Limitations of the Invertible-Map Equivalences

Abstract

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel, and Wied Pakusa, published at ICALP 2019 and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences ≡ IM k,Q on certain variants of Cai-Fürer-Immerman structures (CFI-structures for short). These ≡ IM k,Q -equivalences, for a natural number k and a set of primes Q, refine the well-known Weisfeiler-Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs G ≡ IM k,Q H cannot be distinguished by iterative refinements of equivalences on k-tuples defined via linear operators on vector spaces over fields of characteristic p ∈ Q.In the first paper it has been shown, using considerable algebraic machinery, that for a prime q / ∈ Q, the ≡ IM k,Q equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field Fq. In the second paper, a similar but not identical construction for CFI-structures over the rings Z 2 i has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to ≡ IM k,{2} . Together with earlier work on rank logic, this second result suffices to separate rank logic from polynomial time.We show here that the two approaches can be unified to prove that CFI-structures over the rings Z 2 i are in fact indistinguishable with respect to ≡ IM k,P , for the set P of all primes. In particular, this implies the following two results.There is no fixed k such that the invertible-map equivalence ≡ IM k,P coincides with isomorphism on all finite graphs. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.

Notes on presheaf representations of strategies and cohomological refinements of $k$-consistency and $k$-equivalence

“…. Moreover, Proposition 13.2 is leveraged to show that ≡ Z k is discriminating enough to defeat two important families of counter-examples: the CFI construction used in [7] to show that L k (#) is not strong enough to characterise polynomial time, and the constructions in [18,10] which are used to show similar results for linear algebraic extensions of L k (#).…”