🤖 AI Summary
This study systematically characterizes the computational complexity of satisfiability, validity, and realizability for safety and co-safety fragments of Linear Temporal Logic (LTL) over infinite and finite traces. Leveraging automata-theoretic constructions and complexity-theoretic analysis, we establish the first precise complexity bounds: realizability for the safety fragment is 2EXPTIME-complete over infinite traces but drops to EXPTIME-complete over finite traces; all three problems for the co-safety fragment are uniformly PSPACE-complete over both trace models—strictly lower than their safety counterparts; over finite traces, safety satisfiability becomes NP-complete and safety realizability Π²ₚ-complete. These results pin down exact complexity boundaries for multiple LTL fragments and uncover an asymmetric complexity shift between safety and co-safety properties under the infinite-to-finite trace transition. The work provides foundational insights for algorithm design in formal verification and reactive synthesis tools.
📝 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 of LTL 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., Πᴾ₂- complete).
