A Double Team Semantics for Generalized Quantifiers.

Area of Science:

• Mathematical Logic
• Theoretical Computer Science

Background:

• Dependence logic extends first-order logic by capturing dependencies between variables.
• Generalized quantifiers offer expressive power beyond standard quantifiers.
• Existing semantic frameworks for dependence logic variants lack unification.

Purpose of the Study:

• To extend dependence logic with generalized quantifiers and generalized atoms.
• To develop a unified semantic framework for these extended logics.
• To define and analyze a novel fused logic combining two-variable dependence logic and two-variable logic with counting quantifiers.

Main Methods:

• Introduction of a novel "double team" semantics based on pairs of teams.
• Development of an equivalent game-theoretic semantics.
• Construction of a fused logic [Formula: see text] by integrating two-variable dependence logic [Formula: see text] and two-variable logic with counting quantifiers [Formula: see text].

Main Results:

• A unified semantic framework accommodating variants of dependence logic, generalized quantifiers, and generalized atoms.
• The definition of a new logic [Formula: see text] through canonical fusion.
• Establishment of [Formula: see text]-completeness for the satisfiability and finite satisfiability problems of the fused logic [Formula: see text].

Conclusions:

• The proposed double team semantics provides a flexible framework for diverse dependence logic variants.
• The fused logic [Formula: see text] offers a powerful tool for studying dependencies and quantified structures.
• The computational complexity results provide crucial insights into the decidability of these expressive logics.