We give a categorical semantics to the call-by-name and call-by-value versions of Parigot's λμ-calculus with disjunction types. We introduce the class of control categories, which combine a cartesian-closed structure with a premonoidal structure in the sense of Power and Robinson. We prove, via a categorical structure theorem, that the categorical semantics is equivalent to a CPS semantics in the style of Hofmann and Streicher. We show that the call-by-name λμ-calculus forms an internal language for control categories, and that the call-by-value λμ-calculus forms an internal language for the dual co-control categories. As a corollary, we obtain a syntactic duality result: there exist syntactic translations between call-by-name and call-by-value that are mutually inverse and preserve the operational semantics. This answers a question of Streicher and Reus.

Proofnets may be seen as orientable surfaces with boundary. We investigate how the topology of such nets relates to the number of exchange rules in corresponding proofs.

We consider the propositional modal mu-calculus, a logic proposed by Kozen in 1983. In this logic two operators μ and v are present, which express the least and greatest fixpoints of monotone operators on sets. Bradfield in 1998 proved for any n the existence of a mu-calculus formula that requires n alternations of μ and v. In this paper we consider the particular case n = 3 and we exhibit a new formula requiring 3 alternations. Our proof is independent of the technique of Bradfield, and is based on a new kind of game on infinite trees.