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Linear equations and recursively enumerable sets

Juha Honkala

TL;DR

The work investigates how linear equations in semigroups can encode recursively enumerable sets, presenting two variants that replace Matiyasevich's multivariate Diophantine framework with single-variable linear equations over a morphism monoid and over a matrix monoid. It introduces M-triples to compute polynomials and constructs two upper triangular morphisms $g_1,g_2$ to realize a polynomial equality as a linear equation in a generated monoid $\mathcal{H}$, establishing equivalences to membership in recursively enumerable sets. By mapping morphisms to matrices via $\Psi$ and studying the matrix monoid $\mathcal{M}$, it transfers the undecidability results to linear equations over matrix semigroups as well. The results demonstrate that undecidability of RE-set membership transfers to the solvability of simple linear equations in these algebraic structures, revealing fundamental algorithmic limits in semigroup and matrix settings.

Abstract

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers established by Matiyasevich. These variants use linear equations with one unkwown instead of polynomial equations with several unknowns. As a corollary we get undecidability results for linear equations over morphism semigoups and over matrix semigroups.

Linear equations and recursively enumerable sets

TL;DR

The work investigates how linear equations in semigroups can encode recursively enumerable sets, presenting two variants that replace Matiyasevich's multivariate Diophantine framework with single-variable linear equations over a morphism monoid and over a matrix monoid. It introduces M-triples to compute polynomials and constructs two upper triangular morphisms to realize a polynomial equality as a linear equation in a generated monoid , establishing equivalences to membership in recursively enumerable sets. By mapping morphisms to matrices via and studying the matrix monoid , it transfers the undecidability results to linear equations over matrix semigroups as well. The results demonstrate that undecidability of RE-set membership transfers to the solvability of simple linear equations in these algebraic structures, revealing fundamental algorithmic limits in semigroup and matrix settings.

Abstract

We study connections between linear equations over various semigroups and recursively enumerable sets of positive integers. We give variants of the universal Diophantine representation of recursively enumerable sets of positive integers established by Matiyasevich. These variants use linear equations with one unkwown instead of polynomial equations with several unknowns. As a corollary we get undecidability results for linear equations over morphism semigoups and over matrix semigroups.
Paper Structure (5 sections, 16 theorems, 90 equations)

This paper contains 5 sections, 16 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. M-triples and how to use them to compute values of polynomials
  4. Linear equations over a morphism monoid and recursively enumerable sets
  5. Linear equations over a matrix monoid and recursively enumerable sets

Key Result

Proposition 1

Let $t$ be a positive integer. Let $F=(A,f_1,f_2)$ and $G=(B,g_1,g_2)$ be M-triples of dimension $t$. Assume that $A\cap B=\emptyset$. Let $p: {\mathbb Z}_+^t\to {\mathbb Z}_+$ and $q:{\mathbb Z}_+^t\to {\mathbb Z}_+$ be mappings such that $p$ is computable by $F$ and $q$ is computable by $G$. Assum

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Proposition 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 8 more