# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1902.06648