Partially ordered connectives and Sigma-1-1 on finite models
Merlijn Sevenster, Tero Tulenheimo
Abstract:
In this paper we take up the study of Henkin quantifiers with boolean
variables also known as partially ordered connectives. We consider
first-order formulae prefixed by partially ordered connectives,
denoted D, on finite structures. We characterize D as a fragment of
second-order existential logic Sigma-1-1-hearts whose formulae do not
allow for existential variables being argument of predicate
variables. We show that \Sigma-1-1-hearts harbors a strict hierarchy
induced by the arity of predicate variables and that it is not closed
under complementation, by means of a game-theoretical argument.
Admitting for at most one existential variable to appear as the
argument of a predicate variable already yields a logic coinciding
with full Sigma-1-1 , thus we show.