Equivalence in Finite-Variable Logics is Complete for Polynomial Time

of first-order logic whose formulas contain at most k variables (for some ). We show that for each , equivalence in the logic is complete for polynomial time. Moreover, we show that the same completeness result holds for the powerful extension of with counting quantifiers (for every ).

The k-dimensional Weisfeiler–Lehman algorithm is a combinatorial approach to graph isomorphism that generalizes the naive color-refinement method (for ). Cai, Fürer and Immerman [6] proved that two finite graphs are equivalent in the logic if, and only if, they can be distinguished by the k-dimensional Weisfeiler-Lehman algorithm. Thus a corollary of our main result is that the question of whether two finite graphs can be distinguished by the k-dimensional Weisfeiler–Lehman algorithm is P-complete for each .

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Received: March 23, 1998

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Grohe, M. Equivalence in Finite-Variable Logics is Complete for Polynomial Time. Combinatorica 19, 507–532 (1999). https://doi.org/10.1007/s004939970004

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  • AMS Subject Classification (1991) Classes:  03C13, 68Q15, 05C60