18 Search Results for "Cohen, Cyril"

Document
DOI: 10.4230/LIPIcs.ESA.2024.66
Shortest Path Separators in Unit Disk Graphs

Authors: Elfarouk Harb, Zhengcheng Huang, and Da Wei Zheng

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
We introduce a new balanced separator theorem for unit-disk graphs involving two shortest paths combined with the 1-hop neighbours of those paths and two other vertices. This answers an open problem of Yan, Xiang and Dragan [CGTA '12] and improves their result that requires removing the 3-hop neighbourhood of two shortest paths. Our proof uses very different ideas, including Delaunay triangulations and a generalization of the celebrated balanced separator theorem of Lipton and Tarjan [J. Appl. Math. '79] to systems of non-intersecting paths.
Cite as

Elfarouk Harb, Zhengcheng Huang, and Da Wei Zheng. Shortest Path Separators in Unit Disk Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{harb_et_al:LIPIcs.ESA.2024.66,
  author =	{Harb, Elfarouk and Huang, Zhengcheng and Zheng, Da Wei},
  title =	{{Shortest Path Separators in Unit Disk Graphs}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{66:1--66:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.66},
  URN =		{urn:nbn:de:0030-drops-211375},
  doi =		{10.4230/LIPIcs.ESA.2024.66},
  annote =	{Keywords: Balanced shortest path separators, unit disk graphs, crossings}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2024.44
Type Tailoring

Authors: Ashton Wiersdorf, Stephen Chang, Matthias Felleisen, and Ben Greenman

Published in: LIPIcs, Volume 313, 38th European Conference on Object-Oriented Programming (ECOOP 2024)

Abstract
Type systems evolve too slowly to keep up with the quick evolution of libraries - especially libraries that introduce abstractions. Type tailoring offers a lightweight solution by equipping the core language with an API for modifying the elaboration of surface code into the internal language of the typechecker. Through user-programmable elaboration, tailoring rules appear to improve the precision and expressiveness of the underlying type system. Furthermore, type tailoring cooperates with the host type system by expanding to code that the host then typechecks. In the context of a hygienic metaprogramming system, tailoring rules can even harmoniously compose with one another. Type tailoring has emerged as a theme across several languages and metaprogramming systems, but never with direct support and rarely in the same shape twice. For example, both OCaml and Typed Racket enable forms of tailoring, but in quite different ways. This paper identifies key dimensions of type tailoring systems and tradeoffs along each dimension. It demonstrates the usefulness of tailoring with examples that cover sized vectors, database queries, and optional types. Finally, it outlines a vision for future research at the intersection of types and metaprogramming.
Cite as

Ashton Wiersdorf, Stephen Chang, Matthias Felleisen, and Ben Greenman. Type Tailoring. In 38th European Conference on Object-Oriented Programming (ECOOP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 313, pp. 44:1-44:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{wiersdorf_et_al:LIPIcs.ECOOP.2024.44,
  author =	{Wiersdorf, Ashton and Chang, Stephen and Felleisen, Matthias and Greenman, Ben},
  title =	{{Type Tailoring}},
  booktitle =	{38th European Conference on Object-Oriented Programming (ECOOP 2024)},
  pages =	{44:1--44:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-341-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{313},
  editor =	{Aldrich, Jonathan and Salvaneschi, Guido},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2024.44},
  URN =		{urn:nbn:de:0030-drops-208933},
  doi =		{10.4230/LIPIcs.ECOOP.2024.44},
  annote =	{Keywords: Types, Metaprogramming, Macros, Partial Evaluation}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.4
Taming Differentiable Logics with Coq Formalisation

Authors: Reynald Affeldt, Alessandro Bruni, Ekaterina Komendantskaya, Natalia Ślusarz, and Kathrin Stark

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

Abstract
For performance and verification in machine learning, new methods have recently been proposed that optimise learning systems to satisfy formally expressed logical properties. Among these methods, differentiable logics (DLs) are used to translate propositional or first-order formulae into loss functions deployed for optimisation in machine learning. At the same time, recent attempts to give programming language support for verification of neural networks showed that DLs can be used to compile verification properties to machine-learning backends. This situation is calling for stronger guarantees about the soundness of such compilers, the soundness and compositionality of DLs, and the differentiability and performance of the resulting loss functions. In this paper, we propose an approach to formalise existing DLs using the Mathematical Components library in the Coq proof assistant. Thanks to this formalisation, we are able to give uniform semantics to otherwise disparate DLs, give formal proofs to existing informal arguments, find errors in previous work, and provide formal proofs to missing conjectured properties. This work is meant as a stepping stone for the development of programming language support for verification of machine learning.
Cite as

Reynald Affeldt, Alessandro Bruni, Ekaterina Komendantskaya, Natalia Ślusarz, and Kathrin Stark. Taming Differentiable Logics with Coq Formalisation. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{affeldt_et_al:LIPIcs.ITP.2024.4,
  author =	{Affeldt, Reynald and Bruni, Alessandro and Komendantskaya, Ekaterina and \'{S}lusarz, Natalia and Stark, Kathrin},
  title =	{{Taming Differentiable Logics with Coq Formalisation}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{4:1--4:19},
  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.2024.4},
  URN =		{urn:nbn:de:0030-drops-207325},
  doi =		{10.4230/LIPIcs.ITP.2024.4},
  annote =	{Keywords: Machine Learning, Loss Functions, Differentiable Logics, Logic and Semantics, Interactive Theorem Proving}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.5
A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq

Authors: Reynald Affeldt and Zachary Stone

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

Abstract
Formalization of real analysis offers a chance to rebuild traditional proofs of important theorems as unambiguous theories that can be interactively explored. This paper provides a comprehensive overview of the Lebesgue Differentiation Theorem formalized in the Coq proof assistant, from which the first Fundamental Theorem of Calculus (FTC) for the Lebesgue integral is obtained as a corollary. Proving the first FTC in this way has the advantage of decomposing into loosely-coupled theories of moderate size and of independent interest that lend themselves well to incremental and collaborative development. We explain how we formalize all the topological constructs and all the standard lemmas needed to eventually relate the definitions of derivability and of Lebesgue integration of MathComp-Analysis, a formalization of analysis developed on top of the Mathematical Components library. In the course of this experiment, we substantially enrich MathComp-Analysis and even devise a new proof for Urysohn’s lemma.
Cite as

Reynald Affeldt and Zachary Stone. A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{affeldt_et_al:LIPIcs.ITP.2024.5,
  author =	{Affeldt, Reynald and Stone, Zachary},
  title =	{{A Comprehensive Overview of the Lebesgue Differentiation Theorem in Coq}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{5:1--5:19},
  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.2024.5},
  URN =		{urn:nbn:de:0030-drops-207330},
  doi =		{10.4230/LIPIcs.ITP.2024.5},
  annote =	{Keywords: Coq proof assistant, Mathematical Components library, Lebesgue integral, fundamental theorem of calculus}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.6
Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem

Authors: Dagur Asgeirsson

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

Abstract
Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved idea, namely that of a condensed set. Central to the theory are solid abelian groups and liquid vector spaces, analogues of complete topological groups. Nöbeling’s theorem, a surprising result from the 1960s about the structure of the abelian group of continuous maps from a profinite space to the integers, is a crucial ingredient in the theory of solid abelian groups; without it one cannot give any nonzero examples of solid abelian groups. We discuss a recently completed formalisation of this result in the Lean theorem prover, and give a more detailed proof than those previously available in the literature. The proof is somewhat unusual in that it requires induction over ordinals - a technique which has not previously been used to a great extent in formalised mathematics.
Cite as

Dagur Asgeirsson. Towards Solid Abelian Groups: A Formal Proof of Nöbeling’s Theorem. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{asgeirsson:LIPIcs.ITP.2024.6,
  author =	{Asgeirsson, Dagur},
  title =	{{Towards Solid Abelian Groups: A Formal Proof of N\"{o}beling’s Theorem}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{6:1--6:17},
  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.2024.6},
  URN =		{urn:nbn:de:0030-drops-207347},
  doi =		{10.4230/LIPIcs.ITP.2024.6},
  annote =	{Keywords: Condensed mathematics, N\"{o}beling’s theorem, Lean, Mathlib, Interactive theorem proving}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.11
A Formalization of the General Theory of Quaternions

Authors: Thaynara Arielly de Lima, André Luiz Galdino, Bruno Berto de Oliveira Ribeiro, and Mauricio Ayala-Rincón

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

Abstract
This paper discusses the formalization of the theory of quaternions in the Prototype Verification System (PVS). The general approach in this mechanization relies on specifying quaternion structures using any arbitrary field as a parameter. The approach allows the inheritance of formalized properties on quaternions when the parameters of the general theory are instantiated with specific fields such as reals or rationals. The theory includes characterizing algebraic properties that lead to constructing quaternions as division rings. In particular, we illustrate how the general theory is applied to formalize Hamilton’s quaternions using the field of reals as a parameter, for which we also mechanized theorems that show the completeness of three-dimensional rotations, proving that Hamilton’s quaternions mimic any 3D rotation.
Cite as

Thaynara Arielly de Lima, André Luiz Galdino, Bruno Berto de Oliveira Ribeiro, and Mauricio Ayala-Rincón. A Formalization of the General Theory of Quaternions. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{delima_et_al:LIPIcs.ITP.2024.11,
  author =	{de Lima, Thaynara Arielly and Galdino, Andr\'{e} Luiz and de Oliveira Ribeiro, Bruno Berto and Ayala-Rinc\'{o}n, Mauricio},
  title =	{{A Formalization of the General Theory of Quaternions}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{11:1--11:18},
  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.2024.11},
  URN =		{urn:nbn:de:0030-drops-207398},
  doi =		{10.4230/LIPIcs.ITP.2024.11},
  annote =	{Keywords: Theory of quaternions, Hamilton’s quaternions, Algebraic formalizations, PVS}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.15
Typed Compositional Quantum Computation with Lenses

Authors: Jacques Garrigue and Takafumi Saikawa

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

Abstract
We propose a type-theoretic framework for describing and proving properties of quantum computations, in particular those presented as quantum circuits. Our proposal is based on an observation that, in the polymorphic type system of Coq, currying on quantum states allows one to apply quantum gates directly inside a complex circuit. By introducing a discrete notion of lens to control this currying, we are further able to separate the combinatorics of the circuit structure from the computational content of gates. We apply our development to define quantum circuits recursively from the bottom up, and prove their correctness compositionally.
Cite as

Jacques Garrigue and Takafumi Saikawa. Typed Compositional Quantum Computation with Lenses. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{garrigue_et_al:LIPIcs.ITP.2024.15,
  author =	{Garrigue, Jacques and Saikawa, Takafumi},
  title =	{{Typed Compositional Quantum Computation with Lenses}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{15:1--15:18},
  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.2024.15},
  URN =		{urn:nbn:de:0030-drops-207431},
  doi =		{10.4230/LIPIcs.ITP.2024.15},
  annote =	{Keywords: quantum programming, semantics, lens, currying, Coq, MathComp}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.26
The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant

Authors: Yann Leray, Gaëtan Gilbert, Nicolas Tabareau, and Théo Winterhalter

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

Abstract
In dependently typed proof assistants, users can declare axioms to extend the ambient logic locally with new principles and propositional equalities governing them. Additionally, rewrite rules have recently been proposed to allow users to extend the logic with new definitional equalities, enabling them to handle new principles with a computational behaviour. While axioms can only break consistency, the addition of arbitrary rewrite rules can break other important metatheoretical properties such as type preservation. In this paper, we present an implementation of rewrite rules on top of the Coq proof assistant, together with a modular criterion to ensure that the added rewrite rules preserve typing. This criterion, based on bidirectional type checking, is formally expressed in PCUIC - the type theory of Coq recently developed in the MetaCoq project.
Cite as

Yann Leray, Gaëtan Gilbert, Nicolas Tabareau, and Théo Winterhalter. The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{leray_et_al:LIPIcs.ITP.2024.26,
  author =	{Leray, Yann and Gilbert, Ga\"{e}tan and Tabareau, Nicolas and Winterhalter, Th\'{e}o},
  title =	{{The Rewster: Type Preserving Rewrite Rules for the Coq Proof Assistant}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{26:1--26:18},
  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.2024.26},
  URN =		{urn:nbn:de:0030-drops-207545},
  doi =		{10.4230/LIPIcs.ITP.2024.26},
  annote =	{Keywords: type theory, dependent types, rewrite rules, type preservation, Coq}
}
Document
DOI: 10.4230/LIPIcs.ITP.2024.30
A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations

Authors: Sewon Park and Holger Thies

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

Abstract
In exact real computation real numbers are manipulated exactly without round-off errors, making it well-suited for high precision verified computation. In recent work we propose an axiomatic formalization of exact real computation in the Coq theorem prover. The formalization admits an extended extraction mechanism that lets us extract computational content from constructive parts of proofs to efficient programs built on top of AERN, a Haskell library for exact real computation. Many processes in science and engineering are modeled by ordinary differential equations (ODEs), and often safety-critical applications depend on computing their solutions correctly. The primary goal of the current work is to extend our framework to spaces of functions and to support computation of solutions to ODEs and other essential operators. In numerical mathematics, the most common way to represent continuous functions is to use polynomial approximations. This can be modeled by so-called Taylor models, that encode a function as a polynomial and a rigorous error-bound over some domain. We define types of classical functions that do not hold any computational content and formalize Taylor models to computationally approximate those classical functions. Classical functions are defined in a way to admit classical principles in their constructions and verification. We define various basic operations on Taylor models and verify their correctness based on the classical functions that they approximate. We then shift our interest to analytic functions as a generalization of Taylor models where polynomials are replaced by infinite power series. We use the formalization to develop a theory of non-linear polynomial ODEs. From the proofs we can extract certified exact real computation programs that compute solutions of ODEs on some time interval up to any precision.
Cite as

Sewon Park and Holger Thies. A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 30:1-30:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{park_et_al:LIPIcs.ITP.2024.30,
  author =	{Park, Sewon and Thies, Holger},
  title =	{{A Coq Formalization of Taylor Models and Power Series for Solving Ordinary Differential Equations}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{30:1--30:19},
  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.2024.30},
  URN =		{urn:nbn:de:0030-drops-207581},
  doi =		{10.4230/LIPIcs.ITP.2024.30},
  annote =	{Keywords: Exact real computation, Taylor models, Analytic functions, Computable analysis, Program extraction}
}
Document
DOI: 10.4230/LIPIcs.CONCUR.2024.17
Invariants for One-Counter Automata with Disequality Tests

Authors: Dmitry Chistikov, Jérôme Leroux, Henry Sinclair-Banks, and Nicolas Waldburger

Published in: LIPIcs, Volume 311, 35th International Conference on Concurrency Theory (CONCUR 2024)

Abstract
We study the reachability problem for one-counter automata in which transitions can carry disequality tests. A disequality test is a guard that prohibits a specified counter value. This reachability problem has been known to be NP-hard and in PSPACE, and characterising its computational complexity has been left as a challenging open question by Almagor, Cohen, Pérez, Shirmohammadi, and Worrell (2020). We reduce the complexity gap, placing the problem into the second level of the polynomial hierarchy, namely into the class coNP^NP. In the presence of both equality and disequality tests, our upper bound is at the third level, P^NP^NP. To prove this result, we show that non-reachability can be witnessed by a pair of invariants (forward and backward). These invariants are almost inductive. They aim to over-approximate only a "core" of the reachability set instead of the entire set. The invariants are also leaky: it is possible to escape the set. We complement this with separate checks as the leaks can only occur in a controlled way.
Cite as

Dmitry Chistikov, Jérôme Leroux, Henry Sinclair-Banks, and Nicolas Waldburger. Invariants for One-Counter Automata with Disequality Tests. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{chistikov_et_al:LIPIcs.CONCUR.2024.17,
  author =	{Chistikov, Dmitry and Leroux, J\'{e}r\^{o}me and Sinclair-Banks, Henry and Waldburger, Nicolas},
  title =	{{Invariants for One-Counter Automata with Disequality Tests}},
  booktitle =	{35th International Conference on Concurrency Theory (CONCUR 2024)},
  pages =	{17:1--17:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-339-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{311},
  editor =	{Majumdar, Rupak and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2024.17},
  URN =		{urn:nbn:de:0030-drops-207898},
  doi =		{10.4230/LIPIcs.CONCUR.2024.17},
  annote =	{Keywords: Inductive invariant, Vector addition system, One-counter automaton}
}
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)

Copy BibTex To Clipboard

@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.FSCD.2024.22
Automating Boundary Filling in Cubical Agda

Authors: Maximilian Doré, Evan Cavallo, and Anders Mörtberg

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

Abstract
When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy Type Theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to "contort" a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case - Kan solving is even undecidable - so we focus on heuristics that perform well on practical examples. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.
Cite as

Maximilian Doré, Evan Cavallo, and Anders Mörtberg. Automating Boundary Filling in Cubical Agda. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dore_et_al:LIPIcs.FSCD.2024.22,
  author =	{Dor\'{e}, Maximilian and Cavallo, Evan and M\"{o}rtberg, Anders},
  title =	{{Automating Boundary Filling in Cubical Agda}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{22:1--22:18},
  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.2024.22},
  URN =		{urn:nbn:de:0030-drops-203514},
  doi =		{10.4230/LIPIcs.FSCD.2024.22},
  annote =	{Keywords: Cubical Agda, Automated Reasoning, Constraint Satisfaction Programming}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.101
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Authors: Tsvi Kopelowitz, Ariel Korin, and Liam Roditty

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

Abstract
For an undirected unweighted graph G = (V,E) with n vertices and m edges, let d(u,v) denote the distance from u ∈ V to v ∈ V in G. An (α,β)-stretch approximate distance oracle (ADO) for G is a data structure that given u,v ∈ V returns in constant (or near constant) time a value dˆ(u,v) such that d(u,v) ≤ dˆ(u,v) ≤ α⋅ d(u,v) + β, for some reals α > 1, β. Thorup and Zwick [Mikkel Thorup and Uri Zwick, 2005] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one can obtain stretch 2 using O(m^{1/3}n^{4/3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is Δ_G ≤ O(n^{1/k-ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2-(kε)/3) space and has a (2,1-k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with Δ_G ≤ O(n^{1/2-ε}). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-(k+2)/k stretch for graphs with Δ_G = Θ(n^{1/k}) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n^{1/2}), obtaining a sub-2 stretch requires Ω̃(n²) space.
Cite as

Tsvi Kopelowitz, Ariel Korin, and Liam Roditty. On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{kopelowitz_et_al:LIPIcs.ICALP.2024.101,
  author =	{Kopelowitz, Tsvi and Korin, Ariel and Roditty, Liam},
  title =	{{On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{101:1--101:18},
  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.2024.101},
  URN =		{urn:nbn:de:0030-drops-202443},
  doi =		{10.4230/LIPIcs.ICALP.2024.101},
  annote =	{Keywords: Graph algorithms, Approximate distance oracle, data structures, shortest path}
}
Document
DOI: 10.4230/LIPIcs.ITP.2021.8
Unsolvability of the Quintic Formalized in Dependent Type Theory

Authors: Sophie Bernard, Cyril Cohen, Assia Mahboubi, and Pierre-Yves Strub

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
In this paper, we describe an axiom-free Coq formalization that there does not exists a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois' Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group.
Cite as

Sophie Bernard, Cyril Cohen, Assia Mahboubi, and Pierre-Yves Strub. Unsolvability of the Quintic Formalized in Dependent Type Theory. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 8:1-8:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{bernard_et_al:LIPIcs.ITP.2021.8,
  author =	{Bernard, Sophie and Cohen, Cyril and Mahboubi, Assia and Strub, Pierre-Yves},
  title =	{{Unsolvability of the Quintic Formalized in Dependent Type Theory}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{8:1--8:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.8},
  URN =		{urn:nbn:de:0030-drops-139038},
  doi =		{10.4230/LIPIcs.ITP.2021.8},
  annote =	{Keywords: Galois theory, Coq, Mathematical Components, Dependent Type Theory, Abel-Ruffini, General quintic}
}
Document
System Description
DOI: 10.4230/LIPIcs.FSCD.2020.34
Hierarchy Builder: Algebraic hierarchies Made Easy in Coq with Elpi (System Description)

Authors: Cyril Cohen, Kazuhiko Sakaguchi, and Enrico Tassi

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

Abstract
It is nowadays customary to organize libraries of machine checked proofs around hierarchies of algebraic structures. One influential example is the Mathematical Components library on top of which the long and intricate proof of the Odd Order Theorem could be fully formalized. Still, building algebraic hierarchies in a proof assistant such as Coq requires a lot of manual labor and often a deep expertise in the internals of the prover. Moreover, according to our experience, making a hierarchy evolve without causing breakage in client code is equally tricky: even a simple refactoring such as splitting a structure into two simpler ones is hard to get right. In this paper we describe HB, a high level language to build hierarchies of algebraic structures and to make these hierarchies evolve without breaking user code. The key concepts are the ones of factory, builder and abbreviation that let the hierarchy developer describe an actual interface for their library. Behind that interface the developer can provide appropriate code to ensure backward compatibility. We implement the HB language in the hierarchy-builder addon for the Coq system using the Elpi extension language.
Cite as

Cyril Cohen, Kazuhiko Sakaguchi, and Enrico Tassi. Hierarchy Builder: Algebraic hierarchies Made Easy in Coq with Elpi (System Description). In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{cohen_et_al:LIPIcs.FSCD.2020.34,
  author =	{Cohen, Cyril and Sakaguchi, Kazuhiko and Tassi, Enrico},
  title =	{{Hierarchy Builder: Algebraic hierarchies Made Easy in Coq with Elpi}},
  booktitle =	{5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)},
  pages =	{34:1--34:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-155-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{167},
  editor =	{Ariola, Zena M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2020.34},
  URN =		{urn:nbn:de:0030-drops-123562},
  doi =		{10.4230/LIPIcs.FSCD.2020.34},
  annote =	{Keywords: Algebraic Hierarchy, Packed Classes, Coq, Elpi, Metaprogramming, \lambdaProlog}
}
  • Refine by Author
  • 4 Cohen, Cyril
  • 2 Affeldt, Reynald
  • 2 Mörtberg, Anders
  • 1 Asgeirsson, Dagur
  • 1 Ayala-Rincón, Mauricio
  • Show More...

  • Refine by Classification
  • 7 Theory of computation → Type theory
  • 6 Theory of computation → Logic and verification
  • 2 Computing methodologies → Symbolic and algebraic manipulation
  • 2 Software and its engineering → Formal software verification
  • 2 Theory of computation → Automated reasoning
  • Show More...

  • Refine by Keyword
  • 4 Coq
  • 2 Metaprogramming
  • 2 dependent type theory
  • 1 Abel-Ruffini
  • 1 Algebraic Hierarchy
  • Show More...

  • Refine by Type
  • 18 document

  • Refine by Publication Year
  • 13 2024
  • 2 2019
  • 1 2018
  • 1 2020
  • 1 2021

Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact