Completeness and Soundness of Axiomatizations for Temporal Logics. Without Next

We present axiomatizations for Until Temporal Logic (UTL) and for Since/Until Temporal Logic (SUTL). These logics are intended for use in specifying and reasoning about concurrent systems. They employ neither a next nor a previous operator, which obstruct the use of hierarchical abstraction and refinement, and make reasoning about concurrency difficult. We also provide a procedure for translating SUTL formulas into UTL formulas, and demonstrate that the axiomatizations are sound and complete with respect to the class of ω-sequences. We show, however, that the axiomatization of UTL admits models that the axiomatization of SUTL does not, thereby demonstrating that the logics are not equivalent.

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