[Paper Review] KBO as a Satisfaction Problem

Harald Zankl, Aart Middeldorp|arXiv (Cornell University)|Aug 6, 2006

Logic, programming, and type systems15 references3 citations

TL;DR

This paper proposes an efficient method for proving termination of term rewrite systems (TRSs) using the Knuth-Bendix order by encoding weight function and precedence constraints into propositional logic, then checking satisfiability. A satisfying assignment yields a valid weight function and precedence that orient all rules from left to right, enabling automated termination proofs via SAT solving.

ABSTRACT

Abstract This note presents an approach to prove termination of term rewrite systems (TRSs) with the Knuth-Bendix order efficiently. The constraints for the weight function as well as for the precedence are encoded in propositional logic and the resulting formula is tested for satisfiability. Any satisfying assignment represents a weight function and a precedence such the induced Knuth-Bendix order orients the rules of the encoded TRS from left to right. 1.1

Motivation & Objective

• To develop an efficient method for proving termination of term rewrite systems using the Knuth-Bendix order.
• To encode the constraints on weight functions and precedence relations as propositional logic formulas.
• To leverage SAT solvers to automatically determine valid weight functions and precedence orders.
• To ensure that the induced Knuth-Bendix order orients all rules from left to right, confirming termination.

Proposed method

• Represent the constraints for the weight function and precedence as a formula in propositional logic.
• Map each symbol in the TRS to a propositional variable representing its assigned weight and precedence relation.
• Construct a logical formula that encodes the conditions under which the Knuth-Bendix order orients all rules from left to right.
• Use a SAT solver to test the satisfiability of the resulting formula.
• Interpret any satisfying assignment as a valid weight function and precedence relation.
• Verify that the resulting order orients all rules of the TRS from left to right, confirming termination.

Experimental results

Research questions

• RQ1 Can the constraints for the Knuth-Bendix order be effectively encoded in propositional logic?
• RQ2 Can SAT solving be used to efficiently find a valid weight function and precedence for termination proofs?
• RQ3 Does this encoding method reliably produce a Knuth-Bendix order that orients all rules from left to right?
• RQ4 How does this approach compare in efficiency to traditional methods for Knuth-Bendix termination proofs?

Key findings

• The encoding of Knuth-Bendix order constraints into propositional logic enables automated termination proofs via SAT solving.
• Any satisfying assignment of the propositional formula directly yields a valid weight function and precedence.
• The method ensures that the induced Knuth-Bendix order orients all rules of the TRS from left to right.
• The approach provides an efficient alternative to manual or heuristic-based construction of weight functions and precedences.