Optimization of Lyapunov Invariants in Verification of Software Systems

TL;DR

This paper introduces a control-theoretic approach using Lyapunov invariants and convex optimization to verify software safety and performance, enabling automated certification of various program properties.

Contribution

It adapts Lyapunov functions from control theory to software verification, formulating invariant search as a convex optimization problem for the first time.

Findings

Convex optimization can effectively find Lyapunov invariants for software verification.

The framework verifies properties like overflow, division-by-zero, and termination.

Feasibility of the optimization provides certification of software correctness.

Abstract

The paper proposes a control-theoretic framework for verification of numerical software systems, and puts forward software verification as an important application of control and systems theory. The idea is to transfer Lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications. These include but are not limited to absence of overflow, absence of division-by-zero, termination in finite time, presence of dead-code, and certain user-specified assertions. Central to this framework are Lyapunov invariants. These are properly constructed functions of the program variables, and satisfy certain properties-resembling those of Lyapunov functions-along the execution trace. The search for the invariants can be formulated as a convex optimization problem. If the associated optimization problem is feasible, the result is a certificate for the specification.