The Power of Temporal Proofs

Abstract Some methods for reasoning about concurrent programs and hardware devices have been based on proof systems for temporal logic. Unfortunately, all effective proof systems for temporal logic are incomplete for the standard semantics, in the sense that some formulas hold in every intended model but cannot be proved. We evaluate and compare the power of several proof systems for temporal logic. Specifically, we relate temporal systems to classical systems with explicit time parameters. A typical temporal system turns out to be incomplete in a strong sense; we exhibit a short, valid formula it fails to prove. We suggest the addition of new rules to define auxiliary predicates. With these rules, we obtain nonstandard soundness and completeness results. In particular, one of the simple temporal systems we describe is as powerful as Peano Arithmetic.