On the Expressive Power of Counting

We investigate the expressive power of various extensions of rst-order, inductive, and innnitary logic with counting quantiiers. We consider in particular a LOGSPACE extension of rst-order logic, and a PTIME extension of xpoint 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 rst 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 rst-order logic with equality of the cardinalities of relations has a 0/1 law.