Game Semantics of Martin-Löf Type Theory
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We present game semantics of Martin-Löf type theory (MLTT), which solves a long-standing problem open for more than twenty years. More specifically, we introduce a category with families of a novel variant of games, which induces an interpretation of MLTT equipped with one-, zero-, N-, pi- and sigma-types as well as Id-types or a cumulative hierarchy of universes (n.b., the last two types are incompatible with each other in our semantics), and the interpretation is faithful for the (one, pi, sigma)-fragment. Our semantics can be regarded naturally as a mathematical formalization of the standard BHK-interpretation (or the meaning explanation) of MLTT, giving a mathematical, semantic, intensional foundation of constructive mathematics, comparable to the set-theoretic one for classical mathematics. By its conceptual naturality and mathematical precision, the semantics provides useful insights on the syntax as well.
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