
Approximations of Isomorphism and Logics with LinearAlgebraic Operators
Invertible map equivalences are approximations of graph isomorphism that...
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Separating Rank Logic from Polynomial Time
In the search for a logic capturing polynomial time the most promising c...
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Deep Weisfeiler Leman
We introduce the framework of Deep Weisfeiler Leman algorithms (DeepWL),...
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Insignificant Choice Polynomial Time
In the late 1980s Gurevich conjectured that there is no logic capturing ...
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Capturing Polynomial Time using Modular Decomposition
The question of whether there is a logic that captures polynomial time i...
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Choiceless Polynomial Time, Symmetric Circuits and CaiFürerImmerman Graphs
Choiceless Polynomial Time (CPT) is currently the only candidate logic f...
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Nominal CUnification
Nominal unification is an extension of firstorder unification that take...
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Limitations of the InvertibleMap Equivalences
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 invertiblemap equivalences ≡^IM_k,Q on certain variants of CaiFürerImmerman (CFI) structures. These ≡^IM_k,Qequivalences, for a natural number k and a set of primes Q, refine the wellknown WeisfeilerLeman 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 ktuples defined via linear operators on vector spaces over fields of characteristic p ∈ Q. In the first paper it has been shown that for a prime q ∉ Q, the ≡^IM_k,Q equivalences are not strong enough to distinguish between nonisomorphic CFIstructures over the field 𝔽_q. In the second paper, a similar but not identical construction for CFIstructures over the rings ℤ_2^i has 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 CFIstructures over the rings ℤ_2^i are indistinguishable with respect to ≡^IM_k,ℙ, for the set ℙ of all primes. This implies the following two results. 1. There is no fixed k such that the invertiblemap equivalence ≡^IM_k,ℙ coincides with isomorphism on all finite graphs. 2. No extension of fixedpoint logic by linearalgebraic operators over fields can capture polynomial time.
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