60 Search Results for "Geuvers, Herman"


Volume

LIPIcs, Volume 131

4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)

FSCD 2019, June 24-30, 2019, Dortmund, Germany

Editors: Herman Geuvers

Volume

LIPIcs, Volume 97

22nd International Conference on Types for Proofs and Programs (TYPES 2016)

TYPES 2016, May 23-26, 2016, Novi Sad, Serbia

Editors: Silvia Ghilezan, Herman Geuvers, and Jelena Ivetic

Document
Classical Natural Deduction from Truth Tables

Authors: Herman Geuvers and Tonny Hurkens

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


Abstract
In earlier articles we have introduced truth table natural deduction which allows one to extract natural deduction rules for a propositional logic connective from its truth table definition. This works for both intuitionistic logic and classical logic. We have studied the proof theory of the intuitionistic rules in detail, giving rise to a general Kripke semantics and general proof term calculus with reduction rules that are strongly normalizing. In the present paper we study the classical rules and give a term interpretation to classical deductions with reduction rules. As a variation we define a multi-conclusion variant of the natural deduction rules as it simplifies the study of proof term reduction. We show that the reduction is normalizing and gives rise to the sub-formula property. We also compare the logical strength of the classical rules with the intuitionistic ones and we show that if one non-monotone connective is classical, then all connectives become classical.

Cite as

Herman Geuvers and Tonny Hurkens. Classical Natural Deduction from Truth Tables. In 28th International Conference on Types for Proofs and Programs (TYPES 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 269, pp. 2:1-2:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{geuvers_et_al:LIPIcs.TYPES.2022.2,
  author =	{Geuvers, Herman and Hurkens, Tonny},
  title =	{{Classical Natural Deduction from Truth Tables}},
  booktitle =	{28th International Conference on Types for Proofs and Programs (TYPES 2022)},
  pages =	{2:1--2:27},
  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.2022.2},
  URN =		{urn:nbn:de:0030-drops-184450},
  doi =		{10.4230/LIPIcs.TYPES.2022.2},
  annote =	{Keywords: Natural deduction, classical proposition logic, multiple conclusion natural deduction, proof terms, formulas-as-types, proof normalization, subformula property, Curry-Howard isomorphism}
}
Document
Invited Talk
Interactive and Automated Proofs in Modal Separation Logic (Invited Talk)

Authors: Robbert Krebbers

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)


Abstract
In program verification, it is common to embed a high-level object logic into the meta logic of a proof assistant to hide low-level aspects of the verification. To verify imperative and concurrent programs, separation logic hides explicit reasoning about heaps and pointer disjointness. To verify programs with cyclic features such as modules or higher-order state, modal logic provides modalities to hide explicit reasoning about step-indices that are used to stratify recursion. The meta logic of proof assistants such as Coq is well suited to embed high-level object logics and prove their soundness. However, proof assistants such as Coq do not have native infrastructure to facilitate proofs in embedded logics - their proof contexts and built-in tactics for interactive and automated proofs are tailored to the connectives of the meta logic, and do not extend to those of the object logic. This results in proofs that are at a too low level of abstraction because they are cluttered with bookkeeping code related to manipulating the object logic. In this talk I will describe our work in the Iris project to address this problem - first for interactive proofs, and then for semi-automated proofs. The Iris Proof Mode provides high-level tactics for interactive proofs in higher-order concurrent separation logic with modalities. Recent work on RefinedC and Diaframe have built on top of the Iris Proof Mode to obtain proof automation for low-level C programs and fine-grained concurrent programs.

Cite as

Robbert Krebbers. Interactive and Automated Proofs in Modal Separation Logic (Invited Talk). In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{krebbers:LIPIcs.ITP.2023.2,
  author =	{Krebbers, Robbert},
  title =	{{Interactive and Automated Proofs in Modal Separation Logic}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{2:1--2:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.2},
  URN =		{urn:nbn:de:0030-drops-183770},
  doi =		{10.4230/LIPIcs.ITP.2023.2},
  annote =	{Keywords: Program Verification, Separation Logic, Step-Indexing, Modal Logic, Interactive Theorem Proving, Proof Automation, Iris, Coq}
}
Document
The Formal Theory of Monads, Univalently

Authors: Niels van der Weide

Published in: LIPIcs, Volume 260, 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)


Abstract
We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the https://github.com/UniMath/UniMath library.

