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Equivalence in finite-variable logics is complete for polynomial time

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1 Author(s)
Grohe, M. ; Inst. fur Math. Logik, Albert-Ludwigs-Univ., Freiburg, Germany

How difficult is it to decide whether two finite structures can be distinguished in a given logic? For first order logic, this question is equivalent to the graph isomorphism problem with its well-known complexity theoretic difficulties. Somewhat surprisingly, the situation is much clearer when considering the fragments Lk of first-order logic whose formulae contain at most k (free or bound) variables (for some k⩾1). We show that for each k⩾2, equivalence in the k-variable logic Lk is complete for polynomial time under quantifier-free reductions (a weak form of NC0 reductions). Moreover, we show that the same completeness result holds for the powerful extension Ck of Lk with counting quantifiers (for every k⩾2)

Published in:

Foundations of Computer Science, 1996. Proceedings., 37th Annual Symposium on

Date of Conference:

14-16 Oct 1996