Towards the exact complexity of realizability for Safety LTL

We study the realizability and strong satisfiability problems for Safety LTL, 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.