Finite-trace linear temporal logic: coinductive completeness

Linear temporal logic (LTL) is suitable not only for infinite-trace systems, but also for finite-trace systems. In particular, LTL with finite-trace semantics is frequently used as a specification formalism in runtime verification, in artificial intelligence, and in business process modeling. The satisfiability of LTL with finite-trace semantics, a known PSPACE-complete problem, has been recently studied and both indirect and direct decision procedures have been proposed. However, the proof theory of LTL with finite traces is not that well understood. Specifically, complete proof systems of LTL with only infinite or with both infinite and finite traces have been proposed in the literature, but complete proof systems directly for LTL with only finite traces are missing. The only known results are indirect, by translation to other logics, e.g., infinite-trace LTL. This paper proposes a direct sound and complete proof system for finite-trace LTL. The axioms and proof rules are natural and expected, except for one rule of coinductive nature, reminiscent of the Godel-Lob axiom.