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Reports (Research Report) Year : 1992

On the Expressive Power of Counting

Stéphane Grumbach


We investigate the expressive power of various extensions of first-order, inductive, and infinitary logic with counting quantifiers. We consider in particular a LOGSPACE extension of first-order logic, and a PTIME extension of fixpoint logic with counters. Counting is a fundamental tool of algorithms. It is essential in the case of unordered structures. Our aim is to understand the expressive power gained with a limited counting ability. We consider two problems: (i) unnested counters, and (ii) counters with no free variables. We prove a hierarchy result based on the arity of the counters under the first restriction. The proof is based on a game technique that is introduced in the paper. We also establish results on the asymptotic probabilities of sentences with counters under the second restriction. In particular, we show that first-order logic with equality of the cardinalities of relations has a 0/1 law.
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inria-00074344 , version 1 (24-05-2006)



Stéphane Grumbach, Christophe Tollu. On the Expressive Power of Counting. [Research Report] RR-2330, INRIA. 1992, pp.38. ⟨inria-00074344⟩
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