Game semantics of universes

Abstract: This work extends the present author's computational game semantics of Martin-L\"{o}f type theory to the cumulative hierarchy of universes. This extension completes game semantics of all standard types of Martin-L\"{o}f type theory for the first time in the 30 years history of modern game semantics. As a result, the powerful combinatorial reasoning of game semantics becomes available for the study of universes and types generated by them. A main challenge in achieving game semantics of universes comes from a conflict between identity types and universes: Naive game semantics of the encoding of an identity type by a universe induces a decision procedure on the equality between functions, a contradiction to a well-known fact in recursion theory. We overcome this problem by novel games for universes that encode games for identity types without deciding the equality.