91 Search Results for "Kesner, Delia"

Volume

LIPIcs, Volume 303

29th International Conference on Types for Proofs and Programs (TYPES 2023)

TYPES 2023, June 12-16, 2023, ETSInf, Universitat Politècnica de València, Spain

Editors: Delia Kesner, Eduardo Hermo Reyes, and Benno van den Berg

Volume

LIPIcs, Volume 269

28th International Conference on Types for Proofs and Programs (TYPES 2022)

TYPES 2022, June 20-25, 2022, LS2N, University of Nantes, France

Editors: Delia Kesner and Pierre-Marie Pédrot

Volume

LIPIcs, Volume 52

1st International Conference on Formal Structures for Computation and Deduction (FSCD 2016)

FSCD 2016, June 22-26, 2016, Porto, Portugal

Editors: Delia Kesner and Brigitte Pientka

Document
DOI: 10.4230/LIPIcs.ITP.2024.20
Formalizing the Algebraic Small Object Argument in UniMath

Authors: Dennis Hilhorst and Paige Randall North

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
Quillen model category theory forms the cornerstone of modern homotopy theory, and thus the semantics of (and justification for the name of) homotopy type theory / univalent foundations (HoTT/UF). One of the main tools of Quillen model category theory is the small object argument. Indeed, the particular model categories that can interpret HoTT/UF are usually constructed using the small object argument. In this article, we formalize the algebraic small object argument, a modern categorical version of the small object argument originally due to Garner, in the Coq UniMath library. This constitutes a first step in building up the tools required to formalize - in a system based on HoTT/UF - the semantics of HoTT/UF in particular model categories: for instance, Voevodsky’s original interpretation into simplicial sets. More specifically, in this work, we rephrase and formalize Garner’s original formulation of the algebraic small object argument. We fill in details of Garner’s construction and redefine parts of the construction to be more direct and fit for formalization. We rephrase the theory in more modern language, using constructions like displayed categories and a modern, less strict notion of monoidal categories. We point out the interaction between the theory and the foundations, and motivate the use of the algebraic small object argument in lieu of Quillen’s original small object argument from a constructivist standpoint.
Cite as

Dennis Hilhorst and Paige Randall North. Formalizing the Algebraic Small Object Argument in UniMath. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 20:1-20:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hilhorst_et_al:LIPIcs.ITP.2024.20,
  author =	{Hilhorst, Dennis and North, Paige Randall},
  title =	{{Formalizing the Algebraic Small Object Argument in UniMath}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{20:1--20:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.20},
  URN =		{urn:nbn:de:0030-drops-207486},
  doi =		{10.4230/LIPIcs.ITP.2024.20},
  annote =	{Keywords: formalization of mathematics, univalent foundations, model category theory, algebraic small object argument, coq, unimath}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.24
An Isabelle/HOL Formalization of Narrowing and Multiset Narrowing for E-Unifiability, Reachability and Infeasibility

Authors: Dohan Kim

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
We present an Isabelle/HOL formalization of narrowing for E-unifiability, reachability, and infeasibility. Given a semi-complete rewrite system ℛ and two terms s and t, we show a formalized proof that if narrowing terminates, then it provides a decision procedure for ℛ-unifiability for s and t, where ℛ is viewed as a set of equations. Furthermore, we present multiset narrowing and its formalization for multiset reachability and reachability analysis, providing decision procedures using certain restricted conditions on multiset reachability and reachability problems. Our multiset narrowing also provides a complete method for E-unifiability problems consisting of multiple goals if E can be represented by a complete rewrite system.
Cite as

Dohan Kim. An Isabelle/HOL Formalization of Narrowing and Multiset Narrowing for E-Unifiability, Reachability and Infeasibility. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 24:1-24:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kim:LIPIcs.ITP.2024.24,
  author =	{Kim, Dohan},
  title =	{{An Isabelle/HOL Formalization of Narrowing and Multiset Narrowing for E-Unifiability, Reachability and Infeasibility}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{24:1--24:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.24},
  URN =		{urn:nbn:de:0030-drops-207525},
  doi =		{10.4230/LIPIcs.ITP.2024.24},
  annote =	{Keywords: Narrowing, Multiset Narrowing, Unifiability, Reachability}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.38
The Functor of Points Approach to Schemes in Cubical Agda

Authors: Max Zeuner and Matthias Hutzler

Published in: LIPIcs, Volume 309, 15th International Conference on Interactive Theorem Proving (ITP 2024)

