Partially Ordered Connectives and ∑1 1 on Finite Models

Abstract

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. We characterize D as a fragment of second-order existential logic Σ 1 1 ♥ whose formulae do not allow for existential variables being argument of predicate variables. We show that Σ 1 1 ♥ 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 Σ 1 1 , thus we show.

Boolean dependence logic and partially-ordered connectives

“…We show next that the game N π EF r can be used for studying the truth preservation relation ⇛ Nπ[F Or] . This result is essentially the same as Proposition 12 in [ST06], which in turn is a special case of Proposition 7 of [SV92].…”

“…where ϕ is a first-order formula with quantifier rank at most r. We will next define an Ehrenfeucht-Fraïssé game that captures the truth preservation relation A ⇛ Nπ[F Or] B. This game is a straightforward modification of the corresponding game for D[F O] by Sevenster and Tulenheimo [ST06], which in turn is based on the game for F O(D) by Sandu and Väänänen [SV92].…”

Boolean Dependence Logic and Partially-Ordered Connectives

“…This result is essentially the same as Proposition 12 in [ST06], which in turn is a special case of Proposition 7 of [SV92]. Let r be the quantifier rank of ψ.…”

“…This game is a straightforward modification of the corresponding game for D[F O] by Sevenster and Tulenheimo [ST06], which in turn is based on the game for F O(D) by Sandu and Väänänen [SV92].…”

1. Setting the Stage

“…on the structure B would be equal to the ones for ∀x 1 ∃y 1 ∀x 2 ∃y 2 R(x, y). (The sentence ψ characterizes the finite structures whose universes have even cardinality, see [35].) Considering an absentminded 1-cell agent, we see that during neither of his rounds it knows whether the object it choses will be assigned to y 1 or y 2 ; it is aware of the last action though.…”