Completion of Unraveled Term Rewriting Systems toward Program Inversion of Injective Functions

Given a constructor term rewriting system defining injective functions, the inversion compiler proposed by Nishida, Sakai and Sakabe generates a confluent conditional term rewriting system, and unravels the conditional system into an unconditional term rewriting system. In general, the unconditional system is not confluent and thus not computationally equivalent to the conditional system. In this paper, we slightly modify Knuth-Bendix completion procedure in order to transform the unraveled systems into ones that are computationally equivalent to the corresponding conditional systems if the procedure halts successfully. We also show a way to apply the procedure to rewrite systems evaluated by the innermost strategy, and apply it to examples of program inversion.