Scalable linear invariant generation with Farkas’ lemma

Abstract

Invariant generation is a classical problem to automatically generate invariants to aid the formal analysis of programs. In this work, we consider the problem of generating tight linear-invariants over affine programs (i.e., programs with affine guards and updates) without a prescribed goal property. In the literature, the only known sound and complete characterization to solve this problem is via Farkas’ Lemma (FL), and has been implemented through either quantifier elimination or reasonable heuristics. Although FL-based approaches can generate highly accurate linear invariants from the completeness of FL, the main bottleneck to applying these approaches is the scalability issue caused by either non-linear constraints or combinatorial explosion. We base our approach on the only practical FL-based approach [Sankaranarayanan ‍ et al. , SAS 2004] that applies FL with reasonable heuristics, and develop two novel and independent improvements to leverage the scalability. The first improvement is the novel idea to generate invariants at one program location in a single invariant-generation process, so that the invariants for each location are generated separately rather than together in a single computation. This idea naturally leads to a parallel processing that divides the invariant-generation task for all program locations by assigning the locations separately to multiple processors. Moreover, the idea enables us to develop detailed technical improvements to further reduce the combinatorial explosion in the original work [Sankaranarayanan ‍ et al. , SAS 2004]. The second improvement is a segmented subsumption testing in the CNF-to-DNF expansion that allows discovering more local subsumptions in advance. We formally prove that our approach has the same accuracy as the original work and thus does not incur accuracy loss on the generated invariants. Moreover, experimental results on representative benchmarks involving non-trivial linear invariants demonstrate that our approach improves the runtime of the original work by several orders of magnitude, even in the non-parallel scenario that sums up the execution time for all program locations. Hence, our approach constitutes the first significant improvement in FL-based approaches for linear invariant generation after almost two decades.