Complexity of two-variable logic with counting

Let C/sub k//sup 2/ denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) m", for m/spl les/k, and let C/sup 2/ be the union of C/sub k//sup 2/ taken over all integers k. We prove that the problem of satisfiability of sentences of C/sub 1//sup 2/ is NEXPTIME-complete. This strengthens a recent result of E. Gradel, Ph. Kolaitis and M. Vardi (1997) who proved that the satisfiability problem for the first order two-variable logic L/sup 2/ is NEXPTIME-complete and a very recent result by E. Gradel, M. Otto and E. Rosen (1997) who proved the decidability of C/sup 2/. Our result easily implies that the satisfiability problem for C/sup 2/ is in non-deterministic, doubly exponential time. It is interesting that C/sub 1//sup 2/ is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size.

[1]  Martin Otto,et al.  Undecidability Results on Two-Variable Logics , 1997, STACS.

[2]  Michael Mortimer,et al.  On languages with two variables , 1975, Math. Log. Q..

[3]  Warren D. Goldfarb.  The Unsolvability of the Godel Class with Identity , 1984, J. Symb. Log..

[4]  Neil D. Jones,et al.  Turing machines and the spectra of first-order formulas , 1974, Journal of Symbolic Logic.

[5]  Harry R. Lewis,et al.  Complexity Results for Classes of Quantificational Formulas , 1980, J. Comput. Syst. Sci..

[6]  Erich Grädel,et al.  Two-variable logic with counting is decidable , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[7]  Phokion G. Kolaitis,et al.  On the Decision Problem for Two-Variable First-Order Logic , 1997, Bulletin of Symbolic Logic.

[8]  Neil D. Jones,et al.  Turing machines and the spectra of first-order formulas with equality , 1972, STOC.

[9]  Maarten de Rijke,et al.  Counting Objects , 1995, J. Log. Comput..