Dependence Logic with Generalized Quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.Comment: 17 page

Defining a Double Team Semantics for Generalized Quantifiers

In this brief technical report we sketch a semantics for fi rst-order logic with generalized quantifiers based on double teams. We also define the notion of a generalized atom. Such atoms can be used in order to define extensions of first-order logic with a team-based semantics. We then briefly discuss how our double team semantics relates to game semantics based approaches to extensions of first-order logic with generalized quantifiers

Boolean Dependence Logic and Partially-Ordered Connectives

We introduce a new variant of dependence logic called Boolean dependence logic. In Boolean dependence logic dependence atoms are of the type =(x_1,...,x_n,\alpha), where \alpha is a Boolean variable. Intuitively, with Boolean dependence atoms one can express quantification of relations, while standard dependence atoms express quantification over functions. We compare the expressive power of Boolean dependence logic to dependence logic and first-order logic enriched by partially-ordered connectives. We show that the expressive power of Boolean dependence logic and dependence logic coincide. We define natural syntactic fragments of Boolean dependence logic and show that they coincide with the corresponding fragments of first-order logic enriched by partially-ordered connectives with respect to expressive power. We then show that the fragments form a strict hierarchy.Comment: 41 page

Polyteam Semantics

Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define "Polyteam Semantics" in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatisation for the associated implication problem. We also characterise the expressive power of poly-dependence logic by properties of polyteams that are downward closed and definable in existential second-order logic (ESO). The analogous result is shown to hold for poly-independence logic and all ESO-definable properties.Peer reviewe

Dependence Logic in Pregeometries and ω\omega-Stable Theories

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence

Polyteam semantics

Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define Polyteam Semantics in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatization for the associated implication problem. We relate polyteam semantics to team semantics and investigate in which cases logics over the former can be simulated by logics over the latter. We also characterize the expressive power of poly-dependence logic by properties of polyteams that are downwards closed and definable in existential second-order logic (ESO). The analogous result is shown to hold for poly-independence logic and all ESO-definable properties. We also relate poly-inclusion logic to greatest fixed point logic.Peer reviewe