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Volume

LIPIcs, Volume 130

24th International Conference on Types for Proofs and Programs (TYPES 2018)

TYPES 2018, June 18-21, 2018, Braga, Portugal

Editors: Peter Dybjer, José Espírito Santo, and Luís Pinto

Document

DOI: 10.4230/LIPIcs.FSCD.2024.10

Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics

Authors: Ambrus Kaposi and Szumi Xie

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Programming languages can be defined from the concrete to the abstract by abstract syntax trees, well-scoped syntax, well-typed (intrinsic) syntax, algebraic syntax (well-typed syntax quotiented by conversion). Another aspect is the representation of binding structure for which nominal approaches, De Bruijn indices/levels and higher order abstract syntax (HOAS) are available. In HOAS, binders are given by the function space of an internal language of presheaves. In this paper, we show how to combine the algebraic approach with the HOAS approach: following Uemura, we define languages as second-order generalised algebraic theories (SOGATs). Through a series of examples we show that non-substructural languages can be naturally defined as SOGATs. We give a formal definition of SOGAT signatures (using the syntax of a particular SOGAT) and define two translations from SOGAT signatures to GAT signatures (signatures for quotient inductive-inductive types), based on parallel and single substitutions, respectively.

Cite as

Ambrus Kaposi and Szumi Xie. Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kaposi_et_al:LIPIcs.FSCD.2024.10,
  author =	{Kaposi, Ambrus and Xie, Szumi},
  title =	{{Second-Order Generalised Algebraic Theories: Signatures and First-Order Semantics}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{10:1--10:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.10},
  URN =		{urn:nbn:de:0030-drops-203396},
  doi =		{10.4230/LIPIcs.FSCD.2024.10},
  annote =	{Keywords: Type theory, universal algebra, inductive types, quotient inductive types, higher-order abstract syntax, logical framework}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.14

Two-Dimensional Kripke Semantics I: Presheaves

Authors: G. A. Kavvos

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

The study of modal logic has witnessed tremendous development following the introduction of Kripke semantics. However, recent developments in programming languages and type theory have led to a second way of studying modalities, namely through their categorical semantics. We show how the two correspond.

Cite as

G. A. Kavvos. Two-Dimensional Kripke Semantics I: Presheaves. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 14:1-14:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kavvos:LIPIcs.FSCD.2024.14,
  author =	{Kavvos, G. A.},
  title =	{{Two-Dimensional Kripke Semantics I: Presheaves}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{14:1--14:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.14},
  URN =		{urn:nbn:de:0030-drops-203438},
  doi =		{10.4230/LIPIcs.FSCD.2024.14},
  annote =	{Keywords: modal logic, categorical semantics, Kripke semantics, duality, open maps}
}

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2024.130

Function Spaces for Orbit-Finite Sets

Authors: Mikołaj Bojańczyk, Lê Thành Dũng (Tito) Nguyễn, and Rafał Stefański

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

Orbit-finite sets are a generalisation of finite sets, and as such support many operations allowed for finite sets, such as pairing, quotienting, or taking subsets. However, they do not support function spaces, i.e. if X and Y are orbit-finite sets, then the space of finitely supported functions from X to Y is not orbit-finite. We propose a solution to this problem inspired by linear logic.

Cite as

Mikołaj Bojańczyk, Lê Thành Dũng (Tito) Nguyễn, and Rafał Stefański. Function Spaces for Orbit-Finite Sets. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 130:1-130:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bojanczyk_et_al:LIPIcs.ICALP.2024.130,
  author =	{Boja\'{n}czyk, Miko{\l}aj and Nguy\~{ê}n, L\^{e} Th\`{a}nh D\~{u}ng (Tito) and Stefa\'{n}ski, Rafa{\l}},
  title =	{{Function Spaces for Orbit-Finite Sets}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{130:1--130:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.130},
  URN =		{urn:nbn:de:0030-drops-202730},
  doi =		{10.4230/LIPIcs.ICALP.2024.130},
  annote =	{Keywords: Orbit-finite sets, automata, linear types, game semantics}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2022.13

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.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

DOI: 10.4230/LIPIcs.TYPES.2022.14

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.

