A dichotomy result of Sevenster (2014) [29] completely classified the quantifier prefixes of regular Independence-Friendly (IF) logic according to the patterns of quantifier dependence they contain. On one hand, prefixes that contain “Henkin” or “signalling” patterns were shown to characterize fragments of IF logic that capture NP-complete problems; all the remaining prefixes were shown instead to be essentially first-order. In the present paper we develop the machinery which is needed in order to extend the results of Sevenster to non-prenex, regular (...) IF sentences. This involves shifting attention from quantifier prefixes to a (rather general) class of syntactical tree prefixes. We partially classify the fragments of regular IF logic that are thus determined by syntactical trees; in particular, a) we identify four classes of tree prefixes that are neither signaling nor Henkin, and yet express NP-complete problems and other second-order concepts; and b) we give more general criteria for checking the first-orderness of an IF sentence. (shrink)

Dependence logic and its many variants are new logics that aim at establishing a unified logical theory of dependence and independence underlying seemingly unrelated subjects. The area of dependence logic has developed rapidly in the past few years. We will give a short introduction to dependence logic and review some of the recent developments in the area.

We develop a semantics for independence logic with respect to what we will call general models. We then introduce a simpler entailment semantics for the same logic, and we reduce the validity problem in the former to the validity problem in the latter. Then we build a proof system for independence logic and prove its soundness and completeness with respect to entailment semantics.

In this paper, we introduce a logic based on team semantics, called $\mathbf {FOT} $, whose expressive power is elementary, i.e., coincides with first-order logic both on the level of sentences and (possibly open) formulas, and we also show that a sublogic of $\mathbf {FOT} $, called $\mathbf {FOT}^{\downarrow } $, captures exactly downward closed elementary (or first-order) team properties. We axiomatize completely the logic $\mathbf {FOT} $, and also extend the known partial axiomatization of dependence logic to dependence logic (...) enriched with the logical constants in $\mathbf {FOT}^{\downarrow } $. (shrink)