A A PSPACE-Complete First Order Fragment of Computability Logic

Computability Logic (CoL), introduced by Japaridze in [2003; 2009a], is a research program for developing logic as a formal theory of interactive computability. Formulas in it are understood as (interactive) computational problems and logical operators represent operations on such entities. The goal of this program is to define a systematic way to answer the question “what can be computed?” within the confines of a formal logical system. Computational problems are modelled as logical formulas through the use of game semantics. Each problem is understood as a game played between a machine ⊤ and its environment ⊥, and a problem is seen as computable if there exists a machine that has an algorithmic winning strategy in the corresponding game. The closest predecessors to CoL are Hintikka’s game-theoretic semantics [Hintikka 1996] and Blass’s game semantics for Linear Logic [Blass 1992]. In line with its semantics, CoL introduces a rich set of logical connectives. Among those relevant to this paper are the propositional connectives ¬ (negation), ∨ (parallel disjunction), ∧ (parallel conjunction), ⊔ (choice disjunction) and ⊓ (choice conjunction) as well as the “choice” quantifiers⊔ and⊓. Intuitively, ¬ corresponds to switching the roles of the players ⊤ and ⊥ in game to which it is applied: the game ¬A is obtained from A by turning ⊤’s legal moves and wins into legal moves and wins for ⊥ and vice