15 Search Results for "Escardo, Martin"


Document
Inductive Continuity via Brouwer Trees

Authors: Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)


Abstract
Continuity is a key principle of intuitionistic logic that is generally accepted by constructivists but is inconsistent with classical logic. Most commonly, continuity states that a function from the Baire space to numbers, only needs approximations of the points in the Baire space to compute. More recently, another formulation of the continuity principle was put forward. It states that for any function F from the Baire space to numbers, there exists a (dialogue) tree that contains the values of F at its leaves and such that the modulus of F at each point of the Baire space is given by the length of the corresponding branch in the tree. In this paper we provide the first internalization of this "inductive" continuity principle within a computational setting. Concretely, we present a class of intuitionistic theories that validate this formulation of continuity thanks to computations that construct such dialogue trees internally to the theories using effectful computations. We further demonstrate that this inductive continuity principle implies other forms of continuity principles.

Cite as

Liron Cohen, Bruno da Rocha Paiva, Vincent Rahli, and Ayberk Tosun. Inductive Continuity via Brouwer Trees. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 37:1-37:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{cohen_et_al:LIPIcs.MFCS.2023.37,
  author =	{Cohen, Liron and da Rocha Paiva, Bruno and Rahli, Vincent and Tosun, Ayberk},
  title =	{{Inductive Continuity via Brouwer Trees}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{37:1--37:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.37},
  URN =		{urn:nbn:de:0030-drops-185718},
  doi =		{10.4230/LIPIcs.MFCS.2023.37},
  annote =	{Keywords: Continuity, Dialogue trees, Stateful computations, Intuitionistic Logic, Extensional Type Theory, Constructive Type Theory, Realizability, Theorem proving, Agda}
}
Document
Type Theory with Explicit Universe Polymorphism

Authors: Marc Bezem, Thierry Coquand, Peter Dybjer, and Martín Escardó

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)


Abstract
The aim of this paper is to refine and extend proposals by Sozeau and Tabareau and by Voevodsky for universe polymorphism in type theory. In those systems judgments can depend on explicit constraints between universe levels. We here present a system where we also have products indexed by universe levels and by constraints. Our theory has judgments for internal universe levels, built up from level variables by a successor operation and a binary supremum operation, and also judgments for equality of universe levels.

Cite as

Marc Bezem, Thierry Coquand, Peter Dybjer, and Martín Escardó. Type Theory with Explicit Universe Polymorphism. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{bezem_et_al:LIPIcs.TYPES.2022.13,
  author =	{Bezem, Marc and Coquand, Thierry and Dybjer, Peter and Escard\'{o}, Mart{\'\i}n},
  title =	{{Type Theory with Explicit Universe Polymorphism}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{13:1--13:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.13},
  URN =		{urn:nbn:de:0030-drops-184564},
  doi =		{10.4230/LIPIcs.TYPES.2022.13},
  annote =	{Keywords: type theory, universes in type theory, universe polymorphism, level-indexed products, constraint-indexed products}
}
Document
A Univalent Formalization of Constructive Affine Schemes

Authors: Max Zeuner and Anders Mörtberg

Published in: LIPIcs, Volume 269, 28th International Conference on Types for Proofs and Programs (TYPES 2022)


Abstract
We present a formalization of constructive affine schemes in the Cubical Agda proof assistant. This development is not only fully constructive and predicative, it also makes crucial use of univalence. By now schemes have been formalized in various proof assistants. However, most existing formalizations follow the inherently non-constructive approach of Hartshorne’s classic "Algebraic Geometry" textbook, for which the construction of the so-called structure sheaf is rather straightforwardly formalizable and works the same with or without univalence. We follow an alternative approach that uses a point-free description of the constructive counterpart of the Zariski spectrum called the Zariski lattice and proceeds by defining the structure sheaf on formal basic opens and then lift it to the whole lattice. This general strategy is used in a plethora of textbooks, but formalizing it has proved tricky. The main result of this paper is that with the help of the univalence principle we can make this "lift from basis" strategy formal and obtain a fully formalized account of constructive affine schemes.

