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Documents authored by Shulman, Michael


Document
Parametricity, Automorphisms of the Universe, and Excluded Middle

Authors: Auke B. Booij, Martín H. Escardó, Peter LeFanu Lumsdaine, and Michael Shulman

Published in: LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)


Abstract
It is known that one can construct non-parametric functions by assuming classical axioms. Our work is a converse to that: we prove classical axioms in dependent type theory assuming specific instances of non-parametricity. We also address the interaction between classical axioms and the existence of automorphisms of a type universe. We work over intensional Martin-Löf dependent type theory, and for some results assume further principles including function extensionality, propositional extensionality, propositional truncation, and the univalence axiom.

Cite as

Auke B. Booij, Martín H. Escardó, Peter LeFanu Lumsdaine, and Michael Shulman. Parametricity, Automorphisms of the Universe, and Excluded Middle. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 7:1-7:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{booij_et_al:LIPIcs.TYPES.2016.7,
  author =	{Booij, Auke B. and Escard\'{o}, Mart{\'\i}n H. and Lumsdaine, Peter LeFanu and Shulman, Michael},
  title =	{{Parametricity, Automorphisms of the Universe, and Excluded Middle}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.7},
  URN =		{urn:nbn:de:0030-drops-98554},
  doi =		{10.4230/LIPIcs.TYPES.2016.7},
  annote =	{Keywords: relational parametricity, dependent type theory, univalent foundations, homotopy type theory, excluded middle, classical mathematics, constructive mat}
}
Document
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}
}
Document
The Seifert-van Kampen Theorem in Homotopy Type Theory

Authors: Kuen-Bang Hou (Favonia) and Michael Shulman

Published in: LIPIcs, Volume 62, 25th EACSL Annual Conference on Computer Science Logic (CSL 2016)


Abstract
Homotopy type theory is a recent research area connecting type theory with homotopy theory by interpreting types as spaces. In particular, one can prove and mechanize type-theoretic analogues of homotopy-theoretic theorems, yielding "synthetic homotopy theory". Here we consider the Seifert-van Kampen theorem, which characterizes the loop structure of spaces obtained by gluing. This is useful in homotopy theory because many spaces are constructed by gluing, and the loop structure helps distinguish distinct spaces. The synthetic proof showcases many new characteristics of synthetic homotopy theory, such as the "encode-decode" method, enforced homotopy-invariance, and lack of underlying sets.

Cite as

Kuen-Bang Hou (Favonia) and Michael Shulman. The Seifert-van Kampen Theorem in Homotopy Type Theory. In 25th EACSL Annual Conference on Computer Science Logic (CSL 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 62, pp. 22:1-22:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{hou(favonia)_et_al:LIPIcs.CSL.2016.22,
  author =	{Hou (Favonia), Kuen-Bang and Shulman, Michael},
  title =	{{The Seifert-van Kampen Theorem in Homotopy Type Theory}},
  booktitle =	{25th EACSL Annual Conference on Computer Science Logic (CSL 2016)},
  pages =	{22:1--22:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-022-4},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{62},
  editor =	{Talbot, Jean-Marc and Regnier, Laurent},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2016.22},
  URN =		{urn:nbn:de:0030-drops-65626},
  doi =		{10.4230/LIPIcs.CSL.2016.22},
  annote =	{Keywords: homotopy type theory, fundamental group, homotopy pushout, mechanized reasoning}
}
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