Knuth-Bendix Completion with Modern Termination Checking, Master\u27s Thesis, August 2006

Knuth-Bendix completion is a technique for equational automated theorem proving based on term rewriting. This classic procedure is parametrized by an equational theory and a (well-founded) reduction order used at runtime to ensure termination of intermediate rewriting systems. Any reduction order can be used in principle, but modern completion tools typically implement only a few classes of such orders (e.g., recursive path orders and polynomial orders). Consequently, the theories for which completion can possibly succeed are limited to those compatible with an instance of an implemented class of orders. Finding and specifying a compatible order, even among a small number of classes, is challenging in practice and crucial to the success of the method. In this thesis, a new variant on the Knuth-Bendix completion procedure is developed in which no order is provided by the user. Modern termination-checking methods are instead used to verify termination of rewriting systems. We prove the new method correct and also present an implementation called Slothrop which obtains solutions for theories that do not admit typical orders and that have not previously been solved by a fully automatic tool

Mining Propositional Simplification Proofs for Small Validating Clauses

The problem of obtaining small conflict clauses in SMT systems has received a great deal of attention recently. We report work in progress to find small subsets of the current partial assignment that imply the goal formula when it has been propositionally simplified to a boolean value. The approach used is algebraic proof mining. Proofs from a propositional reasoner that the goal is equivalent to a boolean value (in the current assignment) are viewed as first-order terms. An equational theory between proofs is then defined, which is sound with respect to the quasiorder “proves a more general set theorems. ” The theory is completed to obtain a convergent rewrite system that puts proofs into a canonical form. While our canonical form does not use the smallest subset of the current assignment, it does drop many unnecessary parts of the proof. The paper concludes with discussion of the complexity of the problem and effectiveness of the approach. Key words: SAT, SMT, algebraic proof mining, term rewriting.