Cite as

Max Zeuner and Anders Mörtberg. A Univalent Formalization of Constructive Affine Schemes. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{zeuner_et_al:LIPIcs.TYPES.2022.14,
  author =	{Zeuner, Max and M\"{o}rtberg, Anders},
  title =	{{A Univalent Formalization of Constructive Affine Schemes}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{14:1--14:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-285-3},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{269},
  editor =	{Kesner, Delia and P\'{e}drot, Pierre-Marie},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2022.14},
  URN =		{urn:nbn:de:0030-drops-184574},
  doi =		{10.4230/LIPIcs.TYPES.2022.14},
  annote =	{Keywords: Affine Schemes, Homotopy Type Theory and Univalent Foundations, Cubical Agda, Constructive Mathematics}
}
Document
Sheaf Semantics of Termination-Insensitive Noninterference

Authors: Jonathan Sterling and Robert Harper

Published in: LIPIcs, Volume 228, 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)


Abstract
We propose a new sheaf semantics for secure information flow over a space of abstract behaviors, based on synthetic domain theory: security classes are open/closed partitions, types are sheaves, and redaction of sensitive information corresponds to restricting a sheaf to a closed subspace. Our security-aware computational model satisfies termination-insensitive noninterference automatically, and therefore constitutes an intrinsic alternative to state of the art extrinsic/relational models of noninterference. Our semantics is the latest application of Sterling and Harper’s recent re-interpretation of phase distinctions and noninterference in programming languages in terms of Artin gluing and topos-theoretic open/closed modalities. Prior applications include parametricity for ML modules, the proof of normalization for cubical type theory by Sterling and Angiuli, and the cost-aware logical framework of Niu et al. In this paper we employ the phase distinction perspective twice: first to reconstruct the syntax and semantics of secure information flow as a lattice of phase distinctions between "higher" and "lower" security, and second to verify the computational adequacy of our sheaf semantics with respect to a version of Abadi et al.’s dependency core calculus to which we have added a construct for declassifying termination channels.

Cite as

Jonathan Sterling and Robert Harper. Sheaf Semantics of Termination-Insensitive Noninterference. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{sterling_et_al:LIPIcs.FSCD.2022.5,
  author =	{Sterling, Jonathan and Harper, Robert},
  title =	{{Sheaf Semantics of Termination-Insensitive Noninterference}},
  booktitle =	{7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022)},
  pages =	{5:1--5:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-233-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{228},
  editor =	{Felty, Amy P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2022.5},
  URN =		{urn:nbn:de:0030-drops-162869},
  doi =		{10.4230/LIPIcs.FSCD.2022.5},
  annote =	{Keywords: information flow, noninterference, denotational semantics, phase distinction, Artin gluing, modal type theory, topos theory, synthetic domain theory, synthetic Tait computability}
}
Document
Gardening with the Pythia A Model of Continuity in a Dependent Setting

Authors: Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)


Abstract
We generalize to a rich dependent type theory a proof originally developed by Escardó that all System 𝚃 functionals are continuous. It relies on the definition of a syntactic model of Baclofen Type Theory, a type theory where dependent elimination must be strict, into the Calculus of Inductive Constructions. The model is given by three translations: the axiom translation, that adds an oracle to the context; the branching translation, based on the dialogue monad, turning every type into a tree; and finally, a layer of algebraic binary parametricity, binding together the two translations. In the resulting type theory, every function f : (ℕ → ℕ) → ℕ is externally continuous.

