Guarded Cubical Type Theory

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 c...

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Published in:	Journal of automated reasoning Vol. 63; no. 2; pp. 211 - 253
Main Authors:	Birkedal, Lars, Bizjak, Aleš, Clouston, Ranald, Grathwohl, Hans Bugge, Spitters, Bas, Vezzosi, Andrea
Format:	Journal Article
Language:	English
Published:	Dordrecht Springer Netherlands 01.08.2019 Springer Nature B.V
Subjects:	Artificial Intelligence Computation Computer Science Cubical type theory Guarded recursion Homotopy type theory Mathematical Logic and Formal Languages Mathematical Logic and Foundations Number theory Semantics Symbolic and Algebraic Manipulation Guarded recursion Homotopy type theory Cubical type theory
ISSN:	0168-7433, 1573-0670, 1573-0670
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Abstract	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–Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory ( GCTT ), 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. We show that CTT can be given semantics in presheaves on C × D , where C is the cube category, and D is any small category with an initial object. We then show that the category of presheaves on C × ω provides semantics for GCTT .
AbstractList	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–Löf type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory ( GCTT ), 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. We show that CTT can be given semantics in presheaves on C × D , where C is the cube category, and D is any small category with an initial object. We then show that the category of presheaves on C × ω provides semantics for GCTT . This paper improves the treatment of equality in guarded dependent type theory ((Formula presented.)), by combining it with cubical type theory ((Formula presented.)). (Formula presented.) 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 (Formula presented.) with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. (Formula presented.) is a variation of Martin–Löf type theory in which the identity type is replaced by abstract paths between terms. (Formula presented.) provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory ((Formula presented.)), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of (Formula presented.) 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. We show that (Formula presented.) can be given semantics in presheaves on (Formula presented.), where (Formula presented.) is the cube category, and (Formula presented.) is any small category with an initial object. We then show that the category of presheaves on (Formula presented.) provides semantics for (Formula presented.). This paper improves the treatment of equality in guarded dependent type theory (\[\mathsf {GDTT}\]), by combining it with cubical type theory (\[\mathsf {CTT}\]). \[\mathsf {GDTT}\] 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 \[\mathsf {GDTT}\] with decidable type checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. \[\mathsf {CTT}\] is a variation of Martin–Löf type theory in which the identity type is replaced by abstract paths between terms. \[\mathsf {CTT}\] provides a computational interpretation of functional extensionality, enjoys canonicity for the natural numbers type, and is conjectured to support decidable type-checking. Our new type theory, guarded cubical type theory (\[\mathsf {GCTT}\]), provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of \[\mathsf {CTT}\] 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. We show that \[\mathsf {CTT}\] can be given semantics in presheaves on \[\mathcal {C}\times \mathbb {D}\], where \[\mathcal {C}\] is the cube category, and \[\mathbb {D}\] is any small category with an initial object. We then show that the category of presheaves on \[\mathcal {C}\times \omega \] provides semantics for \[\mathsf {GCTT}\].
Author	Grathwohl, Hans Bugge Vezzosi, Andrea Birkedal, Lars Clouston, Ranald Spitters, Bas Bizjak, Aleš
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References_xml	– reference: McBrideCPatersonRApplicative programming with effectsJ. Funct. Program.200818111310.1017/S09567968070063261128.68020 – reference: Mac LaneSMoerdijkISheaves in Geometry and Logic1992BerlinSpringer0822.18001 – reference: Spitters, B.: Cubical sets as a classifying topos. In: TYPES (2015) – reference: The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations for Mathematics. Institute for Advanced Study (2013) – reference: Coquand, T.: Internal version of the uniform Kan filling condition. http://www.cse.chalmers.se/~coquand/shape.pdf (2015). Accessed 13 June 2018 – reference: Kapulkin, C., Lumsdaine, P.L.: The simplicial model of univalent foundations (after Voevodsky). arXiv:1211.2851 (2012) – reference: Phoa, W.: An introduction to fibrations, topos theory, the effective topos and modest sets. Technical Report ECS-LFCS-92-208, LFCS, University of Edinburgh (1992) – reference: Orton, I., Pitts, A.M.: Axioms for modelling cubical type theory in a topos. In: CSL (2016) – reference: Voevodsky, V.: Martin-Lof identity types in the C-systems defined by a universe category. arXiv:1505.06446 (2015) – reference: Birkedal, L., Reus, B., Schwinghammer, J., Støvring, K., Thamsborg, J., Yang, H.: Step-indexed Kripke models over recursive worlds. In: POPL, pp. 119–132 (2011) – reference: CornishWFowlerPCoproducts of de Morgan algebrasBull. Aust. Math. Soc.197716111343490710.1017/S00049727000229660329.06005 – reference: VickersSAielloMPratt-HartmannIEvan BenthemJFAKLocales and toposes as spacesHandbook of Spatial Logics2007BerlinSpringer42949610.1007/978-1-4020-5587-4_8 – reference: Bizjak, A., Møgelberg, R.E.: A model of guarded recursion with clock synchronisation. In: MFPS, pp. 83–101 (2015) – reference: Abel, A., Vezzosi, A.: A formalized proof of strong normalization for guarded recursive types. In: APLAS, pp. 140–158 (2014) – reference: Altenkirch, T., McBride, C., Swierstra, W.: Observational equality, now! In: PLPV, pp. 57–68 (2007) – reference: Bizjak, A., Grathwohl, H.B., Clouston, R., Møgelberg, R.E., Birkedal, L.: Guarded dependent type theory with coinductive types. In: FoSSaCS, pp. 20–35 (2016) – reference: Birkedal, L., Rasmus, E.M.: Intensional type theory with guarded recursive types qua fixed points on universes. In: LICS, pp. 213–222 (2013) – reference: Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical type theory: a constructive interpretation of the univalence axiom. In: Post-proceedings of the 21st International Conference on Types for Proofs and Programs, TYPES 2015 (2016) – reference: Dybjer, P.: Internal type theory. 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Snippet	This paper improves the treatment of equality in guarded dependent type theory ( GDTT ), by combining it with cubical type theory ( CTT ). GDTT is an... This paper improves the treatment of equality in guarded dependent type theory (\[\mathsf {GDTT}\]), by combining it with cubical type theory (\[\mathsf... This paper improves the treatment of equality in guarded dependent type theory ((Formula presented.)), by combining it with cubical type theory ((Formula...
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SubjectTerms	Artificial Intelligence Computation Computer Science Cubical type theory Guarded recursion Homotopy type theory Mathematical Logic and Formal Languages Mathematical Logic and Foundations Number theory Semantics Symbolic and Algebraic Manipulation
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Title	Guarded Cubical Type Theory
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