MPI-I-1999-3-005

This paper presents a complete axiomatization of fully decidable propositional real-time linear temporal logics with past: the Event Clock Logic (\ECL) and the Metric Interval Temporal Logic with past (\MITL). The completeness proof consists of an effective proof building procedure for \ECL. >From this result we obtain a complete axiomatization of \MITL\ by providing axioms translating $\MITL$ formulae into $\ECL$ formulae, the two logics being equally expressive. Our proof is structured to yield a similar axiomatization and procedure for interesting fragments of these logics, such as the linear temporal logic of the real numbers (\LTR).