Cite as

Martin Baillon, Assia Mahboubi, and Pierre-Marie Pédrot. Gardening with the Pythia A Model of Continuity in a Dependent Setting. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{baillon_et_al:LIPIcs.CSL.2022.5,
  author =	{Baillon, Martin and Mahboubi, Assia and P\'{e}drot, Pierre-Marie},
  title =	{{Gardening with the Pythia A Model of Continuity in a Dependent Setting}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.5},
  URN =		{urn:nbn:de:0030-drops-157256},
  doi =		{10.4230/LIPIcs.CSL.2022.5},
  annote =	{Keywords: Type theory, continuity, syntactic model}
}
Document
Realising Intensional S4 and GL Modalities

Authors: Liang-Ting Chen and Hsiang-Shang Ko

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)


Abstract
There have been investigations into type-theoretic foundations for metaprogramming, notably Davies and Pfenning’s (2001) treatment in S4 modal logic, where code evaluating to values of type A is given the modal type Code A (□A in the original paper). Recently Kavvos (2017) extended PCF with Code A and intensional recursion, understood as the deductive form of the GL (Gödel-Löb) axiom in provability logic, but the resulting type system is logically inconsistent. Inspired by staged computation, we observe that a term of type Code A is, in general, code to be evaluated in a next stage, whereas S4 modal type theory is a special case where code can be evaluated in the current stage, and the two types of code should be discriminated. Consequently, we use two separate modalities ⊠ and □ to model S4 and GL respectively in a unified categorical framework while retaining logical consistency. Following Kavvos’ (2017) novel approach to the semantics of intensionality, we interpret the two modalities in the P-category of assemblies and trackable maps. For the GL modality □ in particular, we use guarded type theory to articulate what it means by a “next” stage and to model intensional recursion by guarded recursion together with Kleene’s second recursion theorem. Besides validating the S4 and GL axioms, our model better captures the essence of intensionality by refuting congruence (so that two extensionally equal terms may not be intensionally equal) and internal quoting (both A → □A and A → ⊠A). Our results are developed in (guarded) homotopy type theory and formalised in Agda.

Cite as

Liang-Ting Chen and Hsiang-Shang Ko. Realising Intensional S4 and GL Modalities. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{chen_et_al:LIPIcs.CSL.2022.14,
  author =	{Chen, Liang-Ting and Ko, Hsiang-Shang},
  title =	{{Realising Intensional S4 and GL Modalities}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.14},
  URN =		{urn:nbn:de:0030-drops-157341},
  doi =		{10.4230/LIPIcs.CSL.2022.14},
  annote =	{Keywords: provability, guarded recursion, realisability, modal types, metaprogramming}
}
Document
Predicative Aspects of Order Theory in Univalent Foundations

Authors: Tom de Jong and Martín Hötzel Escardó

Published in: LIPIcs, Volume 195, 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)


Abstract
We investigate predicative aspects of order theory in constructive univalent foundations. By predicative and constructive, we respectively mean that we do not assume Voevodsky’s propositional resizing axioms or excluded middle. Our work complements existing work on predicative mathematics by exploring what cannot be done predicatively in univalent foundations. Our first main result is that nontrivial (directed or bounded) complete posets are necessarily large. That is, if such a nontrivial poset is small, then weak propositional resizing holds. It is possible to derive full propositional resizing if we strengthen nontriviality to positivity. The distinction between nontriviality and positivity is analogous to the distinction between nonemptiness and inhabitedness. We prove our results for a general class of posets, which includes directed complete posets, bounded complete posets and sup-lattices, using a technical notion of a δ_V-complete poset. We also show that nontrivial locally small δ_V-complete posets necessarily lack decidable equality. Specifically, we derive weak excluded middle from assuming a nontrivial locally small δ_V-complete poset with decidable equality. Moreover, if we assume positivity instead of nontriviality, then we can derive full excluded middle. Secondly, we show that each of Zorn’s lemma, Tarski’s greatest fixed point theorem and Pataraia’s lemma implies propositional resizing. Hence, these principles are inherently impredicative and a predicative development of order theory must therefore do without them. Finally, we clarify, in our predicative setting, the relation between the traditional definition of sup-lattice that requires suprema for all subsets and our definition that asks for suprema of all small families.

