Optimization of Farkas' Lemma-based linear invariant generation using divide-and-conquer with pruning

Abstract

Formal verification plays a critical role in contemporary computer science, offering mathematically rigorous methods to ensure the correctness, reliability, and security of programs. Loops, due to their complexity and uncertainty, have become a major challenge in program verification. Loop invariants are often employed to abstract the properties of loops within a program, making the automatic generation of such invariants a pivotal challenge. Among the various methods, template-based frameworks grounded in Farkas' Lemma are recognized for their effectiveness in generating tight invariants in the realm of constraint solving. Recent advances have identified the conversion from conjunctive normal form (CNF) to disjunctive normal form (DNF) as a major bottleneck, leading to a combinatorial explosion. In this study, we introduce an optimized algorithm to address the combinatorial explosion by trading off space for time efficiency. Our approach employs two key strategies, divide-and-conquer, and pruning, to boost speed. First, we apply a divide-and-conquer strategy to decompose a complex problem into smaller, more manageable subproblems that can be solved quickly and in parallel. Second, we intelligently apply a pruning strategy, navigating the depth-first search process to avoid unnecessary checks. These improvements maintain the accuracy and speed up the analysis. We constructed a small dataset to showcase the superiority of our tool, which achieved an average speedup of 9.27x on this dataset. The experiments demonstrate that our method provides significant acceleration while maintaining accuracy and indicate that our approach outperforms the state-of-the-art methods.