6,972 research outputs found

    Guarded Cubical Type Theory: Path Equality for Guarded Recursion

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    This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-L\"of type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.Comment: 17 pages, to be published in proceedings of CSL 201

    Classical Predicative Logic-Enriched Type Theories

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    A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTTO and LTTO*, which we claim correspond closely to the classical predicative systems of second order arithmetic ACAO and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTTW, which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACAO has also been claimed to correspond to Weyl's foundation. By casting ACAO and ACA as LTTs, we are able to compare them with LTTW. It is a consequence of the work in this paper that LTTW is strictly stronger than ACAO. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.Comment: 49 pages. Accepted for publication in special edition of Annals of Pure and Applied Logic on Computation in Classical Logic. v2: Minor mistakes correcte

    On Irrelevance and Algorithmic Equality in Predicative Type Theory

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    Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To separate static and dynamic code, several static analyses and type systems have been put forward. We consider Pfenning's type theory with irrelevant quantification which is compatible with a type-based notion of equality that respects eta-laws. We extend Pfenning's theory to universes and large eliminations and develop its meta-theory. Subject reduction, normalization and consistency are obtained by a Kripke model over the typed equality judgement. Finally, a type-directed equality algorithm is described whose completeness is proven by a second Kripke model.Comment: 36 pages, superseds the FoSSaCS 2011 paper of the first author, titled "Irrelevance in Type Theory with a Heterogeneous Equality Judgement

    W-types in setoids

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    W-types and their categorical analogue, initial algebras for polynomial endofunctors, are an important tool in predicative systems to replace transfinite recursion on well-orderings. Current arguments to obtain W-types in quotient completions rely on assumptions, like Uniqueness of Identity Proofs, or on constructions that involve recursion into a universe, that limit their applicability to a specific setting. We present an argument, verified in Coq, that instead uses dependent W-types in the underlying type theory to construct W-types in the setoid model. The immediate advantage is to have a proof more type-theoretic in flavour, which directly uses recursion on the underlying W-type to prove initiality. Furthermore, taking place in intensional type theory and not requiring any recursion into a universe, it may be generalised to various categorical quotient completions, with the aim of finding a uniform construction of extensional W-types.Comment: 17 pages, formalised in Coq; v2: added reference to formalisatio

    Categorical structures for deduction

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    We begin by introducing categorized judgemental theories and their calculi as a general framework to present and study deductive systems. As an exemplification of their expressivity, we approach dependent type theory and first-order logic as special kinds of categorized judgemental theories. We believe our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. For first-order logic we offer a deep analysis of structural rules, describing some of their properties, and putting them into context. We then put one of the main constructions introduced, namely that of categorized judgemental dependent type theories, to the test: we frame it in the general context of categorical models for dependent types, describe a few examples, study its properties, and use it to model subtyping and as a tool to prove intrinsic properties hidden in other models. Somehow orthogonally, then, we show a different side as to how categories can help the study of deductive systems: we transport a known model from set-based categories to enriched categories, and use the information naturally encoded into it to describe a theory of fuzzy types. We recover structural rules, observe new phenomena, and study different possible enrichments and their interpretation. We open the discussion to include different takes on the topic of definitional equality
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