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Documents authored by Licata, Daniel R.

Document
DOI: 10.4230/LIPIcs.FSCD.2018.22
Internal Universes in Models of Homotopy Type Theory

Authors: Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract
We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mörtberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.
Cite as

Daniel R. Licata, Ian Orton, Andrew M. Pitts, and Bas Spitters. Internal Universes in Models of Homotopy Type Theory. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{licata_et_al:LIPIcs.FSCD.2018.22,
  author =	{Licata, Daniel R. and Orton, Ian and Pitts, Andrew M. and Spitters, Bas},
  title =	{{Internal Universes in Models of Homotopy Type Theory}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.22},
  URN =		{urn:nbn:de:0030-drops-91929},
  doi =		{10.4230/LIPIcs.FSCD.2018.22},
  annote =	{Keywords: cubical sets, dependent type theory, homotopy type theory, internal language, modalities, univalent foundations, universes}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2018.24
Call-by-Name Gradual Type Theory

Authors: Max S. New and Daniel R. Licata

Published in: LIPIcs, Volume 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

Abstract
We present gradual type theory, a logic and type theory for call-by-name 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. These dynamism judgements build on prior work in blame calculi and on the "gradual guarantee" theorem of gradual typing. Combined with the ordinary extensionality (eta) 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 non-standard 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 2-category representing non-gradual types and programs, which is a semantic analogue of the interpretation of gradual typing using contracts, and use it to build some concrete domain-theoretic models of gradual typing.
Cite as

Max S. New and Daniel R. Licata. Call-by-Name Gradual Type Theory. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{new_et_al:LIPIcs.FSCD.2018.24,
  author =	{New, Max S. and Licata, Daniel R.},
  title =	{{Call-by-Name Gradual Type Theory}},
  booktitle =	{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)},
  pages =	{24:1--24:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-077-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{108},
  editor =	{Kirchner, H\'{e}l\`{e}ne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.24},
  URN =		{urn:nbn:de:0030-drops-91944},
  doi =		{10.4230/LIPIcs.FSCD.2018.24},
  annote =	{Keywords: Gradual Typing, Type Systems, Program Logics, Category Theory, Denotational Semantics}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2017.25
A Fibrational Framework for Substructural and Modal Logics

Authors: Daniel R. Licata, Michael Shulman, and Mitchell Riley

Published in: LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

Abstract
We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones.
Cite as

Daniel R. Licata, Michael Shulman, and Mitchell Riley. A Fibrational Framework for Substructural and Modal Logics. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 25:1-25:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{licata_et_al:LIPIcs.FSCD.2017.25,
  author =	{Licata, Daniel R. and Shulman, Michael and Riley, Mitchell},
  title =	{{A Fibrational Framework for Substructural and Modal Logics}},
  booktitle =	{2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)},
  pages =	{25:1--25:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-047-7},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{84},
  editor =	{Miller, Dale},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.25},
  URN =		{urn:nbn:de:0030-drops-77400},
  doi =		{10.4230/LIPIcs.FSCD.2017.25},
  annote =	{Keywords: type theory, modal logic, substructural logic, homotopy type theory}
}
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