Constraint-Based Multi-Completion Procedures for Term Rewriting Systems

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In equational theorem proving, convergent term rewriting systems play a crucial role. In order to compute convergent term rewriting systems, the standard completion procedure (KB) was proposed by Knuth and Bendix and has been improved in various ways. The multi-completion system MKB developed by Kurihara and Kondo accepts as input a set of reduction orders in addition to equations and efficiently simulates parallel processes each of which executes the KB procedure with one of the given orderings. Wehrman and Stump also developed a new variant of completion procedure, constraint-based completion, in which reduction orders need not be given by using automated modern termination checkers. As a result, the constraint-based procedures simulate the execution of parallel KB processes in a sequential way, but naive search algorithms sometimes cause serious inefficiency when the number of the potential reduction orders is large. In this paper, we present a new procedure, called a constraint-based multi-completion procedure MKBcs, by augmenting the constraint-based completion with the framework of the multi-completion for suppressing the combinatorial explosion by sharing inferences among the processes. The existing constraint-based system Slothrop, which basically employs the best-first search, is more efficient when its built-in heuristics for process selection are appropriate, but when they are not, our system is more efficient. Therefore, both systems have their role to play.