Limitations of the Invertible-Map Equivalences

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Gr\"adel, 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 $\equiv^\text{IM}_{k,Q}$ on certain variants of Cai-F\"urer-Immerman (CFI) structures. These $\equiv^\text{IM}_{k,Q}$-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 $G \equiv^\text{IM}_{k,Q} H$ cannot be distinguished by iterative refinements of equivalences on $k$-tuples defined via linear operators on vector spaces over fields of characteristic $p \in Q$. In the first paper it has been shown that for a prime $q \notin Q$, the $\equiv^\text{IM}_{k,Q}$ equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field $\mathbb{F}_q$. In the second paper, a similar but not identical construction for CFI-structures over the rings $\mathbb{Z}_{2^i}$ has been shown to be indistinguishable with respect to $\equiv^\text{IM}_{k,\{2\}}$. 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 $\mathbb{Z}_{2^i}$ are indistinguishable with respect to $\equiv^\text{IM}_{k,\mathbb{P}}$, for the set $\mathbb{P}$ of all primes. This implies the following two results. 1. There is no fixed $k$ such that the invertible-map equivalence $\equiv^\text{IM}_{k,\mathbb{P}}$ coincides with isomorphism on all finite graphs. 2. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.