Abstract
Abstract We consider fragments of first-order logic (with and without equality) defined by means of standard quantifier prefix specifiers, over signatures containing no function symbols. We determine, in each case, the decidability and complexity of the (finite) satisfiability problem. Two of these results tower above the others. The first concerns the so-called Gödel fragment, where the quantifier prefix features two adjacent universal quantifiers and at least one trailing existential quantifier, but where equality is not allowed. We show that this fragment has the finite model property, and that its satisfiability problem is NExpTime-complete. The second concerns the so-called Goldfarb fragment, which adds equality to the Gödel fragment. We show that the satisfiability and finite satisfiability problems for this fragment are undecidable. We also discuss the effect on lower complexity bounds of making individual constants unavailable.
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