Cite as

Niels van der Weide. The Formal Theory of Monads, Univalently. In 8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 260, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{vanderweide:LIPIcs.FSCD.2023.6,
  author =	{van der Weide, Niels},
  title =	{{The Formal Theory of Monads, Univalently}},
  booktitle =	{8th International Conference on Formal Structures for Computation and Deduction (FSCD 2023)},
  pages =	{6:1--6:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-277-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{260},
  editor =	{Gaboardi, Marco and van Raamsdonk, Femke},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2023.6},
  URN =		{urn:nbn:de:0030-drops-179904},
  doi =		{10.4230/LIPIcs.FSCD.2023.6},
  annote =	{Keywords: bicategory theory, univalent foundations, formalization, monads, Coq}
}
Document
On Model-Theoretic Strong Normalization for Truth-Table Natural Deduction

Authors: Andreas Abel

Published in: LIPIcs, Volume 188, 26th International Conference on Types for Proofs and Programs (TYPES 2020)


Abstract
Intuitionistic truth table natural deduction (ITTND) by Geuvers and Hurkens (2017), which is inherently non-confluent, has been shown strongly normalizing (SN) using continuation-passing-style translations to parallel lambda calculus by Geuvers, van der Giessen, and Hurkens (2019). We investigate the applicability of standard model-theoretic proof techniques and show (1) SN of detour reduction (β) using Girard’s reducibility candidates, and (2) SN of detour and permutation reduction (βπ) using biorthogonals. In the appendix, we adapt Tait’s method of saturated sets to β, clarifying the original proof of 2017, and extend it to βπ.

Cite as

Andreas Abel. On Model-Theoretic Strong Normalization for Truth-Table Natural Deduction. In 26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{abel:LIPIcs.TYPES.2020.1,
  author =	{Abel, Andreas},
  title =	{{On Model-Theoretic Strong Normalization for Truth-Table Natural Deduction}},
  booktitle =	{26th International Conference on Types for Proofs and Programs (TYPES 2020)},
  pages =	{1:1--1:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-182-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{188},
  editor =	{de'Liguoro, Ugo and Berardi, Stefano and 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.TYPES.2020.1},
  URN =		{urn:nbn:de:0030-drops-138805},
  doi =		{10.4230/LIPIcs.TYPES.2020.1},
  annote =	{Keywords: Natural deduction, Permutative conversion, Reducibility, Strong normalization, Truth table}
}
Document
Complete Volume
LIPIcs, Volume 131, FSCD'19, Complete Volume

Authors: Herman Geuvers

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
LIPIcs, Volume 131, FSCD'19, Complete Volume

Cite as

4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@Proceedings{geuvers:LIPIcs.FSCD.2019,
  title =	{{LIPIcs, Volume 131, FSCD'19, Complete Volume}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019},
  URN =		{urn:nbn:de:0030-drops-107734},
  doi =		{10.4230/LIPIcs.FSCD.2019},
  annote =	{Keywords: Theory of computation, Models of computation, Formal languages and automata theory, Logic, Semantics and reasoning}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Herman Geuvers

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


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

Cite as

4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 0:i-0:xx, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{geuvers:LIPIcs.FSCD.2019.0,
  author =	{Geuvers, Herman},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{0:i--0:xx},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.0},
  URN =		{urn:nbn:de:0030-drops-105070},
  doi =		{10.4230/LIPIcs.FSCD.2019.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
A Fresh Look at the lambda-Calculus (Invited Talk)

Authors: Beniamino Accattoli

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
The (untyped) lambda-calculus is almost 90 years old. And yet - we argue here - its study is far from being over. The paper is a bird’s eye view of the questions the author worked on in the last few years: how to measure the complexity of lambda-terms, how to decompose their evaluation, how to implement it, and how all this varies according to the evaluation strategy. The paper aims at inducing a new way of looking at an old topic, focussing on high-level issues and perspectives.

Cite as

Beniamino Accattoli. A Fresh Look at the lambda-Calculus (Invited Talk). In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 1:1-1:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{accattoli:LIPIcs.FSCD.2019.1,
  author =	{Accattoli, Beniamino},
  title =	{{A Fresh Look at the lambda-Calculus}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{1:1--1:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.1},
  URN =		{urn:nbn:de:0030-drops-105083},
  doi =		{10.4230/LIPIcs.FSCD.2019.1},
  annote =	{Keywords: lambda-calculus, sharing, abstract machines, type systems, rewriting}
}
Document
Invited Talk
A Linear Logical Framework in Hybrid (Invited Talk)