Abstract
We present a formalization of quasi-compact and quasi-separated schemes (qcqs-schemes) in the Cubical Agda proof assistant. We follow Grothendieck’s functor of points approach, which defines schemes, the quintessential notion of modern algebraic geometry, as certain well-behaved functors from commutative rings to sets. This approach is often regarded as conceptually simpler than the standard approach of defining schemes as locally ringed spaces, but to our knowledge it has not yet been adopted in formalizations of algebraic geometry. We build upon a previous formalization of the so-called Zariski lattice associated to a commutative ring in order to define the notion of compact open subfunctor. This allows for a concise definition of qcqs-schemes, streamlining the usual presentation as e.g. given in the standard textbook of Demazure and Gabriel. It also lets us obtain a fully constructive proof that compact open subfunctors of affine schemes are qcqs-schemes.
Cite as

Max Zeuner and Matthias Hutzler. The Functor of Points Approach to Schemes in Cubical Agda. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{zeuner_et_al:LIPIcs.ITP.2024.38,
  author =	{Zeuner, Max and Hutzler, Matthias},
  title =	{{The Functor of Points Approach to Schemes in Cubical Agda}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{38:1--38:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-337-9},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{309},
  editor =	{Bertot, Yves and Kutsia, Temur and Norrish, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2024.38},
  URN =		{urn:nbn:de:0030-drops-207667},
  doi =		{10.4230/LIPIcs.ITP.2024.38},
  annote =	{Keywords: Schemes, Algebraic Geometry, Category Theory, Cubical Agda, Homotopy Type Theory and Univalent Foundations, Constructive Mathematics}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.TYPES.2023
LIPIcs, Volume 303, TYPES 2023, Complete Volume

Authors: Delia Kesner, Eduardo Hermo Reyes, and Benno van den Berg

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
LIPIcs, Volume 303, TYPES 2023, Complete Volume
Cite as

29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 1-138, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Proceedings{kesner_et_al:LIPIcs.TYPES.2023,
  title =	{{LIPIcs, Volume 303, TYPES 2023, Complete Volume}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{1--138},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023},
  URN =		{urn:nbn:de:0030-drops-204778},
  doi =		{10.4230/LIPIcs.TYPES.2023},
  annote =	{Keywords: LIPIcs, Volume 303, TYPES 2023, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.TYPES.2023.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Delia Kesner, Eduardo Hermo Reyes, and Benno van den Berg

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 0:i-0:viii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kesner_et_al:LIPIcs.TYPES.2023.0,
  author =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{0:i--0:viii},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.0},
  URN =		{urn:nbn:de:0030-drops-204787},
  doi =		{10.4230/LIPIcs.TYPES.2023.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.1
Classification of Covering Spaces and Canonical Change of Basepoint

Authors: Jelle Wemmenhove, Cosmin Manea, and Jim Portegies

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Using the language of homotopy type theory (HoTT), we 1) prove a synthetic version of the classification theorem for covering spaces, and 2) explore the existence of canonical change-of-basepoint isomorphisms between homotopy groups. There is some freedom in choosing how to translate concepts from classical algebraic topology into HoTT. The final translations we ended up with are easier to work with than the ones we started with. We discuss some earlier attempts to shed light on this translation process. The proofs are mechanized using the Coq proof assistant and closely follow classical treatments like those by Hatcher [Allen Hatcher, 2002].
Cite as

Jelle Wemmenhove, Cosmin Manea, and Jim Portegies. Classification of Covering Spaces and Canonical Change of Basepoint. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{wemmenhove_et_al:LIPIcs.TYPES.2023.1,
  author =	{Wemmenhove, Jelle and Manea, Cosmin and Portegies, Jim},
  title =	{{Classification of Covering Spaces and Canonical Change of Basepoint}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.1},
  URN =		{urn:nbn:de:0030-drops-204795},
  doi =		{10.4230/LIPIcs.TYPES.2023.1},
  annote =	{Keywords: Synthetic Homotopy Theory, Homotopy Type Theory, Covering Spaces, Change-of-Basepoint Isomorphism}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.2
Finite Combinatory Logic with Predicates

Authors: Andrej Dudenhefner, Christoph Stahl, Constantin Chaumet, Felix Laarmann, and Jakob Rehof

