Abstract
We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We also devise a game-theoretic semantics equivalent to the double team semantics. We make use of the double team semantics by defining a logic hbox {DC}^2 which canonically fuses together two-variable dependence logichbox {D}^2 and two-variable logic with counting quantifiershbox {FOC}^2. We establish that the satisfiability and finite satisfiability problems of hbox {DC}^2 are complete for hbox {NEXPTIME}.
Highlights
Independence-friendly logic is an extension of first-order logic motivated by issues concerning Henkin quantifiers and game-theoretic semantics
The syntax of FOC2 contains only first-order atoms, and in light of Propositions 2 and 1, it makes no difference whether we use ordinary Tarskian semantics or double team semantics in the interpretation of FOC2-formulae; if φ is a formula of FOC2 and φ denotes the formula obtained from φ by replacing each symbol ∃≥k by a symbol that denotes the corresponding ordinary generalized quantifier, A, s | FO φ iff A, {s}, ∅ | φ
We have defined the notions of a generalized atom and minor quantifier, and shown how these notions can be used in defining extensions and variants of dependence logic
Summary
Independence-friendly logic is an extension of first-order logic motivated by issues concerning Henkin quantifiers and game-theoretic semantics. The semantic framework based on double teams provides a natural setting for interpretation of the meaning of the strict and lax quantifiers. In order to demonstrate how the double team semantics works in practice, we define a logic DC2 which extends two-variable dependence logic D2 by counting quantifiers ∃≥k. We wish to demonstrate how the double team framework can in practice be used in order to study fragments of team-semantics-based logics extended with generalized quantifiers. It is worth noting that while the double team semantics can be used in investigations related to dependence logic and its variants, it is a canonical semantics for ordinary extensions of first-order logic with generalized quantifiers, i.e., extensions that do not include generalized atoms.
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