This paper presents a new lower bound for the recursive algorithm for solving parity games which is induced by the constructive proof of memoryless determinacy by Zielonka. We outline a family of games of linear size on which the algorithm requires exponential time.

For an arbitrary category, we consider the least class of functorscontaining the projections and closed under finite products, finitecoproducts, parameterized initial algebras and parameterized finalcoalgebras, i.e. the class of functors that are definable byμ-terms. We call the category μ-bicomplete if every μ-termdefines a functor. We provide concrete examples of such categories andexplicitly characterize this class of functors for the category ofsets and functions. This goal is achieved through parity games: weassociate to each game an algebraic expression and turn the game intoa term of a categorical theory. We show that μ-terms and paritygames are equivalent, meaning that they define the same property ofbeing μ-bicomplete. Finally, the interpretation of a parity gamein the category of sets is shown to be the set of deterministicwinning strategies for a chosen player.