Game Semantics of Martin-L\"of Type Theory

Abstract

We present new game semantics of Martin-L\"of type theory (MLTT) equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types. Our game semantics interprets MLTT more accurately than existing ones. Another advantage of our game semantics over existing ones is its interpretation of Sigma-types that is direct and compatible with the game semantics of product types . Besides, its mathematical structure is novel and useful; e.g., the category of our games has all finite limits, which is a key step to an extension of the present work to homotopy type theory, and our games interpret subtyping on dependent types for the first time as game semantics. Finally, we provide a new, game-semantic proof of the independence of Markov's principle from MLTT, which demonstrates an advantage of our game semantics over extensional models of MLTT such as the effective topos.