• Corpus ID: 10861013

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… 
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References

SHOWING 1-10 OF 117 REFERENCES
A game semantics for generic polymorphism
Dynamic Games and Strategies
TLDR
A new game semantics of a prototypical programming language that distinguishes terms with the same value yet different algorithms, capturing intensionality of computation is given, equipped with the hiding operation on strategies that exactly corresponds to the (small-step) operational semantics of the programming language.
Extensional and Intensional Semantic Universes: A Denotational Model of Dependent Types
TLDR
An operational semantics for intensional terms is defined, giving a functional programming language based on the type theory, and it is proved that the semantics for it is computationally adequate.
On Full Abstraction for PCF: I, II, and III
TLDR
An order-extensional, order (or inequationally) fully abstract model for Scott's language pcf, based on a kind of game in which each play consists of a dialogue of questions and answers between two players who observe the following principles of civil conversation.
On Universes in Type Theory
The notion of a universe of types was introduced into constructive type theory by Martin-Lof (1975). According to the propositions-as-types principle inherent in type theory, the notion plays two
Full Abstraction for PCF
TLDR
The effective version of the model is considered and it is proved that every element of the effective extensional model is definable in PCF, which is the first syntax-independent description of the fully abstract model for PCF.
Games and Full Abstraction for a Functional Metalanguage with Recursive Types
TLDR
This book discusses FPC and its Models, Rational Categories and Recursive Types, and Semantics of the Recursion Combinator, as well as games and Strategies, and the author's own contributions.
Games and full completeness for multiplicative linear logic
TLDR
It is shown that this semantics yields a categorical model of Linear Logic and proves full completeness for Multiplicative Linear Logic with the MIX rule: every winning strategy is the denotation of a unique cut-free proof net.
A formulation of the simple theory of types
  • A. Church
  • Mathematics
    Journal of Symbolic Logic
  • 1940
TLDR
A formulation of the simple theory oftypes which incorporates certain features of the calculus of λ-conversion into the theory of types and is offered as being of interest on this basis.
...
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