# Game Semantics for Martin-Löf Type Theory

@article{Yamada2016GameSF, title={Game Semantics for Martin-L{\"o}f Type Theory}, author={Norihiro Yamada}, journal={ArXiv}, year={2016}, volume={abs/1610.01669} }

We present a new game semantics for Martin-L\"of type theory (MLTT), our aim is to give a mathematical and intensional explanation of MLTT. Specifically, we propose a category with families of a novel variant of games, which induces a surjective and injective (when Id-types are excluded) interpretation of the intensional variant of MLTT equipped with unit-, empty-, N-, dependent product, dependent sum and Id-types as well as the cumulative hierarchy of universes for the first time in the…

## 7 Citations

Game-theoretic Investigation of Intensional Equalities

- MathematicsArXiv
- 2017

A first game semantics for MLTT is obtained that refutes the principle of uniqueness of identity proofs (UIP) and validates univalence axiom (UA) though it does not model non-trivial higher equalities.

A game-semantic model of computation

- Computer ScienceResearch in the Mathematical Sciences
- 2018

This work shows, as a main technical achievement, that viable strategies in game semantics are Turing complete and has given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higher-order computation, in a more abstract fashion.

Proofs and Strategies A Characterization of Classical and Intuitionistic Logic using Games with Explicit Strategies MSc Thesis (Afstudeerscriptie)

- Computer Science
- 2020

This work enrichs the language of first-order logic with two force markers denoting assertion and challenge with a close correspondence between composition and the cut-rule, giving soundness and completeness for classical and intuitionistic logic.

Game semantics of Martin-Löf type theory, part III: its consistency with Church's thesis

- Computer ScienceArXiv
- 2020

This work proves consistency of intensional Martin-Lof type theory with formal Church's thesis (CT) by novel realizability a la game semantics, which is based on the author's previous work.

Parametric Church's Thesis: Synthetic Computability Without Choice

- PhilosophyLFCS
- 2022

This work introduces various parametric strengthenings of CTφ, which are equivalent to assuming CT φ and an S n operator for φ like in the S n theorem, and explains the novel axioms and proofs of Rice’s theorem.

In Search of Effectful Dependent Types

- Computer ScienceArXiv
- 2017

This thesis develops a game semantics for dependent type theory, which had so far been missing altogether and explores a generalisation of Levy’s call-by-push-value to encompass dependent types.

Dependent Cartesian Closed Categories

- Computer ScienceArXiv
- 2017

DCCCs accomplish mathematical elegance as well as a direct interpretation of the syntax of Martin-L\"{o}f type theory (MLTT) by capturing the categorical counterpart of the generalization of the simply-typed lambda-calculus to MLTT in syntax.

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