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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.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

DOI: 10.4230/LIPIcs.TYPES.2019.8

Coherence for Monoidal Groupoids in HoTT

Authors: Stefano Piceghello

Published in: LIPIcs, Volume 175, 25th International Conference on Types for Proofs and Programs (TYPES 2019)

Abstract

We present a proof of coherence for monoidal groupoids in homotopy type theory. An important role in the formulation and in the proof of coherence is played by groupoids with a free monoidal structure; these can be represented by 1-truncated higher inductive types, with constructors freely generating their defining objects, natural isomorphisms and commutative diagrams. All results included in this paper have been formalised in the proof assistant Coq.

Cite as

Stefano Piceghello. Coherence for Monoidal Groupoids in HoTT. In 25th International Conference on Types for Proofs and Programs (TYPES 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 175, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{piceghello:LIPIcs.TYPES.2019.8,
  author =	{Piceghello, Stefano},
  title =	{{Coherence for Monoidal Groupoids in HoTT}},
  booktitle =	{25th International Conference on Types for Proofs and Programs (TYPES 2019)},
  pages =	{8:1--8:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-158-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{175},
  editor =	{Bezem, Marc and Mahboubi, Assia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2019.8},
  URN =		{urn:nbn:de:0030-drops-130722},
  doi =		{10.4230/LIPIcs.TYPES.2019.8},
  annote =	{Keywords: homotopy type theory, coherence, monoidal categories, groupoids, higher inductive types, formalisation, Coq}
}

Document

Complete Volume

DOI: 10.4230/LIPIcs.TYPES.2018

LIPIcs, Volume 130, TYPES'18, Complete Volume

Authors: Peter Dybjer, José Espírito Santo, and Luís Pinto

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

LIPIcs, Volume 130, TYPES'18, Complete Volume

Cite as

24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{dybjer_et_al:LIPIcs.TYPES.2018,
  title =	{{LIPIcs, Volume 130, TYPES'18, Complete Volume}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018},
  URN =		{urn:nbn:de:0030-drops-114507},
  doi =		{10.4230/LIPIcs.TYPES.2018},
  annote =	{Keywords: Theory of computation,Type theory; Constructive mathematics; Logic and verification; Program verification, Software and its engineering}
}

Document

Front Matter

DOI: 10.4230/LIPIcs.TYPES.2018.0

Front Matter, Table of Contents, Preface, Conference Organization

Authors: Peter Dybjer, José Espírito Santo, and Luís Pinto

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

Front Matter, Table of Contents, Preface, Conference Organization

Cite as

24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 0:i-0:x, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dybjer_et_al:LIPIcs.TYPES.2018.0,
  author =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{0:i--0:x},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.0},
  URN =		{urn:nbn:de:0030-drops-114045},
  doi =		{10.4230/LIPIcs.TYPES.2018.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.1

Martin Hofmann’s Case for Non-Strictly Positive Data Types

Authors: Ulrich Berger, Ralph Matthes, and Anton Setzer

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

We describe the breadth-first traversal algorithm by Martin Hofmann that uses a non-strictly positive data type and carry out a simple verification in an extensional setting. Termination is shown by implementing the algorithm in the strongly normalising extension of system F by Mendler-style recursion. We then analyze the same algorithm by alternative verifications first in an intensional setting using a non-strictly positive inductive definition (not just a non-strictly positive data type), and subsequently by two different algebraic reductions. The verification approaches are compared in terms of notions of simulation and should elucidate the somewhat mysterious algorithm and thus make a case for other uses of non-strictly positive data types. Except for the termination proof, which cannot be formalised in Coq, all proofs were formalised in Coq and some of the algorithms were implemented in Agda and Haskell.

