A sound and complete axiomatic system and a tableau-based decision procedure are presented for propositional temporal logic over linear and discrete time models. The axiomatic system and decision procedure are presented for the full logic, including the past operators, but contain a clear identification of the parts whose omission yields axiomatization and a decision procedure for the future fragment. The paper summarizes work of over 20 years and is intended to provide a definitive reference to the version of propositional temporal (...) logic used for the specification and verification of reactive systems. (shrink)

We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cut-elimination procedure and also no known non-trivial bound on the proof-depth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside these trees. Thus we call this system “deep” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows one to give (...) a straightforward syntactic cut-elimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cut-elimination procedure for the shallow system. For both systems we thus obtain an upper bound of φ20 on the depth of proofs, where φ is the Veblen function. (shrink)