Complexity of Safety and coSafety Fragments of Linear Temporal Logic
Alessandro Artale, Luca Geatti, Nicola Gigante, Andrea Mazzullo, Angelo Montanari
TL;DR
This work analyzes the satisfiability, validity, and realizability of safety and cosafety fragments of Linear Temporal Logic (LTL) under both infinite and finite trace semantics. It shows that most cosafety fragments retain the same complexity as full LTL for satisfiability and validity on infinite traces ($\mathsf{PSPACE}$-complete), while realizability spans from $\mathsf{2EXPTIME}$ to $\mathsf{EXPTIME}$ depending on the fragment; crucially, cosafety fragments preserve equivalence between finite and infinite trace semantics. For finite traces, safety fragments exhibit a substantial drop in complexity (satisfiability in $\mathsf{NP}$, realizability in $\mathsf{\Pi^P_2}$), whereas cosafety fragments keep the complexity stable, enabling finite-trace reasoning without heavy determinization. The paper also establishes dualities between cosafety and safety fragments to transfer results, and it introduces general finite-trace theorems (suffix independence and a small-model property) that underpin these findings. Overall, the results illuminate the computational boundaries of practical fragments, with implications for runtime verification and synthesis.
Abstract
Linear Temporal Logic (LTL) is the de-facto standard temporal logic for system specification, whose foundational properties have been studied for over five decades. Safety and cosafety properties define notable fragments of LTL, where a prefix of a trace suffices to establish whether a formula is true or not over that trace. In this paper, we study the complexity of the problems of satisfiability, validity, and realizability over infinite and finite traces for the safety and cosafety fragments of LTL. As for satisfiability and validity over infinite traces, we prove that the majority of the fragments have the same complexity as full LTL, that is, they are PSPACE-complete. The picture is radically different for realizability: we find fragments with the same expressive power whose complexity varies from 2EXPTIME-complete (as full LTL) to EXPTIME-complete. Notably, for all cosafety fragments, the complexity of the three problems does not change passing from infinite to finite traces, while for all safety fragments the complexity of satisfiability (resp., realizability) over finite traces drops to NP-complete (resp., $Π^P_2$-complete).