Certifying Differential Equation Solutions from Computer Algebra Systems in Isabelle/HOL

Abstract

The Isabelle/HOL proof assistant has a powerful library for continuous analysis, which provides the foundation for verification of hybrid systems. However, Isabelle lacks automated proof support for continuous artifacts, which means that verification is often manual. In contrast, Computer Algebra Systems (CAS), such as Mathematica and SageMath, contain a wealth of efficient algorithms for matrices, differential equations, and other related artifacts. Nevertheless, these algorithms are not verified, and thus their outputs cannot, of themselves, be trusted for use in a safety critical system. In this paper we integrate two CAS systems into Isabelle, with the aim of certifying symbolic solutions to ordinary differential equations. This supports a verification technique that is both automated and trustworthy.

Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

“…Existing verification techniques for CAS often focus on specific subroutines or functions [6,7,13,21,25,26,30,31], such as a specific theorems [28], differential equations [23], or the implementation of the math.h library [29]. Most common are verification approaches that rely on intermediate verification languages [6,21,23,25,26], such as Boogie [2,30] or Why3 [5,26], which, in turn, rely on proof assistants and theorem provers, such as Coq [4,6], Isabelle [23,33], or HOL Light [20,21,25]. Kaliszyk and Wiedijk [25] proposed on entire new CAS which is built on top of the proof assistant HOL Light so that each simplification step can be proven by the underlying architecture.…”

“…For [11, (18.17.47)], WED (Mathematica's kernel) ran into a segmentation fault (core dumped) for n > 1. The kernel of the full version of Mathematica gracefully died without returning an output 23 .…”

Hybrid Systems Verification with Isabelle/HOL: Simpler Syntax, Better Models, Faster Proofs

“…While the proof methods so far describe the second workflow using framed versions of the inference rule of dL, the first workflow of the framework [4] supports verification with certified solutions (flows), which can be supplied using a CAS [37]. We have developed a proof method called local-flow for certifying that a flow is the unique solution to an ODE.…”