Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis

TL;DR

This paper presents a novel approach combining policy iteration and semi-definite relaxation to compute precise numerical invariants in static analysis, extending linear templates to non-linear functions for better program analysis accuracy.

Contribution

It introduces a new domain based on level sets of non-linear functions and applies semi-definite relaxation with policy iteration for precise invariant computation.

Findings

Effective in analyzing filters and integration schemes

Handles degenerate cases like symplectic schemes

Provides accurate fixpoint solutions for non-linear invariants

Abstract

We introduce a new domain for finding precise numerical invariants of programs by abstract interpretation. This domain, which consists of level sets of non-linear functions, generalizes the domain of linear "templates" introduced by Manna, Sankaranarayanan, and Sipma. In the case of quadratic templates, we use Shor's semi-definite relaxation to derive computable yet precise abstractions of semantic functionals, and we show that the abstract fixpoint equation can be solved accurately by coupling policy iteration and semi-definite programming. We demonstrate the interest of our approach on a series of examples (filters, integration schemes) including a degenerate one (symplectic scheme).