Topological Observations on Multiplicative Additive Linear Logic

As an attempt to uncover the topological nature of composition of strategies in game semantics, we present a “topological” game for Multiplicative Additive Linear Logic without propositional variables, including cut moves. We recast the notion of (winning) strategy and the question of cut elimination in this context, and prove a cut elimination theorem. Finally, we prove soundness and completeness. The topology plays a crucial role, in particular through the fact that strategies form a sheaf. 1. Overview The notion of a game between two players (P and O) has become fundamental in proof theory and programming language theory. A natural way to think of such a game is as a directed graph, whose edges represent moves between positions, together with some information about who plays the moves. Game semantics (Abramsky 1997; Hyland 1997) has widened this notion of game, by providing means to connect two such games together. In game semantics, each player takes part in two distinct games, and acts as P in one and as O in the other. Connection, or interaction, then happens by letting two players respectively play P and O on a common game. By making several such connections, one obtains a sequence of games, subject to topological considerations. For example, one may see the involved games as edges in a graph with the players as vertices, as in game 0 player 1 game 1 player 2 game 2 etc., and decree that an open neighborhood of player i is the sequence game i− 1 player i game i . The topology here is simplistic, but arguably, this is only due to the requirement that game semantics be categorical, i.e., each player sees only two games. This is most striking in the game semantics of sequent calculi, where sequents A1, . . . , An ⊢ B1, . . . , Bm are interpreted as games A1 ∧ · · · ∧An → B1 ∨ · · · ∨Bm. [Copyright notice will appear here once ’preprint’ option is removed.] Let us instead allow each player to see more than two games, i.e., lie in an open neighborhood like