92 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

**1**answer

324 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

**2**answers

552 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**

votes

**0**answers

155 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**

votes

**0**answers

78 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 \...

**35**

votes

**3**answers

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

**1**answer

413 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

**1**answer

346 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

**1**answer

112 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

**1**answer

579 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

**0**answers

84 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 $S^{\otimes n}$....

**3**

votes

**1**answer

137 views

### Relationship between computational undecidability and axiomatic undecidability

On surface, these seem two completely different class of problems. One class represent statements which can't be proved or disproved in an axiomatic theory. For example
One can write down a ...

**5**

votes

**0**answers

262 views

### Is the two variable fragment of arithmetic, i.e., theory of ($\mathbb{N}, + ,\times$), decidable?

Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).

**6**

votes

**1**answer

238 views

### Deciding isomorphism between graphs which interpret in the pure set

I am interested in the following decision problem:
Given descriptions of two graphs $G,H$ which interpret in the pure set $\mathbb N=(\{0,1,2,\ldots\},=)$, decide whether $G$ and $H$ are isomorphic....

**13**

votes

**1**answer

446 views

### Is this theory decidable?

It is well-known that both Presburger arithmetic (by contrast with Peano arithmetic) and Tarski geometry are decidable. I was in the shower this morning and wondered whether there exists an elegant ...

**0**

votes

**0**answers

179 views

### Is the positive existential theory undecidable?

Could you tell if the positive existential theory of $\mathbb{C}[e^{\mu x} \mid \mu \in \mathbb{C}]$ is undecidable in the language $\{+, \cdot , \frac{d}{dx} , 0, 1, e^x\}$ ?
How can we prove the (...

**2**

votes

**1**answer

367 views

### Complexity of Deciding Feasibility of a system of linear inequalities over restricted variables

I am working out an interesting problem and would like some help with this particular sub problem:
Suppose we have a matrix $ M =\left\lbrace a_{ij}\right\rbrace $ of size $n\times m$ where $ a_{ij}\...

**6**

votes

**1**answer

422 views

### Show that the positive existential theory is undecidable

To show that the positive existential theory of $\mathbb{C}[t, e^{\lambda t} \mid \lambda \in \mathbb{C}]$ in the language $\{+, \cdot , ' , 0 , 1, t\}$ is undecidable we have to prove the following: $...

**20**

votes

**3**answers

1k views

### Is being rational decidable?

Given a real number uniquely defined by a finite system of equations and inequalities with rational coefficients involving the standard elementary functions only. Is it decidable whether this number ...

**8**

votes

**1**answer

458 views

### List of finitely presented groups with undecidable word problem

Is there any reasonably updated list of (representative) examples of finitely presented groups with undecidable word problem?
By "representative" I mean "avoiding obvious redundancy", i.e. examples ...

**7**

votes

**0**answers

445 views

### Proving Richardson's theorem for constants

(I asked this a little over 3 months ago on math.SE, and when I initially re-asked here, no one had responded there. $\:$ After I re-asked here, Eric Towers responded there, since I had forgotten to ...

**2**

votes

**1**answer

273 views

### Recurrence relations with polynomial coefficients: an undecidable problem

I read once (somewhere) that solving a recurrence relation with polynomial coefficients, in the general case, is an undecidable problem. I can't remember the exact reference and I've been trying to ...