We introduce a new decidable fragment of first-order logic with equality, which strictly generalizes two already well-known ones-the Bernays-Schönfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The defining principle is the syntactic separation of universally quantified variables from existentially quantified ones at the level of atoms. Thus, our classification neither rests on restrictions on quantifier prefixes (as in the BSR case) nor on restrictions on the arity of predicate symbols (as in the monadic case). We demonstrate that the new fragment exhibits the finite model property and derive a non-elementary upper bound on the computing time required for deciding satisfiability in the new fragment. For the subfragment of prenex sentences with the quantifier prefix ∃ * ∀ * ∃ * the satisfiability problem is shown to be complete for NEXPTIME. Finally, we discuss how automated reasoning procedures can take advantage of our results.

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Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SF's satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence ∃ z ∀ x1∃ y1 . . . ∀ xn∃ yn. ψ in which ψ is quantifier free, satisfiability can be decided in nondeterministic n-fold exponential time. In the present paper, we conduct a more fine-grained analysis of the complexity of SF-satisfiability. We derive an upper and a lower bound in terms of the degree ∂ of interaction of existential variables (short: degree)-a novel measure of how many separate existential quantifier blocks in a sentence are connected via joint occurrences of variables in atoms. Our main result is the k-NExpTime-completeness of the satisfiability problem for the set SF ∂≤k of all SF sentences that have degree k or smaller. Consequently, we show that SF-satisfiability is non-elementary in general, since SF is defined without restrictions on the degree. Beyond trivial lower bounds, nothing has been known about the hardness of SF-satisfiability so far.