Cite as

Ulrich Berger, Ralph Matthes, and Anton Setzer. Martin Hofmann’s Case for Non-Strictly Positive Data Types. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{berger_et_al:LIPIcs.TYPES.2018.1,
  author =	{Berger, Ulrich and Matthes, Ralph and Setzer, Anton},
  title =	{{Martin Hofmann’s Case for Non-Strictly Positive Data Types}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{1:1--1:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.1},
  URN =		{urn:nbn:de:0030-drops-114052},
  doi =		{10.4230/LIPIcs.TYPES.2018.1},
  annote =	{Keywords: non strictly-positive data types, breadth-first traversal, program verification, Mendler-style recursion, System F, theorem proving, Coq, Agda, Haskell}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.2

A Simpler Undecidability Proof for System F Inhabitation

Authors: Andrej Dudenhefner and Jakob Rehof

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

Provability in the intuitionistic second-order propositional logic (resp. inhabitation in the polymorphic lambda-calculus) was shown by Löb to be undecidable in 1976. Since the original proof is heavily condensed, Arts in collaboration with Dekkers provided a fully unfolded argument in 1992 spanning approximately fifty pages. Later in 1997, Urzyczyn developed a different, syntax oriented proof. Each of the above approaches embeds (an undecidable fragment of) first-order predicate logic into second-order propositional logic. In this work, we develop a simpler undecidability proof by reduction from solvability of Diophantine equations (is there an integer solution to P(x_1, ..., x_n) = 0 where P is a polynomial with integer coefficients?). Compared to the previous approaches, the given reduction is more accessible for formalization and more comprehensible for didactic purposes. Additionally, we formalize soundness and completeness of the reduction in the Coq proof assistant under the banner of "type theory inside type theory".

Cite as

Andrej Dudenhefner and Jakob Rehof. A Simpler Undecidability Proof for System F Inhabitation. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 2:1-2:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dudenhefner_et_al:LIPIcs.TYPES.2018.2,
  author =	{Dudenhefner, Andrej and Rehof, Jakob},
  title =	{{A Simpler Undecidability Proof for System F Inhabitation}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{2:1--2:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.2},
  URN =		{urn:nbn:de:0030-drops-114061},
  doi =		{10.4230/LIPIcs.TYPES.2018.2},
  annote =	{Keywords: System F, Lambda Calculus, Inhabitation, Propositional Logic, Provability, Undecidability, Coq, Formalization}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.3

Dependent Sums and Dependent Products in Bishop’s Set Theory

Authors: Iosif Petrakis

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

According to the standard, non type-theoretic accounts of Bishop’s constructivism (BISH), dependent functions are not necessary to BISH. Dependent functions though, are explicitly used by Bishop in his definition of the intersection of a family of subsets, and they are necessary to the definition of arbitrary products. In this paper we present the basic notions and principles of CSFT, a semi-formal constructive theory of sets and functions intended to be a minimal, adequate and faithful, in Feferman’s sense, semi-formalisation of Bishop’s set theory (BST). We define the notions of dependent sum (or exterior union) and dependent product of set-indexed families of sets within CSFT, and we prove the distributivity of prod over sum i.e., the translation of the type-theoretic axiom of choice within CSFT. We also define the notions of dependent sum (or interior union) and dependent product of set-indexed families of subsets within CSFT. For these definitions we extend BST with the universe of sets #1 V_0 and the universe of functions #1 V_1.

Cite as

Iosif Petrakis. Dependent Sums and Dependent Products in Bishop’s Set Theory. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{petrakis:LIPIcs.TYPES.2018.3,
  author =	{Petrakis, Iosif},
  title =	{{Dependent Sums and Dependent Products in Bishop’s Set Theory}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.3},
  URN =		{urn:nbn:de:0030-drops-114070},
  doi =		{10.4230/LIPIcs.TYPES.2018.3},
  annote =	{Keywords: Bishop’s constructive mathematics, Martin-L\"{o}f’s type theory, dependent sums, dependent products, type-theoretic axiom of choice}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.4

Semantic Subtyping for Non-Strict Languages

Authors: Tommaso Petrucciani, Giuseppe Castagna, Davide Ancona, and Elena Zucca

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

Semantic subtyping is an approach to define subtyping relations for type systems featuring union and intersection type connectives. It has been studied only for strict languages, and it is unsound for non-strict semantics. In this work, we study how to adapt this approach to non-strict languages: in particular, we define a type system using semantic subtyping for a functional language with a call-by-need semantics. We do so by introducing an explicit representation for divergence in the types, so that the type system distinguishes expressions that are results from those which are computations that might diverge.

