Guards, Bounds, and Generalized Semantics

Some initial motivations for the Guarded Fragment still seem of interest in carrying its program further. First, we stress the equivalence between two perspectives: (a) satisfiability on standard models for guarded first-order formulas, and (b) satisfiability on general assignment models for arbitrary firstorder formulas. In particular, we give a new straightforward reduction from the former notion to the latter. We also show how a perspective shift to general assignment models provides a new look at the fixed-point extension LFP(FO) of first-order logic, making it decidable. Next, we relate guarded syntax to earlier quantifier restriction strategies for the purpose of achieving effective axiomatizability in second-order logic – pointing at analogies with 'persistent' formulas, which are essentially in the Bounded Fragment of many-sorted first-order logic. Finally, we look at some further unexplored directions, including the systematic use of 'quasi-models' as a semantics by itself. 1 Basics of the Guarded Fragment 1.1 Guarded syntax The Guarded Fragment of Andréka, van Benthem & Németi 1998 is a decidable part of first-order syntax with a semantic philosophy: quantifiers only access the total domain of individual objects 'locally' by means of predicates over objects. But there is more to the motivation and ambitions of guarding, as will be shown in this paper. But first, we make a quick tour of some known results and proof methods. Here are some syntactic preliminaries. In what follows, mostly for convenience, we consider only languages with predicate symbols and variables: no function symbols or identity predicates occur. But we do allow so-called polyadic first-order quantifiers x , x over tuples of variables x, with their obvious interpretation. Finally, we also use polyadic notations [u/y] for simultaneous substitutions. These are taken in the standard syntactic sense that the substitution is performed provided the u are free for the y. If not, some suitable alphabetic variant is taken first for . Our key idea is that objects y can only be introduced relative to given objects x, as expressed by a 'guard atom' G(x, y) where objects can occur in any order and multiplicity – and that the subsequent statement refers only to those guarded x, y.