Enhancing Xed Point Logic with Cardinality Quantiiers
نویسندگان
چکیده
Let Q IFP be any quantiier such that FO(Q IFP), rst order logic enhanced with Q IFP and its vectorizations, equals inductive xed point logic, IFP in expressive power. It is known that for certain quantiiers Q, the equivalence FO(Q IFP) IFP is no longer true if Q is added on both sides 12, 13]. Rather, we have FO(Q IFP ; Q) < IFP(Q) in such cases. We extend these results to a great variety of quantiiers, namely all unbounded simple cardinality quantiiers. Our argument also applies to partial xed point logic, PFP. In order to establish an analogous result for least xed point logic, LFP, we exhibit a general method to pass from arbitrary quantiiers to monotone quantiiers. Our proof shows that the tree isomorphism problem is not deen-able in L ! 1! (Q 1) ! , innnitary logic extended with all monadic quanti-ers and their vectorizations, where a nite bound is imposed to the number of variables as well as to the number of nested quantiiers in Q 1. This strengthens a result of Etessami and Immerman 5] by which tree isomorphism is not deenable in TC + COUNTING.
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