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DOI: 10.4230/LIPIcs.ITP.2021.14

A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm

Authors: Katherine Cordwell, Yong Kiam Tan, and André Platzer

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

Abstract

We formalize the univariate fragment of Ben-Or, Kozen, and Reif’s (BKR) decision procedure for first-order real arithmetic in Isabelle/HOL. BKR’s algorithm has good potential for parallelism and was designed to be used in practice. Its key insight is a clever recursive procedure that computes the set of all consistent sign assignments for an input set of univariate polynomials while carefully managing intermediate steps to avoid exponential blowup from naively enumerating all possible sign assignments (this insight is fundamental for both the univariate case and the general case). Our proof combines ideas from BKR and a follow-up work by Renegar that are well-suited for formalization. The resulting proof outline allows us to build substantially on Isabelle/HOL’s libraries for algebra, analysis, and matrices. Our main extensions to existing libraries are also detailed.

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Katherine Cordwell, Yong Kiam Tan, and André Platzer. A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cordwell_et_al:LIPIcs.ITP.2021.14,
  author =	{Cordwell, Katherine and Tan, Yong Kiam and Platzer, Andr\'{e}},
  title =	{{A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{14:1--14:20},
  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.2021.14},
  URN =		{urn:nbn:de:0030-drops-139099},
  doi =		{10.4230/LIPIcs.ITP.2021.14},
  annote =	{Keywords: quantifier elimination, matrix, theorem proving, real arithmetic}
}

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DOI: 10.4230/LIPIcs.FSCD.2020.14

Refining Constructive Hybrid Games

Authors: Rose Bohrer and André Platzer

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

Abstract

We extend the constructive differential game logic (CdGL) of hybrid games with a refinement connective that relates two hybrid games. In addition to CdGL’s ability to prove the existence of winning strategies for specific postconditions of hybrid games, game refinements relate two games to one another. That makes it possible to prove that any winning strategy for any postcondition of one game carries over to a winning strategy for the other. Since CdGL is constructive, a computable winning strategy can be extracted from a proof that a player wins a game. A folk theorem says that any such winning strategy for a hybrid game gives rise to a corresponding hybrid system satisfying the same property. We make this precise using CdGL’s game refinements and prove correct the construction of hybrid systems from winning strategies of hybrid games.

Cite as

Rose Bohrer and André Platzer. Refining Constructive Hybrid Games. In 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 167, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bohrer_et_al:LIPIcs.FSCD.2020.14,
  author =	{Bohrer, Rose and Platzer, Andr\'{e}},
  title =	{{Refining Constructive Hybrid Games}},
  booktitle =	{5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)},
  pages =	{14:1--14:19},
  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.2020.14},
  URN =		{urn:nbn:de:0030-drops-123369},
  doi =		{10.4230/LIPIcs.FSCD.2020.14},
  annote =	{Keywords: Hybrid Games, Constructive Logic, Refinement, Game Logic}
}

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A Verified Decision Procedure for Univariate Real Arithmetic with the BKR Algorithm

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Refining Constructive Hybrid Games

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