Completion after Program Inversion of Injective Functions

Given a constructor term rewriting system that defines injective functions, the inversion compiler proposed by Nishida, Sakai and Sakabe generates a conditional term rewriting system that defines the inverse relations of the injective functions, and then the compiler unravels the conditional system into an unconditional term rewriting system. In general, the resulting unconditional system is not (innermost-)confluent even if the conditional system is (innermost-)confluent. In this paper, we propose a modification of the Knuth-Bendix completion procedure, which is used as a post-processor of the inversion compiler. Given a confluent and operationally terminating conditional system, the procedure takes the resulting unconditional systems as input. When the procedure halts successfully, it returns convergent systems that are computationally equivalent to the conditional systems. To adapt the modified procedure to the conditional systems that are not confluent but innermost-confluent, we propose a simplified variant of the modified procedure. We report that the implementations of the procedures succeed in generating innermost-convergent inverse systems for all the examples we tried.