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SIAM J. Comput. 29, pp. 1083-1117 (35 pages)
Complexity Results for First-Order Two-Variable Logic with Counting
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Add to FacebookLet $C^2_p$ denote the class of first-order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) $i$" for $i\leq p$, and let $C^2$ be the union of $C^2_p$ taken over all integers $p$. We prove that the satisfiability problem for $C^2_1$ sentences is NEXPTIME-complete. This strengthens the results by [E. Grädel, Ph. Kolaitis, and M. Vardi, Bull. Symbolic Logic, 3 (1997), pp. 53--69], who showed that the satisfiability problem for the first-order two-variable logic $L^2$ is NEXPTIME-complete and by [E. Grädel, M. Otto, and E. Rosen, 12th Annual IEEE Symposium on Logic in Computer Science, 1997, pp. 306--317], who proved the decidability of $C^2$. Our result easily implies that the satisfiability problem for $C^2$ is in nondeterministic, doubly exponential time. It is interesting that $C^2_1$ is in NEXPTIME in spite of the fact that there are sentences whose minimal (and only) models are of doubly exponential size.
It is worth noticing that by a recent result of [E. Grädel, M. Otto, and E. Rosen, Proceedings of 14th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Comput. Sci. 1200, Springer-Verlag, Berlin, 1997], extensions of two-variable logic $L^2$ by a weak access to cardinalities through the Härtig (or equicardinality) quantifier is undecidable. The same is true for extensions of $L^2$ by very weak forms of recursion.
The satisfiability problem for logics with a bounded number of variables has applications in artificial intelligence, notably in modal logics (see, e.g., [W. van der Hoek and M. De Rijke, J. Logic Comput., 5 (1995), pp. 325--345]), where counting comes in the context of graded modalities and in description logics, where counting can be used to express so-called number restrictions (see, e.g., [A. Borgida, Artificial Intelligence, 82 (1996), pp. 353--367]).
© 2000 Society for Industrial and Applied Mathematics
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