Approximations of Isomorphism and Logics with Linear-Algebraic Operators

TL;DR

This paper explores the limitations of linear-algebraic operators in graph isomorphism and fixed-point logic, showing that certain polynomial-time properties are not definable without operators for all prime characteristics.

Contribution

It introduces a new algebraic analysis of invertible map equivalences and demonstrates their limitations in capturing PTIME in fixed-point logic extensions.

Findings

k-Q-IM-equivalence does not coincide with isomorphism unless Q includes all primes

Certain polynomial-time graph properties are not definable in the considered logic

The analysis employs advanced algebraic tools like Maschke's Theorem

Abstract

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parametrised by a number k and a set Q of primes. The intuition is that two graphs G and H which are equivalent with respect to k-Q-IM-equivalence cannot be distinguished by a refinement of k-tuples given by linear operators acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that the k-Q-IM-equivalence is the natural notion of elementary equivalence for this logic. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, F\"urer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that k-Q-IM-equivalence is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in the infinitary logic with all Q-linear-algebraic operators and finitely many variables, which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.