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Solvable Polynomial Ideals: The Ideal Reflection for Program Analysis

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Zachary KincaidAuthors Info & Claims

Proceedings of the ACM on Programming Languages, Volume 8, Issue POPL

Article No.: 25, Pages 724 - 752

https://doi.org/10.1145/3632867

Published: 05 January 2024 Publication History

Abstract

This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to generate all polynomial invariants for a restricted class of programs, but cannot be applied to programs outside of this class---for instance, programs with nested loops, conditional branching, unstructured control flow, etc. There currently lacks approaches to apply these prior methods to the case of general programs. This paper bridges that gap. Instead of restricting the kinds of programs we can handle, our method abstracts every loop into a model that can be solved with prior techniques, bringing to bear prior work on solvable polynomial maps to general programs. While no method can generate all polynomial invariants for arbitrary programs, our method establishes its merit through a monotonicty result. We have implemented our techniques, and tested them on a suite of benchmarks from the literature. Our experiments indicate our techniques show promise on challenging verification tasks requiring non-linear reasoning.

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Cited By

Amy MLunderville J(2025)Linear and Non-linear Relational Analyses for Quantum Program OptimizationProceedings of the ACM on Programming Languages10.1145/37048739:POPL(1072-1103)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704873
Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704862
Bayarmagnai EMohammadi FPrébet R(2024)Algebraic Tools for Computing Polynomial Loop InvariantsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669710(371-381)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669710
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Index Terms

Solvable Polynomial Ideals: The Ideal Reflection for Program Analysis
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials
        Gröbner bases and other special bases
2. Theory of computation
  1. Logic

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cover image Proceedings of the ACM on Programming Languages

Proceedings of the ACM on Programming Languages Volume 8, Issue POPL

January 2024

2820 pages

EISSN:2475-1421

DOI:10.1145/3554315

Editor:
Michael Hicks
Amazon, USA

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Copyright © 2024 Owner/Author.

This work is licensed under a Creative Commons Attribution 4.0 International License.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 January 2024

Published in PACMPL Volume 8, Issue POPL

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Cited By

Amy MLunderville J(2025)Linear and Non-linear Relational Analyses for Quantum Program OptimizationProceedings of the ACM on Programming Languages10.1145/37048739:POPL(1072-1103)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704873
Ait El Manssour RKenison GShirmohammadi MVaronka A(2025)Simple Linear Loops: Algebraic Invariants and ApplicationsProceedings of the ACM on Programming Languages10.1145/37048629:POPL(745-771)Online publication date: 9-Jan-2025
https://dl.acm.org/doi/10.1145/3704862
Bayarmagnai EMohammadi FPrébet R(2024)Algebraic Tools for Computing Polynomial Loop InvariantsProceedings of the 2024 International Symposium on Symbolic and Algebraic Computation10.1145/3666000.3669710(371-381)Online publication date: 16-Jul-2024
https://dl.acm.org/doi/10.1145/3666000.3669710
Wang CLin F(2024)On Polynomial Expressions with C-Finite Recurrences in Loops with Nested Nondeterministic BranchesComputer Aided Verification10.1007/978-3-031-65627-9_20(409-430)Online publication date: 26-Jul-2024
https://doi.org/10.1007/978-3-031-65627-9_20

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