From Hilbert space to Dilbert space: context semantics as a language for games and flow analysis

We give a tutorial and first-principles description of the context semantics of Gonthier, Abadi, and Lévy [5, 4], a computer-science analogue of Girard's geometry of interaction [3]. In the spirit of the invited presentation of Tom Knight (see this Proceedings [7]), the semantics is reversible, and supports pseudo-quantum computation via a superposed sharing of terms and evaluation contexts [2].Context semantics provides a mechanism for modelling ?-calculus, and more generally multiplicative-exponential linear logic (MELL); we explain the the call-by-name (CBN) coding of the ?-calculus, and sketch a proof of the correctness of readback, where the normal form of a ?-term is recovered from its semantics. This analysis yields the algorithmic correctness of Lamping's optimal reduction algorithm [8]. We relate the context semantics to linear logic types and to ideas from game semantics, used to prove full abstraction theorems for PCF and other ?-calculus variants [1, 6, 10]. Readback is essentially a game played by an environment (the Opponent) who wants to discover the Böhm tree (normal form) of a term known by a Player. A type plays the role---using the games jargon---of an arena of possible moves, and a term of that type provides a winning strategy for the Player, permitting the Player to respond correctly to moves made by the Opponent. The interaction between Opponent and Player describes a perfect flow analysis which answers questions like, "can call site a ever call procedure p?" The context semantics provides a low-level coding mechanism for describing such flows, the positions of subexpressions and head variables in Böhm trees, as well as moves in the above described two-player games.