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Type inhabitation in extensions of Finite Combinatory Logic (FCL) is the mechanism underlying various component-oriented synthesis frameworks. In FCL inhabitant sets correspond to regular tree languages and vice versa. Therefore, it is not possible to specify non-regular properties of inhabitants, such as (dis)equality of subterms. Additionally, the monomorphic nature of FCL oftentimes hinders concise specification of components. We propose a conservative extension to FCL by quantifiers and predicates, introducing a restricted form of polymorphism. In the proposed type system (FCLP) inhabitant sets correspond to decidable term languages and vice versa. As a consequence, type inhabitation in FCLP is undecidable. Based on results in tree automata theory, we identify a fragment of FCLP with the following two properties. First, the fragment enjoys decidable type inhabitation. Second, it allows for specification of local (dis)equality constraints for subterms of inhabitants. For empirical evaluation, we implement a semi-decision procedure for type inhabitation in FCLP. We compare specification capabilities, scalability, and performance of the implementation to existing FCL-based approaches. Finally, we evaluate practical applicability via a case study, synthesizing mechanically sound robotic arms.
Cite as

Andrej Dudenhefner, Christoph Stahl, Constantin Chaumet, Felix Laarmann, and Jakob Rehof. Finite Combinatory Logic with Predicates. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{dudenhefner_et_al:LIPIcs.TYPES.2023.2,
  author =	{Dudenhefner, Andrej and Stahl, Christoph and Chaumet, Constantin and Laarmann, Felix and Rehof, Jakob},
  title =	{{Finite Combinatory Logic with Predicates}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.2},
  URN =		{urn:nbn:de:0030-drops-204808},
  doi =		{10.4230/LIPIcs.TYPES.2023.2},
  annote =	{Keywords: combinatory logic, inhabitation, intersection types, program synthesis}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.3
Categorical Models of Subtyping

Authors: Greta Coraglia and Jacopo Emmenegger

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context - and for each type a set of terms of said type. This is the case for categories with families, categories with attributes, and natural models; in particular, all of them can be traced back to certain discrete Grothendieck fibrations. We extend this intuition to the case of general, not necessarily discrete, fibrations, so that over a given context one has not only a set but a category of types. We argue that the added structure can be attributed to a notion of subtyping that shares many features with that of coercive subtyping, in the sense that it is the product of thinking about subtyping as an abbreviation mechanism: we say that a given type A' is a subtype of A if there is a unique coercion from A' to A. Whenever we need a term of type A, then, it suffices to have a term of type A', which we can "plug-in" into A. For this version of subtyping we provide rules, coherences, and explicit models, and we compare and contrast it to coercive subtyping as introduced by Z. Luo and others. We conclude by suggesting how the tools we present can be employed in finding appropriate rules relating subtyping and certain type constructors.
Cite as

Greta Coraglia and Jacopo Emmenegger. Categorical Models of Subtyping. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 3:1-3:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coraglia_et_al:LIPIcs.TYPES.2023.3,
  author =	{Coraglia, Greta and Emmenegger, Jacopo},
  title =	{{Categorical Models of Subtyping}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{3:1--3:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.3},
  URN =		{urn:nbn:de:0030-drops-204811},
  doi =		{10.4230/LIPIcs.TYPES.2023.3},
  annote =	{Keywords: dependent types, subtyping, coercive subtyping, categorical semantics, categories with families, monad}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.4
A Sound and Complete Substitution Algorithm for Multimode Type Theory

Authors: Joris Ceulemans, Andreas Nuyts, and Dominique Devriese

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Multimode Type Theory (MTT) is a generic type theory that can be instantiated with an arbitrary mode theory to model features like parametricity, cohesion and guarded recursion. However, the presence of modalities in MTT significantly complicates the substitution calculus of this system. Moreover, MTT’s syntax has explicit substitutions with an axiomatic system - not an algorithm - governing the connection between an explicitly substituted term and the resulting term in which variables have actually been replaced. So far, the only results on eliminating explicit substitutions in MTT rely on normalisation by evaluation and hence also immediately normalise a term. In this paper, we present a substitution algorithm for MTT that is completely separated from normalisation. To this end, we introduce Substitution-Free Multimode Type Theory (SFMTT): a formulation of MTT without explicit substitutions, but for which we are able to give a structurally recursive substitution algorithm, suitable for implementation in a total programming language or proof assistant. On the usual formulation of MTT, we consider σ-equality, the congruence generated solely by equality rules for explicit substitutions. There is a trivial embedding from SFMTT to MTT, and a converse translation that eliminates the explicit substitutions. We prove soundness and completeness of our algorithm with respect to σ-equivalence and thus establish that MTT with σ-equality has computable σ-normal forms, given by the terms of SFMTT.
Cite as

