
A General Framework for the Semantics of Type Theory
We propose an abstract notion of a type theory to unify the semantics of...
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Dialectica models of type theory
We present two Dialecticalike constructions for models of intensional M...
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Formalizing relations in type theory
Type theory plays an important role in foundations of mathematics as a f...
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Relational Type Theory (All Proofs)
This paper introduces Relational Type Theory (RelTT), a new approach to ...
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CostAware Type Theory
Although computational complexity is a fundamental aspect of program beh...
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Effective Kan fibrations in simplicial sets
We introduce the notion of an effective Kan fibration, a new mathematica...
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Transpension: The Right Adjoint to the Pitype
Presheaf models of dependent type theory have been successfully applied ...
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Callbyname Gradual Type Theory
We present gradual type theory, a logic and type theory for callbyname gradual typing. We define the central constructions of gradual typing (the dynamic type, type casts and type error) in a novel way, by universal properties relative to new judgments for gradual type and term dynamism, which were developed in blame calculi and to state the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality (η) principles that type theory provides, we show that most of the standard operational behavior of casts is uniquely determined by the gradual guarantee. This provides a semantic justification for the definitions of casts, and shows that nonstandard definitions of casts must violate these principles. Our type theory is the internal language of a certain class of preorder categories called equipments. We give a general construction of an equipment interpreting gradual type theory from a 2category representing nongradual types and programs, which is a semantic analogue of Findler and Felleisen's definitions of contracts, and use it to build some concrete domaintheoretic models of gradual typing.
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