Towards the exact complexity of realizability for Safety LTL

Abstract

We study the realizability and strong satisfiability problems for SafetyLTL, a syntactic fragment of Linear Temporal Logic (▪) capturing safe formulas. While it is well-known that realizability for this fragment lies in ▪, the best-known lower bound is ▪-hardness. Surprisingly, closing this gap has proven an elusive task. Previous works have claimed first ▪-completeness [1] and later ▪-completeness [2] for this problem, but both of these proofs turned out to be incorrect.We revisit the problem of the exact classification of the complexity of realizability for ▪ through the lens of seemingly weaker fragments. While we cannot settle the question for ▪, we study a subfragment of it consisting of formulas of the form ▪, where α is a present formula over system variables and ψ contains Next as the only temporal operator. We prove that the realizability problem for this new fragment, which we call ▪, is ▪-complete, and observe that this fragment is equirealizable to existing more expressive fragments, such as the class ▪[3].Furthermore, we revisit the techniques used in the purported proof of ▪-completeness of Arteche and Hermo [1], and observe that, while incorrect in their original claims, their proofs can be modified to classify the complexity of strong satisfiability, a necessary condition for realizability introduced by Kupferman, Sadigh, and Seshia [4]. We prove that, with regards to strong satisfiability, the fragments ▪ and ▪ are in fact equivalent under polynomial-time many-one reductions.