The Complexity of Dual Models Problem of Propositional Linear Temporal Logics

In this paper, the concept of dual models of a propositional linear temporal logic formula is defined: A formula f has dual models if it has two models (namely two ω-sequences of states) such that the assignments to atomic propositions at each position of them are dual. Then for various propositional linear temporal logics, the complexity of the problem deciding whether a formula f has dual models (denoted by DM) and the problem of determination of dual models in a Kripke-structure for a formula f (denoted by KDM) are investigated. It is shown that DM and KDM are NP-complete for the logic with F (“Future”) operator, and they are PSPACE-complete for the logic with F, X (“Next”) operators, the logic with U (“Until”) operator, the logic with U, S, X operators, and the logic with regular operators given by Wolper (known as extended temporal logic, ETL).