
On Higher Inductive Types in Cubical Type Theory
Cubical type theory provides a constructive justification to certain asp...
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Computational Higher Type Theory III: Univalent Universes and Exact Equality
This is the third in a series of papers extending MartinLöf's meaning e...
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Game semantics of MartinLöf type theory, part III: its consistency with Church's thesis
We prove consistency of intensional MartinLöf type theory (MLTT) with f...
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Denotational semantics of recursive types in synthetic guarded domain theory
Just like any other branch of mathematics, denotational semantics of pro...
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Mathematical Game Theory
These lecture notes attempt a mathematical treatment of game theory akin...
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Whither Semantics?
We discuss how mathematical semantics has evolved, and suggest some new ...
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Theory Presentation Combinators
To build a scalable library of mathematics, we need a method which takes...
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Game Semantics of MartinLöf Type Theory
We present game semantics of MartinLöf type theory (MLTT), which solves a longstanding 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 sigmatypes as well as Idtypes 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 BHKinterpretation (or the meaning explanation) of MLTT, giving a mathematical, semantic, intensional foundation of constructive mathematics, comparable to the settheoretic 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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