@MISC{Pratt-hartmann_complexityof, author = {Ian Pratt-hartmann}, title = {Complexity of the Two- Variable Fragment with Counting Quantifiers}, year = {} }

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Abstract The satisfiability and finite satisfiability problems for the two-variable fragment of firstorder logic with counting quantifiers are both in NEXPTIME, even when counting quantifiers are coded succinctly. Key words: two- variable fragment, counting quantifiers, logic, complexity 1. Background The two- variable fragment with counting quantifiers, here denoted C2, is the set of function-free, first-order formulas containing at most two variables, but with the counting quantifiers 3<c, 3>c and 3=c (for every C> 0) allowed. The satisfiability problem, Sat-C2, is the problem of deciding, for a given formula <f> of C2, whether <j> has a model; the finite satisfiability problem, Fin-Sat-C2, is the problem of deciding, for a given formula </> of C2, whether <p has a finite model. It is well-known that C2 lacks the finite model property; hence Sat-C2 and Fin-Sat-C2 do not coincide. The decidability of Sat-C2 and Fin-Sat-C2 was shown by Gradel et al. (1997); the decidability of Sat-C2 was shown independently by Pacholski et al. (1997, 1999). For a general survey, see Gradel and Otto (1999).