Cite as

Tom de Jong and Martín Hötzel Escardó. Predicative Aspects of Order Theory in Univalent Foundations. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{dejong_et_al:LIPIcs.FSCD.2021.8,
  author =	{de Jong, Tom and Escard\'{o}, Mart{\'\i}n H\"{o}tzel},
  title =	{{Predicative Aspects of Order Theory in Univalent Foundations}},
  booktitle =	{6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021)},
  pages =	{8:1--8:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-191-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{195},
  editor =	{Kobayashi, Naoki},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2021.8},
  URN =		{urn:nbn:de:0030-drops-142461},
  doi =		{10.4230/LIPIcs.FSCD.2021.8},
  annote =	{Keywords: order theory, constructivity, predicativity, univalent foundations}
}
Document
Domain Theory in Constructive and Predicative Univalent Foundations

Authors: Tom de Jong and Martín Hötzel Escardó

Published in: LIPIcs, Volume 183, 29th EACSL Annual Conference on Computer Science Logic (CSL 2021)


Abstract
We develop domain theory in constructive univalent foundations without Voevodsky’s resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott’s D_∞ model of the untyped λ-calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe U₁ and suprema of directed families with indexing type in the first universe U₀. Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott’s D_∞ again gives a large, locally small, algebraic dcpo with small directed suprema.

Cite as

Tom de Jong and Martín Hötzel Escardó. Domain Theory in Constructive and Predicative Univalent Foundations. In 29th EACSL Annual Conference on Computer Science Logic (CSL 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 183, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{dejong_et_al:LIPIcs.CSL.2021.28,
  author =	{de Jong, Tom and Escard\'{o}, Mart{\'\i}n H\"{o}tzel},
  title =	{{Domain Theory in Constructive and Predicative Univalent Foundations}},
  booktitle =	{29th EACSL Annual Conference on Computer Science Logic (CSL 2021)},
  pages =	{28:1--28:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-175-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{183},
  editor =	{Baier, Christel and Goubault-Larrecq, Jean},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2021.28},
  URN =		{urn:nbn:de:0030-drops-134625},
  doi =		{10.4230/LIPIcs.CSL.2021.28},
  annote =	{Keywords: domain theory, constructivity, predicativity, univalent foundations}
}
Document
A Gentzen-Style Monadic Translation of Gödel’s System T

Authors: Chuangjie Xu

Published in: LIPIcs, Volume 167, 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)


Abstract
We introduce a syntactic translation of Gödel’s System 𝖳 parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzen’s negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of 𝖳-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.

Cite as

Chuangjie Xu. A Gentzen-Style Monadic Translation of Gödel’s System T. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{xu:LIPIcs.FSCD.2020.25,
  author =	{Xu, Chuangjie},
  title =	{{A Gentzen-Style Monadic Translation of G\"{o}del’s System T}},
  booktitle =	{5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-155-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{167},
  editor =	{Ariola, Zena M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.25},
  URN =		{urn:nbn:de:0030-drops-123472},
  doi =		{10.4230/LIPIcs.FSCD.2020.25},
  annote =	{Keywords: monadic translation, G\"{o}del’s System T, logical relation, negative translation, majorizability, continuity, bar recursion, Agda}
}
Document
Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341)

Authors: Andrej Bauer, Martín Escardó, Peter L. Lumsdaine, and Assia Mahboubi

Published in: Dagstuhl Reports, Volume 8, Issue 8 (2019)


Abstract
Formalized mathematics is mathematical knowledge (definitions, theorems, and proofs) represented in digital form suitable for computer processing. The central goal of this seminar was to identify the theoretical advances and practical improvements needed in the area of formalized mathematics, in order to make it a mature technology, truly useful to a larger community of students and researchers in mathematics. During the seminar, various software systems for formalization were compared, and potential improvements to existing systems were investigated. There have also been discussions on the representation of algebraic structures in formalization systems.

