Formal Methods for Computer Arithmetic

Overview

We are interested in the use of formal methods for verifying algorithms that handle numerical values, be they floating-point numbers as described by the IEEE-754 standard, arbitrary-precision integer as in GMP, or intervals. These algorithms range from basic blocks of mathematical libraries to whole programs for numerical analysis. They are usually simple when it comes to control flow and data structures, but their correctness might depend on some intricate mathematical reasoning in number theorem, real analysis, and so on. This makes their verification out of range of purely automatic approaches.

We cover a wide range of aspects of the field: formalization of arithmetic and analysis, design of dedicated decision procedures, specification and verification of libraries and programs.

Members

Permanent

PhD Students

Louise Ben Salem-Knapp

PostDoc

Selected publications

  • G. Melquiond, R. Rieu-Helft. WhyMP, a formally verified arbitrary-precision integer library. ISSAC, 2020. HAL
  • A. Mahboubi, G. Melquiond, T. Sibut-Pinote. Formally verified approximations of definite integrals. JAR, 2019. HAL
  • S. Boldo, G. Melquiond. Computer arithmetic and formal proofs: Verifying floating-point algorithms with the Coq System. ISTE-Elsevier, 2018.
  • J-M. Muller et al. Handbook of floating-point arithmetic. Birkhaüser, 2010, 2018.
  • S. Conchon, M. Iguernelala, K. Ji, G. Melquiond, C. Fumex. A three-tier strategy for reasoning about floating-point numbers in SMT. CAV, 2017. HAL
  • S. Boldo, J-H. Jourdan, G. Melquiond, X. Leroy. Verified compilation of floating-point computations. JAR, 2015. HAL
  • C. Fumex, C. Marché, Y. Moy. Automating the verification of floating-point programs. VSTTE, 2017. HAL

Selected software

Flocq

Formalization of fixed- and floating-point arithmetic for the Coq proof assistant

CoqInterval

Coq tactics for automatically proving inequalities over real numbers

Gappa

Decision procedure for arithmetic properties of floating-point algorithms

WhyMP

Efficient C library for arbitrary-precision integer computations, formally verified and compatible with GMP

Projects

NuSCAP

Numerical Safety for Computer-Aided Proofs
ANR
2021-2025

EMC2

Extreme-scale Mathematically-based Computational Chemistry
ERC Synergy Grant

FRESCO

Fast and Reliable Symbolic Computation
ERC Consolidator Grant