A Primal-Dual Perspective on Program Verification Algorithms (Extended Version) (2501.01512v1)

Abstract: Many algorithms in verification and automated reasoning leverage some form of duality between proofs and refutations or counterexamples. In most cases, duality is only used as an intuition that helps in understanding the algorithms and is not formalized. In other cases, duality is used explicitly, but in a specially tailored way that does not generalize to other problems. In this paper we propose a unified primal-dual framework for designing verification algorithms that leverage duality. To that end, we generalize the concept of a Lagrangian that is commonly used in linear programming and optimization to capture the domains considered in verification problems, which are usually discrete, e.g., powersets of states, predicates, ranking functions, etc. A Lagrangian then induces a primal problem and a dual problem. We devise an abstract primal-dual procedure that simultaneously searches for a primal solution and a dual solution, where the two searches guide each other. We provide sufficient conditions that ensure that the procedure makes progress under certain monotonicity assumptions on the Lagrangian. We show that many existing algorithms in program analysis, verification, and automated reasoning can be derived from our algorithmic framework with a suitable choice of Lagrangian. The Lagrangian-based formulation sheds new light on various characteristics of these algorithms, such as the ingredients they use to ensure monotonicity and guarantee progress. We further use our framework to develop a new validity checking algorithm for fixpoint logic over quantified linear arithmetic. Our prototype achieves promising results and in some cases solves instances that are not solved by state-of-the-art techniques.

• The paper introduces a unified primal-dual framework for designing program verification algorithms by generalizing Lagrangian duality concepts from linear programming to discrete domains.
• The framework unifies and reinterprets existing verification methods like CEGAR, providing a common formal grounding and algorithmic insights based on primal-dual interactions.
• This primal-dual framework facilitates developing novel algorithms, including a validity checker for fixpoint logic over quantified linear arithmetic with promising empirical results.

The paper "A Primal-Dual Perspective on Program Verification Algorithms (Extended Version)" explores a unified framework for designing verification algorithms through a primal-dual perspective. The authors propose a generalized framework leveraging duality concepts, typically used in linear programming, to create verification algorithms that simultaneously pursue proof and refutation.

Key Contributions and Approach

1. Unified Primal-Dual Framework: The paper introduces a unified primal-dual framework for designing verification algorithms. The authors generalize the concept of a Lagrangian, which is commonly used in linear programming, to discrete domains often encountered in verification problems, such as powersets of states, predicates, and ranking functions. This generalized approach aims to induce primal and dual problems where searches for solutions guide each other.
2. Abstract Primal-Dual Procedure: The authors devise an abstract primal-dual procedure that searches simultaneously for primal and dual solutions. This procedure applies to various verification contexts by considering duality not just as an intuitive concept but as a formalized method that can produce algorithms directly from the primal-dual perspective.
3. Leveraging Monotonicity and Progress: By defining sufficient conditions, the primal-dual procedure is shown to make progress under certain monotonicity assumptions related to the Lagrangian. This ensures that solutions are produced through the guidance between primal and dual searches.
4. Applying the Framework to Existing Algorithms: The framework is used to reinterpret and extend many existing algorithms in program analysis and verification. This includes techniques such as Counterexample-Guided Abstraction Refinement (CEGAR), must-may analysis, ICE learning, and others. In this context, various ingredients of algorithms, like ensuring monotonicity and guaranteeing progress, can be effectively analyzed.
5. Novel Algorithm Development: Besides reinterpreting existing methods, the authors develop a new validity checking algorithm for fixpoint logic over quantified linear arithmetic using their proposed framework. This algorithm achieves promising results in empirical evaluation.
6. Soundness and Comparisons: The framework presents a unique perspective on application cases, letting the authors compare different algorithms based on their characteristics of strong duality, symmetry between primal and dual problems, and accumulation on various sides of the search.

Insights and Implications

• Algorithmic Insight: The paper consolidates understanding among many existing methods used in verification and reasoning under a common formal grounding of primal-dual interactions. By doing so, it offers an insightful lens for viewing verification procedures as closely aligned with dual optimization problems.
• Generalization of Duality: The process of generalizing the linear programming notions to verification problems introduces a structured perspective to leverage duality beyond mere abstraction into verified algorithmic procedures.
• Enhancing Solver Efficacy: The duality framework not only offers reinterpretation but can enhance solver performance for new types of problems as discussed with quantified arithmetic over fixpoint logic.

The authors conclude by emphasizing the potential of future work incorporating Lagrangian duality in contexts such as linear/semidefinite programming applications to verification, possibly inspiring further developments in program verification methodologies and solver tools.