# Approximations of Isomorphism and Logics with Linear-Algebraic Operators

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, Fürer, 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.

## Authors

• 16 publications
• 11 publications
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• ### Limitations of the Invertible-Map Equivalences

This note draws conclusions that arise by combining two recent papers, b...
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A classical result by Lovász asserts that two graphs G and H are isomorp...
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• ### Nominal C-Unification

Nominal unification is an extension of first-order unification that take...
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• ### On system rollback and totalised fields

In system operations it is commonly assumed that arbitrary changes to a ...
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• ### (Dual) Hoops Have Unique Halving

Continuous logic extends the multi-valued Lukasiewicz logic by adding a ...
03/02/2012 ∙ by Rob Arthan, et al. ∙ 0

• ### Symmetric Circuits for Rank Logic

Fixed-point logic with rank (FPR) is an extension of fixed-point logic w...
04/09/2018 ∙ by Anuj Dawar, et al. ∙ 0