Limitations of the invertible-map equivalences

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