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I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that $x^3+y^...
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Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight? For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...
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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 ...
Bogdan Grechuk's user avatar
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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), ...
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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}])...
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(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 ...
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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 ...
Carl-Fredrik Nyberg Brodda's user avatar
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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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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 ...
Bogdan Grechuk's user avatar
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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 ...
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Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).
Thinniyam Srinivasan Ramanatha's user avatar
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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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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{...
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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) ...
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Let $P_1(x),P_2(x),P_3(x),\dotsc$ be a sequence of polynomials, determined by some initial conditions and a finite-length linear recursion with coefficients being polynomials in $x$ and the index. For ...
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I asked this question on math stack exchange, but it didn't get any responses. So, I am asking it here. Suppose we are working in the signature of a single binary operation $*$. We are given a finite ...
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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 ...
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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 ...
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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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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 ...
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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 $ ...
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Following Poonen [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 ...
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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 ...
Max CYLin's user avatar
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Is the first order theory of finite posets known to be undecidable? Does anyone know a survey about such results?
Thinniyam Srinivasan Ramanatha's user avatar
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We know that Hilbert's 10th problem for $\mathbb{Z}$ is undecidable. I was wondering whether there is a strong opinion in the mathematical community on the decidability of Hilbert's 10th for $\mathbb{...
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My colleague Matthias Baaz is looking for a reference for the following question (or possibly theorem): Let T be the "theory of pairs of finite linear orders". That is, consider all finite ...
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The post correspondence problem, as defined by wikipedia, is undecidable. The problem is defined as follows. Let $A$ be an alphabet with at least two symbols. The input of the problem consists of ...
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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}...
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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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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 ...
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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-...
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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 \...
suitangi's user avatar
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I am reading a paper and there is the following theorem: Let $n$ be a fixed integer, and $n >1$. Denote divisibility in $\mathbb{Z}[\frac{1}{n}]$ by $|_n$, thus for all $x, y \in \mathbb{...
Mary Star's user avatar
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I'm not sure about the relation between indexed languages (generated by indexed grammars--Aho) and scattered context languages (generated by scattered context grammars--J Hopcroft). I think that ...
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Since relational monadic first-order logic has finite model property, its SAT problem is decidable. In H.Behmann's paper, he extended this result to fragment of SOL where all predicates, free and ...
Infinity's user avatar
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I've seen several instances where an undecidability/uncomputability result can be used to produce a lower bound complexity result, and I am interested in when the process can go the other way. In ...
Helen's user avatar
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Sums of primes have been studied by number theorists for many years. Goldbach's conjecture is the most famous unsolved problem in this direction. Here I'd like to consider weighted sums of primes. For ...
Zhi-Wei Sun's user avatar
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Let $G = \langle X: R\rangle$ be a finitely presented group. The following problem seems very natural to me, yet I cannot find any reference for it: Decide if $G$ is abelian or not. With a reduction ...
user540172's user avatar
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If we measure the length of a sentence by the number of occurrences of atomic subformulas in it. So, for example in set theory written in $\sf FOL(\in)$, the length of a sentence is the number of ...
Zuhair Al-Johar's user avatar
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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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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,...
Per Alexandersson's user avatar
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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_{...
user978394's user avatar
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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 $$ \sum_{i} [a_i \sin(\sum l_{i,j}\theta_j)+b_i \...
gondolf's user avatar
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Suppose I have some combination of exponentials, logarithms, and arithmetic operations on rational numbers. For example, $e^{e^{r_1} + \log r_2} - r_3$. Under what conditions does an algorithm exist ...
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Given a $d\times d$ complex matrix subspace $S$, we are asking whether there is some finite integer $n$ such that there exists a non-zero positive semi-definite matrix is orthogonal to $S^{\otimes n}$....
gondolf's user avatar
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I suspect that it should be the same as that of the Turing machine halting problem, which wikipedia gives as GenP and attributes this result to Hamkins and Miasnikov, but I am not sure. Is the generic ...
Jim Graber's user avatar
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Suppose $\mathcal{M}$ is an infinite structure which has the property that every type that is realised is realised uniquely. Also assume that every element of $\mathcal{M}$ is definable but there is ...
Thinniyam Srinivasan Ramanatha's user avatar
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In 1972 C. L. Siegel proved that there is an algorithm to decide for any polynomial $P(x_1,\ldots,x_n)\in\mathbb Z[x_1,\ldots,x_n]$ with $\deg P=2$ whether $$P(x_1,\ldots,x_n)=0$$ for some $x_1,\ldots,...
Zhi-Wei Sun's user avatar
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It is known there is an explicit algebraic variety in $\mathbb Z[x_1,\dots,x_t]$ a bounded $t>2$ whose integral zero-set is non-empty is undecidable. If the variety has genus $0$ is there anything ...
Turbo's user avatar
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I want to know if there exists a polynomial $ P(z, x_1,x_2,\ldots,x_n)$ over the rationals such that the set $$ Z_P = \{z | \exists x_1,\ldots,x_n. P(z, x_1,x_2,\ldots,x_n) = 0 \} \subsetneq \mathbb Q ...
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