Two-Level Game Semantics, Intersection Types, and Recursion Schemes

C.-H. Luke Ong, Takeshi Tsukada

Abstract: We introduce a new cartesian closed category of two-level arenas and innocent strategies to model intersection types that are refinements of simple types. Intuitively a property (respectively computation) on the upper level refines that on the lower level. We prove Subject Expansion—any lower-level computation is closely and canonically tracked by the upper-level computation that lies over it—which is a measure of the robustness of the two-level semantics. The game semantics of the type system is fully complete: every winning strategy is the denotation of some derivation. To demonstrate the relevance of the game model, we use it to construct new semantic proofs of non-trivial algorithmic results in higher-order model checking.