Authors: Amy P. Felty

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
We present a linear logical framework implemented within the Hybrid system [Amy P. Felty and Alberto Momigliano, 2012]. Hybrid is designed to support the use of higher-order abstract syntax for representing and reasoning about formal systems, implemented in the Coq Proof Assistant. In this work, we extend the system with two linear specification logics, which provide infrastructure for reasoning directly about object languages with linear features. We originally developed this framework in order to address the challenges of reasoning about the type system of a quantum lambda calculus. In particular, we started by considering the Proto-Quipper language [Neil J. Ross, 2015], which contains the core of Quipper [Green et al., 2013; Peter Selinger and Benoît Valiron, 2006]. Quipper is a relatively new quantum programming language under active development with a linear type system. We have completed a formal proof of type soundness for Proto-Quipper [Mohamed Yousri Mahmoud and Amy P. Felty, 2018]. Our current work includes extending this work to other properties of Proto-Quipper, reasoning about other quantum programming languages [Mohamed Yousri Mahmoud and Amy P. Felty, 2018], and reasoning about other languages such as the meta-theory of low-level abstract machine code. We are also interested in applying this framework to applications outside the domain of meta-theory of programming languages and have focused on two areas - formal reasoning about the proof theory of focused linear sequent calculi and modeling biological processes as transition systems and proving properties about them. We found that a slight extension of the initial linear specification logic allowed us to provide succinct encodings and facilitate reasoning in these new domains. We illustrate by discussing a model of breast cancer progression as a set of transition rules and proving properties about this model [Joëlle Despeyroux et al., 2018]. Current work also includes modeling stem cells as they mature into different types of blood cells. This work illustrates the use of Hybrid as a meta-logical framework for fast prototyping of logical frameworks, which is achieved by defining inference rules of a specification logic inductively in Coq and building a library of definitions and lemmas used to reason about a class of object logics. Our focus here is on linear specification logics and their applications.

Cite as

Amy P. Felty. A Linear Logical Framework in Hybrid (Invited Talk). In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 2:1-2:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{felty:LIPIcs.FSCD.2019.2,
  author =	{Felty, Amy P.},
  title =	{{A Linear Logical Framework in Hybrid}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{2:1--2:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.2},
  URN =		{urn:nbn:de:0030-drops-105099},
  doi =		{10.4230/LIPIcs.FSCD.2019.2},
  annote =	{Keywords: Logical frameworks, proof assistants, linear logic}
}
Document
Invited Talk
Extending Maximal Completion (Invited Talk)

Authors: Sarah Winkler

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
Maximal completion (Klein and Hirokawa 2011) is an elegantly simple yet powerful variant of Knuth-Bendix completion. This paper extends the approach to ordered completion and theorem proving as well as normalized completion. An implementation of the different procedures is described, and its practicality is demonstrated by various examples.

Cite as

Sarah Winkler. Extending Maximal Completion (Invited Talk). In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{winkler:LIPIcs.FSCD.2019.3,
  author =	{Winkler, Sarah},
  title =	{{Extending Maximal Completion}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.3},
  URN =		{urn:nbn:de:0030-drops-105102},
  doi =		{10.4230/LIPIcs.FSCD.2019.3},
  annote =	{Keywords: automated reasoning, completion, theorem proving}
}
Document
Invited Talk
Some Semantic Issues in Probabilistic Programming Languages (Invited Talk)

Authors: Hongseok Yang

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
This is a slightly extended abstract of my talk at FSCD'19 about probabilistic programming and a few semantic issues on it. The main purpose of this abstract is to provide keywords and references on the work mentioned in my talk, and help interested audience to do follow-up study.

Cite as

Hongseok Yang. Some Semantic Issues in Probabilistic Programming Languages (Invited Talk). In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 4:1-4:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{yang:LIPIcs.FSCD.2019.4,
  author =	{Yang, Hongseok},
  title =	{{Some Semantic Issues in Probabilistic Programming Languages}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{4:1--4:6},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.4},
  URN =		{urn:nbn:de:0030-drops-105118},
  doi =		{10.4230/LIPIcs.FSCD.2019.4},
  annote =	{Keywords: Probabilistic Programming, Denotational Semantics, Non-differentiable Models, Bayesian Nonparametrics, Exchangeability}
}
Document
Bicategories in Univalent Foundations

Authors: Benedikt Ahrens, Dan Frumin, Marco Maggesi, and Niels van der Weide

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of those, we develop the notion of "displayed bicategories", an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. Displayed bicategories allow us to construct univalent bicategories in a modular fashion. To demonstrate the applicability of this notion, we prove several bicategories are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Our work is formalized in the UniMath library of univalent mathematics.

