From Parity Games to Circular Proofs

We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. μ-bicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of μ-bicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game. We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of μ-bicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles. This paper is a survey on our recent work lifting results on free μ-lattices [1,2] to a categorical setting. A μ-lattice is a lattice with enough least and greatest fixed points to interpret formal μ-terms. A generalization of this notion leads to consider categories with finite products, finite coproducts, and enough initial algebras and final coalgebras of functors. We call these categories μ-bicomplete. The outcome of this research is so far described in [3,4,5]. A main goal for us is to understand how the algebra of μ-bicomplete categories describes a computational situation through the combinatorics of games; when attempting to achieve this goal, computational logic and proof-theory become unavoidable ingredients. It is the aim of this note to give insights on how these four worlds – categories, games, computation and logic – relate in this context. As the need of a mathematical formalization has too often hidden these relationships, we shall present here only informal arguments. The reader will find formal proofs of the statements in the references cited above. 1 Email: [email protected] c ©2002 Published by Elsevier Science B. V.