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Andrej Bauer, Martín Escardó, Peter L. Lumsdaine, and Assia Mahboubi. Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341). In Dagstuhl Reports, Volume 8, Issue 8, pp. 130-145, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@Article{bauer_et_al:DagRep.8.8.130,
  author =	{Bauer, Andrej and Escard\'{o}, Mart{\'\i}n and Lumsdaine, Peter L. and Mahboubi, Assia},
  title =	{{Formalization of Mathematics in Type Theory (Dagstuhl Seminar 18341)}},
  pages =	{130--145},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2019},
  volume =	{8},
  number =	{8},
  editor =	{Bauer, Andrej and Escard\'{o}, Mart{\'\i}n and Lumsdaine, Peter L. and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.8.8.130},
  URN =		{urn:nbn:de:0030-drops-102370},
  doi =		{10.4230/DagRep.8.8.130},
  annote =	{Keywords: formal methods, formalized mathematics, proof assistant, type theory}
}
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-dev.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
Partial Elements and Recursion via Dominances in Univalent Type Theory

Authors: Martín H. Escardó and Cory M. Knapp

Published in: LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)


Abstract
We begin by revisiting partiality in univalent type theory via the notion of dominance. We then perform first steps in constructive computability theory, discussing the consequences of working with computability as property or structure, without assuming countable choice or Markov’s principle. A guiding question is what, if any, notion of partial function allows the proposition “all partial functions on natural numbers are Turing computable” to be consistent.

Cite as

Martín H. Escardó and Cory M. Knapp. Partial Elements and Recursion via Dominances in Univalent Type Theory. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{escardo_et_al:LIPIcs.CSL.2017.21,
  author =	{Escard\'{o}, Mart{\'\i}n H. and Knapp, Cory M.},
  title =	{{Partial Elements and Recursion via Dominances in Univalent Type Theory}},
  booktitle =	{26th EACSL Annual Conference on Computer Science Logic (CSL 2017)},
  pages =	{21:1--21:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-045-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{82},
  editor =	{Goranko, Valentin and Dam, Mads},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.21},
  URN =		{urn:nbn:de:0030-drops-76822},
  doi =		{10.4230/LIPIcs.CSL.2017.21},
  annote =	{Keywords: univalent type theory, homotopy type theory, partial function, dominance, recursion theory, computability theory}
}
Document
The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation

Authors: Martín Hötzel Escardó and Chuangjie Xu

Published in: LIPIcs, Volume 38, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)


Abstract
If all functions (N -> N) -> N are continuous then 0 = 1. We establish this in intensional (and hence also in extensional) intuitionistic dependent-type theories, with existence in the formulation of continuity expressed as a Sigma type via the Curry-Howard interpretation. But with an intuitionistic notion of anonymous existence, defined as the propositional truncation of Sigma, it is consistent that all such functions are continuous. A model is Johnstone’s topological topos. On the other hand, any of these two intuitionistic conceptions of existence give the same, consistent, notion of uniform continuity for functions (N -> 2) -> N, again valid in the topological topos. It is open whether the consistency of (uniform) continuity extends to homotopy type theory. The theorems of type theory informally proved here are also formally proved in Agda, but the development presented here is self-contained and doesn't show Agda code.

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Martín Hötzel Escardó and Chuangjie Xu. The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation. In 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 38, pp. 153-164, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{hotzelescardo_et_al:LIPIcs.TLCA.2015.153,
  author =	{H\"{o}tzel Escard\'{o}, Mart{\'\i}n and Xu, Chuangjie},
  title =	{{The Inconsistency of a Brouwerian Continuity Principle with the Curry–Howard Interpretation}},
  booktitle =	{13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015)},
  pages =	{153--164},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-87-3},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{38},
  editor =	{Altenkirch, Thorsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.TLCA.2015.153},
  URN =		{urn:nbn:de:0030-drops-51618},
  doi =		{10.4230/LIPIcs.TLCA.2015.153},
  annote =	{Keywords: Dependent type, intensional Martin-L\"{o}f type theory, Curry-Howard interpretation, constructive mathematics, Brouwerian continuity axioms, anonymous exi}
}
Document
System T and the Product of Selection Functions

Authors: Martin Escardo, Paulo Oliva, and Thomas Powell

Published in: LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)


Abstract
We show that the finite product of selection functions (for all finite types) is primitive recursively equivalent to Goedel's higher-type recursor (for all finite types). The correspondence is shown to hold for similar restricted fragments of both systems: The recursor for type level n+1 is primitive recursively equivalent to the finite product of selection functions of type level n. Whereas the recursor directly interprets induction, we show that other classical arithmetical principles such as bounded collection and finite choice are more naturally interpreted via the product of selection functions.

