Game Semantics of Martin-Löf Type Theory

05/03/2019 ∙ by Norihiro Yamada, et al. ∙ 0

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.



There are no comments yet.


This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.