[Paper Review] SAT Solving for Argument Filterings

Michael Codish, Peter Schneider–Kamp+3 more|ArXiv.org|May 17, 2006

Logic, programming, and type systems3 citations

TL;DR

This paper presents a SAT-based encoding that jointly searches for lexicographic path orders (LPO) and argument filterings to prove termination of term rewrite systems via the dependency pair method. By encoding the interdependent constraints of LPO precedence and argument filtering into a single propositional formula, the approach enables efficient SAT solving, resulting in orders-of-magnitude speedups and significantly increased proving power compared to traditional methods.

ABSTRACT

This paper introduces a propositional encoding for lexicographic path orders in connection with dependency pairs. This facilitates the application of SAT solvers for termination analysis of term rewrite systems based on the dependency pair method. We address two main inter-related issues and encode them as satisfiability problems of propositional formulas that can be efficiently handled by SAT solving: (1) the combined search for a lexicographic path order together with an \emph{argument filtering} to orient a set of inequalities; and (2) how the choice of the argument filtering influences the set of inequalities that have to be oriented. We have implemented our contributions in the termination prover AProVE. Extensive experiments show that by our encoding and the application of SAT solvers one obtains speedups in orders of magnitude as well as increased termination proving power.

Motivation & Objective

• To address the automation bottleneck in dependency pair-based termination analysis caused by the vast search space of argument filterings.
• To integrate the search for lexicographic path orders (LPO) and argument filterings into a unified, efficient framework.
• To improve the proving power and performance of automated termination tools by leveraging SAT solving on combined constraints.
• To demonstrate that SAT solving with the proposed encoding outperforms existing non-SAT-based approaches in both speed and success rate.

Proposed method

• Propose a propositional encoding that captures all constraints from LPO definition and argument filtering choices.
• Model the dependency between argument filterings and the set of usable rules in the dependency pair framework.
• Encode the precedence relations on function symbols and filtering choices into a single SAT formula that is satisfiable iff a valid LPO and filtering pair exist.
• Integrate the encoding into the AProVE termination prover to enable SAT-based search for LPO and argument filterings.
• Optimize the encoding with pruning techniques to reduce CNF size and improve SAT solver performance.
• Use SAT solvers to efficiently explore the combined search space of LPO and argument filterings, with each model yielding a valid termination proof.

Experimental results

Research questions

• RQ1 Can the combined search for lexicographic path orders and argument filterings be effectively encoded as a propositional satisfiability problem?
• RQ2 How does the choice of argument filtering influence the set of inequalities that must be oriented in the dependency pair method?
• RQ3 Can SAT solving be leveraged to achieve significant performance improvements in automated termination analysis with argument filterings?
• RQ4 Does the proposed encoding increase the proving power of termination tools compared to non-SAT-based configurations?
• RQ5 What is the impact of optimization techniques on the efficiency of the SAT-based encoding in practice?

Key findings

• The SAT-based configuration in AProVE achieved speedups of orders of magnitude compared to non-SAT-based configurations, with total runtimes reduced from over 30,000 seconds to just 115 seconds on the same benchmark set.
• The SAT-based approach proved termination for 5 hard examples that were unsolved by any tool in the 2005 International Termination Competition, including one in just 0.95 seconds.
• The number of successful proofs increased significantly with the SAT-based configuration, with 348 out of 773 examples proved in strict-LPO mode versus 338 with the non-SAT version.
• The encoding reduced the total runtime by a factor of 16 when key optimizations were enabled, dropping from 1968 seconds to 119 seconds on a test set with 10-minute timeouts.
• The SAT-based configuration had almost no time-outs (0 out of 773 examples), whereas the non-SAT version timed out on 368 examples in strict-LPO mode.
• The largest CNF formula generated during proofs contained 29,794 literals, and SAT solving was completed in under 50 milliseconds on average, demonstrating scalability.