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Martin Escardo, Paulo Oliva, and Thomas Powell. System T and the Product of Selection Functions. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 233-247, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)


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@InProceedings{escardo_et_al:LIPIcs.CSL.2011.233,
  author =	{Escardo, Martin and Oliva, Paulo and Powell, Thomas},
  title =	{{System T and the Product of Selection Functions}},
  booktitle =	{Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL},
  pages =	{233--247},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-32-3},
  ISSN =	{1868-8969},
  year =	{2011},
  volume =	{12},
  editor =	{Bezem, Marc},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.233},
  URN =		{urn:nbn:de:0030-drops-32341},
  doi =		{10.4230/LIPIcs.CSL.2011.233},
  annote =	{Keywords: primitive recursion, product of selection functions, finite choice, dialectica interpretation}
}
Document
Tutorial
Theory and Practice of Higher-type Computation (Tutorial)

Authors: Martin Escardó

Published in: OASIcs, Volume 11, 6th International Conference on Computability and Complexity in Analysis (CCA'09) (2009)


Abstract
In higher-type computation, established by Kleene and Kreisel in the late 1950's (independently), one works with the data types obtained from the discrete natural numbers by closing under finite products and function spaces. For the theory of higher-type programming languages, it is natural to work with a corresponding hierarchy, or type structure, of domains, identified by Ershov and Scott in the late 1960's (again independently). The Kleene-Kreisel and Ershov-Scott hierarchies account for total and partial computation respectively. In this tutorial I'll explain the theory and practice of higher-type computation and programming languages, and develop old and new applications. From a theoretical point of view, I'll present Kleene-Kreisel spaces and Ershov-Scott domains, and relate the two. Moreover, I'll discuss common generalizations, chiefly QCB spaces and equilogical spaces, which admit further useful closure properties, and their relationship to TTE (Schroeder, Simpson. Scott, Bauer, Weihrauch and many others). I'll also present a natural higher-type model of computation/programming language, namely PCF (Platek, Scott, Plotkin). From a practical point of view, I'll introduce a fragment of the language Haskell as a faithful implementation of PCF. Moreover, I'll develop and run several examples (and prove theorems about them), pertaining to (i) exhaustive search of infinite sets in finite time in particular Ulrich Berger's algorithm and generalizations), and (ii) computation with real numbers (in particular Alex Simpson's integration algorithm and generalizations).

Cite as

Martin Escardó. Theory and Practice of Higher-type Computation (Tutorial). In 6th International Conference on Computability and Complexity in Analysis (CCA'09). Open Access Series in Informatics (OASIcs), Volume 11, p. 21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)


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@InProceedings{escardo:OASIcs.CCA.2009.2254,
  author =	{Escard\'{o}, Martin},
  title =	{{Theory and Practice of Higher-type Computation}},
  booktitle =	{6th International Conference on Computability and Complexity in Analysis (CCA'09)},
  pages =	{21--21},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-939897-12-5},
  ISSN =	{2190-6807},
  year =	{2009},
  volume =	{11},
  editor =	{Bauer, Andrej and Hertling, Peter and Ko, Ker-I},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/OASIcs.CCA.2009.2254},
  URN =		{urn:nbn:de:0030-drops-22540},
  doi =		{10.4230/OASIcs.CCA.2009.2254},
  annote =	{Keywords: Higher-type computation, domain theory, Kleene-Kreisel spaces, Ershov-Scott domains, QCB spaces, equilogical spaces, PCF}
}
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