Joris Ceulemans, Andreas Nuyts, and Dominique Devriese. A Sound and Complete Substitution Algorithm for Multimode Type Theory. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{ceulemans_et_al:LIPIcs.TYPES.2023.4,
  author =	{Ceulemans, Joris and Nuyts, Andreas and Devriese, Dominique},
  title =	{{A Sound and Complete Substitution Algorithm for Multimode Type Theory}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.4},
  URN =		{urn:nbn:de:0030-drops-204826},
  doi =		{10.4230/LIPIcs.TYPES.2023.4},
  annote =	{Keywords: dependent type theory, modalities, multimode type theory, explicit substitutions, substitution algorithm}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.5
Consistent Ultrafinitist Logic

Authors: Michał J. Gajda

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Ultrafinitism postulates that we can only compute on relatively short objects, and numbers beyond a certain value are not available. This approach would also forbid many forms of infinitary reasoning and allow removing certain paradoxes stemming from enumeration theorems. For a computational application of ultrafinitist logic, we need more than a proof system, but a logical framework to express both proofs, programs, and theorems in a single framework. We present its inference rules, reduction relation, and self-encoding to allow direct proving of the properties of ultrafinitist logic within itself. We also provide a justification why it can express all bounded Turing programs, and thus serve as a "logic of computability".
Cite as

Michał J. Gajda. Consistent Ultrafinitist Logic. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajda:LIPIcs.TYPES.2023.5,
  author =	{Gajda, Micha{\l} J.},
  title =	{{Consistent Ultrafinitist Logic}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.5},
  URN =		{urn:nbn:de:0030-drops-204833},
  doi =		{10.4230/LIPIcs.TYPES.2023.5},
  annote =	{Keywords: ultrafinitism, bounded Turing completeness, logic of computability, decidable logic, explicit complexity, strict finitism}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.6
A Reflection Principle for Potential Infinite Models of Type Theory

Authors: Matthias Eberl

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Denotational models of type theory, such as set-theoretic, domain-theoretic, or category-theoretic models use (actual) infinite sets of objects in one way or another. The potential infinite, seen as an extensible finite, requires a dynamic understanding of the infinite sets of objects. It follows that the type nat cannot be interpreted as a set of all natural numbers, [[nat]] = ℕ, but as an increasing family of finite sets ℕ_i = {0, … , i-1}. Any reference to [[nat]], either by the formal syntax or by meta-level concepts, must be a reference to a (sufficiently large) set ℕ_i. We present the basic concepts for interpreting a fragment of the simply typed λ-calculus within such a dynamic model. A type ϱ is thereby interpreted as a process, which is formally a factor system together with a limit of it. A factor system is very similar to a direct or an inverse system, and its limit is also defined by a universal property. It is crucial to recognize that a limit is not necessarily an unreachable end beyond the process. Rather, it can be regarded as an intermediate state within the factor system, which can still be extended. The logical type bool plays an important role, which we interpret classically as the set {true, false}. We provide an interpretation of simply typed λ-terms in these factor systems and limits. The main result is a reflection principle, which states that an element in the limit has a "full representative" at a sufficiently large stage within the factor system. For propositions, that is, terms of type bool, this implies that statements about the limit are true if and only if they are true at that sufficiently large stage.
Cite as

Matthias Eberl. A Reflection Principle for Potential Infinite Models of Type Theory. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eberl:LIPIcs.TYPES.2023.6,
  author =	{Eberl, Matthias},
  title =	{{A Reflection Principle for Potential Infinite Models of Type Theory}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.6},
  URN =		{urn:nbn:de:0030-drops-204845},
  doi =		{10.4230/LIPIcs.TYPES.2023.6},
  annote =	{Keywords: Indefinite extensibility, Potential infinite, Reflection principle}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2024.4
Univalent Enriched Categories and the Enriched Rezk Completion

Authors: Niels van der Weide

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

Abstract
Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk completion for enriched categories to construct univalent enriched Kleisli categories.
Cite as

Niels van der Weide. Univalent Enriched Categories and the Enriched Rezk Completion. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{vanderweide:LIPIcs.FSCD.2024.4,
  author =	{van der Weide, Niels},
  title =	{{Univalent Enriched Categories and the Enriched Rezk Completion}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{4:1--4:19},
  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.2024.4},
  URN =		{urn:nbn:de:0030-drops-203337},
  doi =		{10.4230/LIPIcs.FSCD.2024.4},
  annote =	{Keywords: enriched categories, univalent categories, homotopy type theory, univalent foundations, Rezk completion}
}
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