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Journal of Logic and Computation, Volume 8, Issue 1: February 1998.
Enhancing fixed point logic with cardinality quantifiers
L Hella1 and H Imhof2
1Department of Mathematics, PO. Box 4 (Yliopistonkatu 5), 00014 University of Helsinki, Finland. E-mail: firstname.lastname@example.org, 2, Department of Computer Science, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK. Current address: atraxis/SAirGroup, CGAE, CH-8058 Zurich Airport, Switzerland
Let QIFP be any quantifier such that FO (QIFP), first-order logic enhanced with QIFP and its vectorizations, equals inductive fixed point logic, IFP in expressive power. It is known that for certain quantifiers the equivalence FO (QIFP)[equiv] IFP is no longer true if Q is added on both sides. Rather, we have FO (QIFP, Q) < IFP(Q) in such cases. We extend these results to a great variety of quantifiers, namely all unbounded simple cardinality quantifiers. Our argument also applies to partial fixed point logic, PFP. In order to establish an analogous result for least fixed point logic, LFP, we exhibit a general method to pass from arbitrary quantifiers to monotone quantifiers. Our proof shows that the tree isomorphism problem is not definable in L[ohgr][infin][ohgr](Q1)[ohgr], infinitary logic extended with all monadic quantifiers and their vectorizations, where a finite bound is imposed to the number of variables as well as to the number of nested quantifiers in Q1. This strengthens a result of Etessami and Immerman by which tree isomorphism is not definable in TC + COUNTING.
Keywords: Finite model theory, generalized quantifiers, fixed point logics, descriptive complexity theory.