Categorical Combinatorics for Innocent Strategies

International audienceWe show how to construct the category of games and innocent strategies from a more primitive category of games. On that category we define a comonad and monad with the former distributing over the latter. Innocent strategies are the maps in the induced two-sided Kleisli category. Thus the problematic composition of innocent strategies reflects the use of the distributive law. The composition of simple strategies, and the combinatorics of pointers used to give the comonad and monad are themselves described in categorical terms. The notions of view and of legal play arise naturally in the explanation of the distributivity. The category-theoretic perspective provides a clear discipline for the necessary combinatorics

Game semantics for first-order logic

We refine HO/N game semantics with an additional notion of pointer (mu-pointers) and extend it to first-order classical logic with completeness results. We use a Church style extension of Parigot's lambda-mu-calculus to represent proofs of first-order classical logic. We present some relations with Krivine's classical realizability and applications to type isomorphisms

A graphical foundation for schedules

AbstractIn 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a schedule. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, a proof that the composition of schedules is associative involves cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of schedule, prove that it is equivalent to Harmer et al.ʼs definition, and illustrate its value by giving a proof of associativity of composition

A graphical foundation for interleaving in game semantics

In 2007, Harmer, Hyland and Melliès gave a formal mathematical foundation for game semantics using a notion they called a {multimap}-schedule, and the similar notion of ⊗-schedule, both structures describing interleavings of plays in games. Their definition was combinatorial in nature, but researchers often draw pictures when describing schedules in practice. Moreover, several proofs of key properties, such as that the composition of {multimap}-schedules is associative, involve cumbersome combinatorial detail, whereas in terms of pictures the proof is straightforward, reflecting the geometry of the plane. Here, we give a geometric formulation of {multimap}-schedules and ⊗-schedules, prove that they are isomorphic to Harmer et al.'s definitions, and illustrate their value by giving such geometric proofs. Harmer et al.'s notions may be combined to describe plays in multi-component games, and researchers have similarly developed intuitive graphical representations of plays in these games. We give a characterisation of these diagrams and explicitly describe how they relate to the underlying schedules, finally using this relation to provide new, intuitive proofs of key categorical properties

The anatomy of innocence revisited

International audienceWe refine previous analyses of Hyland-Ong game semantics and its relation to lambda- and lambda-mu-calculi and present improved factorization results for bracketing and rigidity that can be combined in any order

The anatomy of innocence revisited

Abstract. We refine previous analyses of Hyland-Ong game semantics and its relation to λ- and λµ-calculi and present improved factorization results for bracketing and rigidity that can be combined in any order.