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Tommaso Petrucciani, Giuseppe Castagna, Davide Ancona, and Elena Zucca. Semantic Subtyping for Non-Strict Languages. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 4:1-4:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{petrucciani_et_al:LIPIcs.TYPES.2018.4,
  author =	{Petrucciani, Tommaso and Castagna, Giuseppe and Ancona, Davide and Zucca, Elena},
  title =	{{Semantic Subtyping for Non-Strict Languages}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{4:1--4:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.4},
  URN =		{urn:nbn:de:0030-drops-114083},
  doi =		{10.4230/LIPIcs.TYPES.2018.4},
  annote =	{Keywords: Semantic subtyping, non-strict semantics, call-by-need, union types, intersection types}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.5

New Formalized Results on the Meta-Theory of a Paraconsistent Logic

Authors: Anders Schlichtkrull

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

Classical logics are explosive, meaning that everything follows from a contradiction. Paraconsistent logics are logics that are not explosive. This paper presents the meta-theory of a paraconsistent infinite-valued logic, in particular new results showing that while the question of validity for a given formula can be reduced to a consideration of only finitely many truth values, this does not mean that the logic collapses to a finite-valued logic. All definitions and theorems are formalized in the Isabelle/HOL proof assistant.

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Anders Schlichtkrull. New Formalized Results on the Meta-Theory of a Paraconsistent Logic. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{schlichtkrull:LIPIcs.TYPES.2018.5,
  author =	{Schlichtkrull, Anders},
  title =	{{New Formalized Results on the Meta-Theory of a Paraconsistent Logic}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{5:1--5:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.5},
  URN =		{urn:nbn:de:0030-drops-114098},
  doi =		{10.4230/LIPIcs.TYPES.2018.5},
  annote =	{Keywords: Paraconsistent logic, Many-valued logic, Formalization, Isabelle proof assistant, Paraconsistency}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2018.6

Normalization by Evaluation for Typed Weak lambda-Reduction

Authors: Filippo Sestini

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract

Weak reduction relations in the lambda-calculus are characterized by the rejection of the so-called xi-rule, which allows arbitrary reductions under abstractions. A notable instance of weak reduction can be found in the literature under the name restricted reduction or weak lambda-reduction. In this work, we attack the problem of algorithmically computing normal forms for lambda-wk, the lambda-calculus with weak lambda-reduction. We do so by first contrasting it with other weak systems, arguing that their notion of reduction is not strong enough to compute lambda-wk-normal forms. We observe that some aspects of weak lambda-reduction prevent us from normalizing lambda-wk directly, thus devise a new, better-behaved weak calculus lambda-ex, and reduce the normalization problem for lambda-w to that of lambda-ex. We finally define type systems for both calculi and apply Normalization by Evaluation to lambda-ex, obtaining a normalization proof for lambda-wk as a corollary. We formalize all our results in Agda, a proof-assistant based on intensional Martin-Löf Type Theory.

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Filippo Sestini. Normalization by Evaluation for Typed Weak lambda-Reduction. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sestini:LIPIcs.TYPES.2018.6,
  author =	{Sestini, Filippo},
  title =	{{Normalization by Evaluation for Typed Weak lambda-Reduction}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.6},
  URN =		{urn:nbn:de:0030-drops-114101},
  doi =		{10.4230/LIPIcs.TYPES.2018.6},
  annote =	{Keywords: normalization, lambda-calculus, reduction, term-rewriting, Agda}
}

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