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Verified Quadratic Virtual Substitution for Real Arithmetic

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Formal Methods (FM 2021)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 13047))

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Abstract

This paper presents a formally verified quantifier elimination (QE) algorithm for first-order real arithmetic by linear and quadratic virtual substitution (VS) in Isabelle/HOL. The Tarski-Seidenberg theorem established that the first-order logic of real arithmetic is decidable by QE. However, in practice, QE algorithms are highly complicated and often combine multiple methods for performance. VS is a practically successful method for QE that targets formulas with low-degree polynomials. To our knowledge, this is the first work to formalize VS for quadratic real arithmetic including inequalities. The proofs necessitate various contributions to the existing multivariate polynomial libraries in Isabelle/HOL. Our framework is modularized and easily expandable (to facilitate integrating future optimizations), and could serve as a basis for developing practical general-purpose QE algorithms. Further, as our formalization is designed with practicality in mind, we export our development to SML and test the resulting code on 378 benchmarks from the literature, comparing to Redlog, Z3, Wolfram Engine, and SMT-RAT. This identified inconsistencies in some tools, underscoring the significance of a verified approach for the intricacies of real arithmetic.

This material is based upon work supported by the National Science Foundation under Grant No. CNS-1739629, a National Science Foundation Graduate Research Fellowship under Grants Nos. DGE1252522 and DGE1745016, and by the AFOSR under grant number FA9550-16–1-0288. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or of AFOSR.

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Notes

  1. 1.

    Notice that if \(y = 0\), then both \(\sqrt{2}y^2 + 2y + 1 = 0\) and \(y^2\cdot (2y + 1) \le 0 \wedge 2y^4 - (2y + 1)^2= 0\) are false. If instead \(y \ne 0\), then \(\sqrt{2}y^2 + 2y + 1 = 0\) is true exactly when \(\sqrt{2} = -(2y + 1)/y^2\), or exactly when \(-(2y + 1)/y^2 \ge 0 \wedge 2y^4 - (2y + 1)^2= 0\), which is logically equivalent to \(y^2\cdot (2y + 1) \le 0 \wedge 2y^4 - (2y + 1)^2= 0\), as desired.

  2. 2.

    MacBook Pro 2019 with 2.6 GHz Intel Core i7 (model 9750H) and 32 GB memory (2667 MHz DDR4 SDRAM).

  3. 3.

    https://sourceforge.net/projects/reduce-algebra/files/snapshot_2021-04-13/.

  4. 4.

    https://sourceforge.net/projects/reduce-algebra/files/snapshot_2021-07-16/.

  5. 5.

    https://github.com/ths-rwth/smtrat/releases/tag/21.05.

  6. 6.

    https://github.com/Z3Prover/z3/releases/tag/z3-4.8.10.

  7. 7.

    http://mlton.org/.

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Acknowledgment

We wish to thank Fabian Immler for his substantial contributions at CMU to the polynomial theories of Isabelle/HOL and regret that his current industry position precludes our ability to include him as a coauthor. Thank you also to the anonymous FM reviewers for their useful feedback.

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Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Matias Scharager, Katherine Cordwell, Stefan Mitsch & André Platzer

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  1. Matias Scharager
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Correspondence to Matias Scharager .

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Editors and Affiliations

  1. University of Twente, Enschede, The Netherlands

    Marieke Huisman

  2. NASA Ames Research Center, Moffett Field, CA, USA

    Corina Păsăreanu

  3. Institute of Software, Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Scharager, M., Cordwell, K., Mitsch, S., Platzer, A. (2021). Verified Quadratic Virtual Substitution for Real Arithmetic. In: Huisman, M., Păsăreanu, C., Zhan, N. (eds) Formal Methods. FM 2021. Lecture Notes in Computer Science(), vol 13047. Springer, Cham. https://doi.org/10.1007/978-3-030-90870-6_11

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