Definability of linear equation systems over groups and rings

TL;DR

This paper investigates the logical definability of linear equation systems over finite groups and rings, revealing that certain solvability problems are decidable in polynomial time but not expressible in fixed-point logic with counting, and establishing reductions and closure properties.

Contribution

It introduces a logical framework for analyzing solvability of linear systems over rings, showing reductions among problems and extending definability results to commutative rings.

Findings

All considered problems are in P but not definable in fixed-point logic with counting.

Solvability problems over rings can be reduced to those over commutative rings.

Closure properties for queries reducible to ring solvability are established.

Abstract

Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings, generalising known results on the logical definability of linear-algebraic problems over finite fields.