Completeness for Logics on Finite Traces

Temporal logics over finite traces are not the same as temporal logics over potentially infinite traces. We propose that existing methods for proving deductive completeness for infinite-trace logics are effective on their finite counterparts. To adapt proofs for infinite-trace logics, we “inject” finiteness: that is, we alter the proof structure to ensure that models are finite. As evidence for this claim, we offer deductive completeness results for two finite temporal logics: linear temporal logic over finite traces (LTLf ) and linear dynamic logic over finite traces (LDLf ). Both proofs of completeness follow a conventional, graph based, least fixed point structure. Roşu first proved completeness for LTLf with a novel coinductive axiom; our proof uses fewer and more conventional axioms [12]. The proof for LDLf is novel; it largely follows our LTLf proof, using Brzozowski derivatives to define a transition function.