Complexity of primality in n-nested simulation logics

Establish the exact computational complexity of the primality decision problem for the modal logics that characterize Groote–Vaandrager’s n-nested simulation preorders over finite, loop-free labelled transition systems. Specifically, prove or refute the conjecture that deciding whether a formula is prime in the logic for 2-nested simulation (2S) is coNP-complete, and that deciding whether a formula is prime in the logics for n-nested simulation (nS) with n ≥ 3 is PSPACE-complete.

Background

The paper analyzes the complexity of deciding whether modal formulas are characteristic (i.e., consistent and prime) across several simulation-based semantics in van Glabbeek’s branching-time spectrum. It establishes polynomial-time algorithms for S and CS, and various hardness results: coNP-hardness for 2S and PSPACE-hardness for 3S primality, along with corresponding satisfiability bounds.

Despite these results, the exact classification of the primality problem for the logics that characterize the n-nested simulation preorders remains unresolved. The authors put forward a conjecture proposing coNP-completeness for 2S and PSPACE-completeness for all n ≥ 3, aiming to complete the spectrum’s complexity landscape.