Partially ordered connectives and Σ11 on finite models

Abstract

In this paper we take up the study of Henkin quantifiers with boolean variables [4], also known as partially ordered connectives [19]. We consider first-order formulae prefixed by partially ordered connectives, denoted D, on finite structures. D is characterized as a fragment of second-order existential logic Σ11heart sign, whose formulae do not allow existential variables as arguments of predicate variables. By means of a game theoretic argument, it is shown that Σ11heart sign harbors a strict hierarchy induced by the arity of predicate variables, and that it is not closed under complementation. It is further shown that allowing at most one existential variable to appear as an argument of a predicate variable, already yields a logic coinciding with full Σ11. © Springer-Verlag Berlin Heidelberg 2006.