Perfect IFG-Formulas

Abstract

IFG logic is a variant of the independence-friendly logic of Hintikka and Sandu. We answer the question: "Which IFG-formulas are equivalent to ordinary first-order formulas?" We use the answer to prove the ordinary cylindric set algebra over a structure can be embedded into a reduct of the IFG-cylindric set algebra over the structure. Mathematics Subject Classification (2000). Primary 03G25; Secondary 03B60, 03G15.Independence-friendly logic (IF logic) is a conservative extension of first-order logic that has the same expressive power as existential second-order logic [6,7]. In IF logic the truth of a sentence is defined via a game between two players, Abélard (∀) and Eloïse (∃). The additional expressivity is obtained by modifying the quantifiers and connectives of an ordinary first-order sentence in order to restrict the information available to the existential player, Eloïse, in the associated semantic game.In IF logic, only the information available to Eloïse is restricted, which means existential quantifiers are not dual to universal quantifiers. To compensate, negation symbols are only allowed before atomic formulas. Generalized independencefriendly logic (IFG logic) is a variant of independence-friendly logic in which the information available to both players can be restricted, making existential quantifiers dual to universal quantifiers and allowing any formula to be negated [2]. Definition 1.1. Given a first-order signature σ, an atomic IFG-formula is a pair φ, X where φ is an atomic first-order formula and X is a finite set of variables that includes every variable that appears in φ (and possibly more).Definition 1.2. Given a first-order signature σ, the language L σ IFG is the smallest set of formulas such that: