3 Search Results for "Rybalchenko, Andrey"

Document
DOI: 10.4230/LIPIcs.ITP.2024.35
Formal Verification of the Empty Hexagon Number

Authors: Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule

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

Abstract
A recent breakthrough in computer-assisted mathematics showed that every set of 30 points in the plane in general position (i.e., no three points on a common line) contains an empty convex hexagon. Heule and Scheucher solved this problem with a combination of geometric insights and automated reasoning techniques by constructing CNF formulas ϕ_n, with O(n⁴) clauses, such that if ϕ_n is unsatisfiable then every set of n points in general position must contain an empty convex hexagon. An unsatisfiability proof for n = 30 was then found with a SAT solver using 17 300 CPU hours of parallel computation. In this paper, we formalize and verify this result in the Lean theorem prover. Our formalization covers ideas in discrete computational geometry and SAT encoding techniques by introducing a framework that connects geometric objects to propositional assignments. We see this as a key step towards the formal verification of other SAT-based results in geometry, since the abstractions we use have been successfully applied to similar problems. Overall, we hope that our work sets a new standard for the verification of geometry problems relying on extensive computation, and that it increases the trust the mathematical community places in computer-assisted proofs.
Cite as

Bernardo Subercaseaux, Wojciech Nawrocki, James Gallicchio, Cayden Codel, Mario Carneiro, and Marijn J. H. Heule. Formal Verification of the Empty Hexagon Number. In 15th International Conference on Interactive Theorem Proving (ITP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 309, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{subercaseaux_et_al:LIPIcs.ITP.2024.35,
  author =	{Subercaseaux, Bernardo and Nawrocki, Wojciech and Gallicchio, James and Codel, Cayden and Carneiro, Mario and Heule, Marijn J. H.},
  title =	{{Formal Verification of the Empty Hexagon Number}},
  booktitle =	{15th International Conference on Interactive Theorem Proving (ITP 2024)},
  pages =	{35:1--35:19},
  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.2024.35},
  URN =		{urn:nbn:de:0030-drops-207633},
  doi =		{10.4230/LIPIcs.ITP.2024.35},
  annote =	{Keywords: Empty Hexagon Number, Discrete Computational Geometry, Erd\H{o}s-Szekeres}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2024.139
T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations

Authors: Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah

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

Abstract
We introduce a powerful termination algorithm for structurally recursive functions that improves on the core ideas behind lexicographic termination algorithms for functional programs. The algorithm generates linear-lexicographic combinations of primitive measure functions measuring the recursive structure of terms. We introduce a measure language that enables the simplification and comparison of measures and we prove meta-theoretic properties of our measure language. Moreover, we demonstrate our algorithm, on an untyped first-order functional language and prove its soundness and that it runs in polynomial time. We also provide a Haskell implementation. As part of this work, we also show how to solve the maximisation of negative vector-components as a linear program.
Cite as

Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah. T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{giles_et_al:LIPIcs.ICALP.2024.139,
  author =	{Giles, Raphael Douglas and Jackson, Vincent and Rizkallah, Christine},
  title =	{{T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{139:1--139:19},
  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.2024.139},
  URN =		{urn:nbn:de:0030-drops-202827},
  doi =		{10.4230/LIPIcs.ICALP.2024.139},
  annote =	{Keywords: Termination, Recursive functions}
}
Document
DOI: 10.4230/DagRep.3.4.1
Formal Verification of Distributed Algorithms (Dagstuhl Seminar 13141)

Authors: Bernadette Charron-Bost, Stephan Merz, Andrey Rybalchenko, and Josef Widder

Published in: Dagstuhl Reports, Volume 3, Issue 4 (2013)

Abstract
The Dagstuhl Seminar 13141 "Formal Verification of Distributed Algorithms" brought together researchers from the areas of distributed algorithms, model checking, and semi-automated proofs with the goal to establish a common base for approaching the many open problems in verification of distributed algorithms. In order to tighten the gap between the involved communities, who have been quite separated in the past, the program contained tutorials on the basics of the concerned fields. In addition to technical talks, we also had several discussion sessions, whose goal was to identify the most pressing research challenges. This report describes the program and the outcomes of the seminar.
Cite as

Bernadette Charron-Bost, Stephan Merz, Andrey Rybalchenko, and Josef Widder. Formal Verification of Distributed Algorithms (Dagstuhl Seminar 13141). In Dagstuhl Reports, Volume 3, Issue 4, pp. 1-16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{charronbost_et_al:DagRep.3.4.1,
  author =	{Charron-Bost, Bernadette and Merz, Stephan and Rybalchenko, Andrey and Widder, Josef},
  title =	{{Formal Verification of Distributed Algorithms (Dagstuhl Seminar 13141)}},
  pages =	{1--16},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2013},
  volume =	{3},
  number =	{4},
  editor =	{Charron-Bost, Bernadette and Merz, Stephan and Rybalchenko, Andrey and Widder, Josef},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.4.1},
  URN =		{urn:nbn:de:0030-drops-40747},
  doi =		{10.4230/DagRep.3.4.1},
  annote =	{Keywords: Distributed algorithms; semi-automated proofs; model checking}
}
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