{"title":"Nominal Presentation of Cubical Sets Models of Type Theory","authors":"A. Pitts","doi":"10.4230/LIPIcs.TYPES.2014.202","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.202","url":null,"abstract":"The cubical sets model of Homotopy Type Theory introduced by Bezem, Coquand and Huber uses a particular category of presheaves. We show that this presheaf category is equivalent to a category of sets equipped with an action of a monoid of name substitutions for which a finite support property holds. That category is in turn isomorphic to a category of nominal sets equipped with operations for substituting constants 0 and 1 for names. This formulation of cubical sets brings out the potentially useful connection that exists between the homotopical notion of path and the nominal sets notion of name abstraction. The formulation in terms of actions of monoids of name substitutions also encompasses a variant category of cubical sets with diagonals, equivalent to presheaves on Grothendieck's \"smallest test category.\" We show that this category has the pleasant property that path objects given by name abstraction are exponentials with respect to an interval object.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116732957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Extensional Kleene Realizability Semantics for the Minimalist Foundation","authors":"M. Maietti, Samuele Maschio","doi":"10.4230/LIPIcs.TYPES.2014.162","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.162","url":null,"abstract":"We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove that both levels of MF are consistent with the formal Church Thesis CT. Since MF consists of two levels, an intensional one, called mtt, and an extensional one, called emtt, linked by an interpretation, it is enough to build a realizability semantics for the intensional level mtt to get one for the extensional one emtt, too. Moreover, both levels consists of type theories based on versions of Martin-Lof's type theory. Our realizability semantics for mtt is a modification of the realizability semantics by Beeson in 1985 for extensional first order Martin-Lof's type theory with one universe. So it is formalized in Feferman's classical arithmetic theory of inductive definitions, called ID1^. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson's one. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+CT+AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the Axiom of Choice AC on generic types. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-Lof's theory, but it does not validate CT. The semantics we present here seems to be the best approximation of Kleene realizability for the extensional level emtt. Indeed Beeson's semantics is not an option for emtt since AC on generic sets added to it entails the excluded middle.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122801257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The General Universal Property of the Propositional Truncation","authors":"Nicolai Kraus","doi":"10.4230/LIPIcs.TYPES.2014.111","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.111","url":null,"abstract":"In a type-theoretic fibration category in the sense of Shulman (representing a dependent type theory with at least 1, Sigma, Pi, and identity types), we define the type of constant functions from A to B. This involves an infinite tower of coherence conditions, and we therefore need the category to have Reedy limits of diagrams over omega. Our main result is that, if the category further has propositional truncations and satisfies function extensionality, the type of constant function is equivalent to the type ||A|| -> B. \u0000If B is an n-type for a given finite n, the tower of coherence conditions becomes finite and the requirement of nontrivial Reedy limits vanishes. The whole construction can then be carried out in Homotopy Type Theory and generalises the universal property of the truncation. This provides a way to define functions ||A|| -> B if B is not known to be propositional, and it streamlines the common approach of finding a proposition Q with A -> Q and Q -> B.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129122543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Objects and Subtyping in the Lambda-Pi-Calculus Modulo","authors":"Raphaël Cauderlier, Catherine Dubois","doi":"10.4230/LIPIcs.TYPES.2014.47","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.47","url":null,"abstract":"We present a shallow embedding of the Object Calculus of Abadi and Cardelli in the lambda-Pi-calculus modulo, an extension of the lambda-Pi-calculus with rewriting. This embedding may be used as an example of translation of subtyping. We prove this embedding correct with respect to the operational semantics and the type system of the Object Calculus. We implemented a translation tool from the Object Calculus to Dedukti, a type-checker for the lambda-Pi-calculus modulo.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126302159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Calculus of Constructions with Explicit Subtyping","authors":"Ali Assaf","doi":"10.4230/LIPIcs.TYPES.2014.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.27","url":null,"abstract":"The calculus of constructions can be extended with an infinite hierarchy of universes and cumulative subtyping. In this hierarchy, each universe is contained in a higher universe. Subtyping is usually left implicit in the typing rules. We present an alternative version of the calculus of constructions where subtyping is explicit. This new system avoids problems related to coercions and dependent types by using the Tarski style of universes and by introducing additional equations to reflect equality.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123743426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Terminal Semantics for Codata Types in Intensional Martin-Löf Type Theory","authors":"B. Ahrens, Régis Spadotti","doi":"10.4230/LIPIcs.TYPES.2014.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.1","url":null,"abstract":"In this work, we study the notions of relative comonad and comodule over a relative comonad, and use these notions to give a terminal coalgebra semantics for the coinductive type families of streams and of infinite triangular matrices, respectively, in intensional Martin-L\"of type theory. Our results are mechanized in the proof assistant Coq.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132295708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extensionality of lambda-","authors":"A. Polonsky","doi":"10.4230/LIPIcs.TYPES.2014.221","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2014.221","url":null,"abstract":"We prove an extensionality theorem for the \"type-in-type\" dependent type theory with Sigma-types. We suggest that the extensional equality type be identified with the logical equivalence relation on the free term model of type theory.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132997449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Montagovian Generative Lexicon Lambda Ty_n: a Type Theoretical Framework for Natural Language Semantics","authors":"C. Retoré","doi":"10.4230/LIPIcs.TYPES.2013.202","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2013.202","url":null,"abstract":"We present a framework, named the Montagovian generative lexicon, for computing the semantics of natural language sentences, expressed in many-sorted higher order logic. Word meaning is described by several lambda terms of second order lambda calculus (Girard’s system F): the principal lambda term encodes the argument structure, while the other lambda terms implement meaning transfers. The base types include a type for propositions and many types for sorts of a many-sorted logic for expressing restriction of selection. This framework is able to integrate a proper treatment of lexical phenomena into a Montagovian compositional semantics, like the (im)possible arguments of a predicate, and the adaptation of a word meaning to some contexts. Among these adaptations of a word meaning to contexts, ontological inclusions are handled by coercive subtyping, an extension of system F introduced in the present paper. The benefits of this framework for lexical semantics and pragmatics are illustrated on meaning transfers and coercions, on possible and impossible copredication over different senses, on deverbal ambiguities, and on “fictive motion”. Next we show that the compositional treatment of determiners, quantifiers, plurals, and other semantic phenomena is richer in our framework. We then conclude with the linguistic, logical and computational perspectives opened by the Montagovian generative lexicon.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128892356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators","authors":"Danko Ilik, Keiko Nakata","doi":"10.4230/LIPIcs.TYPES.2013.188","DOIUrl":"https://doi.org/10.4230/LIPIcs.TYPES.2013.188","url":null,"abstract":"First, we reconstruct Wim Veldman's result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov's Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov's Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski.","PeriodicalId":131421,"journal":{"name":"Types for Proofs and Programs","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131577727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}