Cite as

Benedikt Ahrens, Dan Frumin, Marco Maggesi, and Niels van der Weide. Bicategories in Univalent Foundations. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{ahrens_et_al:LIPIcs.FSCD.2019.5,
  author =	{Ahrens, Benedikt and Frumin, Dan and Maggesi, Marco and van der Weide, Niels},
  title =	{{Bicategories in Univalent Foundations}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{5:1--5:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.5},
  URN =		{urn:nbn:de:0030-drops-105124},
  doi =		{10.4230/LIPIcs.FSCD.2019.5},
  annote =	{Keywords: bicategory theory, univalent mathematics, dependent type theory, Coq}
}
Document
Modular Specification of Monads Through Higher-Order Presentations

Authors: Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
In their work on second-order equational logic, Fiore and Hur have studied presentations of simply typed languages by generating binding constructions and equations among them. To each pair consisting of a binding signature and a set of equations, they associate a category of "models", and they give a monadicity result which implies that this category has an initial object, which is the language presented by the pair. In the present work, we propose, for the untyped setting, a variant of their approach where monads and modules over them are the central notions. More precisely, we study, for monads over sets, presentations by generating ("higher-order") operations and equations among them. We consider a notion of 2-signature which allows to specify a monad with a family of binding operations subject to a family of equations, as is the case for the paradigmatic example of the lambda calculus, specified by its two standard constructions (application and abstraction) subject to beta- and eta-equalities. Such a 2-signature is hence a pair (Sigma,E) of a binding signature Sigma and a family E of equations for Sigma. This notion of 2-signature has been introduced earlier by Ahrens in a slightly different context. We associate, to each 2-signature (Sigma,E), a category of "models of (Sigma,E)"; and we say that a 2-signature is "effective" if this category has an initial object; the monad underlying this (essentially unique) object is the "monad specified by the 2-signature". Not every 2-signature is effective; we identify a class of 2-signatures, which we call "algebraic", that are effective. Importantly, our 2-signatures together with their models enjoy "modularity": when we glue (algebraic) 2-signatures together, their initial models are glued accordingly. We provide a computer formalization for our main results.

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Benedikt Ahrens, André Hirschowitz, Ambroise Lafont, and Marco Maggesi. Modular Specification of Monads Through Higher-Order Presentations. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Copy BibTex To Clipboard

@InProceedings{ahrens_et_al:LIPIcs.FSCD.2019.6,
  author =	{Ahrens, Benedikt and Hirschowitz, Andr\'{e} and Lafont, Ambroise and Maggesi, Marco},
  title =	{{Modular Specification of Monads Through Higher-Order Presentations}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{6:1--6:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.6},
  URN =		{urn:nbn:de:0030-drops-105136},
  doi =		{10.4230/LIPIcs.FSCD.2019.6},
  annote =	{Keywords: free monads, presentation of monads, initial semantics, signatures, syntax, monadic substitution, computer-checked proofs}
}
Document
Towards the Average-Case Analysis of Substitution Resolution in Lambda-Calculus

Authors: Maciej Bendkowski

Published in: LIPIcs, Volume 131, 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)


Abstract
Substitution resolution supports the computational character of beta-reduction, complementing its execution with a capture-avoiding exchange of terms for bound variables. Alas, the meta-level definition of substitution, masking a non-trivial computation, turns beta-reduction into an atomic rewriting rule, despite its varying operational complexity. In the current paper we propose a somewhat indirect average-case analysis of substitution resolution in the classic lambda-calculus, based on the quantitative analysis of substitution in lambda-upsilon, an extension of lambda-calculus internalising the upsilon-calculus of explicit substitutions. Within this framework, we show that for any fixed n >= 0, the probability that a uniformly random, conditioned on size, lambda-upsilon-term upsilon-normalises in n normal-order (i.e. leftmost-outermost) reduction steps tends to a computable limit as the term size tends to infinity. For that purpose, we establish an effective hierarchy (G_n)_n of regular tree grammars partitioning upsilon-normalisable terms into classes of terms normalising in n normal-order rewriting steps. The main technical ingredient in our construction is an inductive approach to the construction of G_{n+1} out of G_n based, in turn, on the algorithmic construction of finite intersection partitions, inspired by Robinson’s unification algorithm. Finally, we briefly discuss applications of our approach to other term rewriting systems, focusing on two closely related formalisms, i.e. the full lambda-upsilon-calculus and combinatory logic.

Cite as

Maciej Bendkowski. Towards the Average-Case Analysis of Substitution Resolution in Lambda-Calculus. In 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 131, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Copy BibTex To Clipboard

@InProceedings{bendkowski:LIPIcs.FSCD.2019.7,
  author =	{Bendkowski, Maciej},
  title =	{{Towards the Average-Case Analysis of Substitution Resolution in Lambda-Calculus}},
  booktitle =	{4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)},
  pages =	{7:1--7:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-107-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{131},
  editor =	{Geuvers, Herman},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2019.7},
  URN =		{urn:nbn:de:0030-drops-105144},
  doi =		{10.4230/LIPIcs.FSCD.2019.7},
  annote =	{Keywords: lambda calculus, explicit substitutions, complexity, combinatorics}
}
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