命题线性时序逻辑的对偶模型问题的复杂性

摘要

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 w-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).%定义了一个命题线性时序逻辑的对偶模型的概念.一个公式f的对偶模型是指f的满足以下条件的两个模型(即状态的w序列):在每个位置上这两个模型对原子命题的赋值都是对偶的.然后,对于确定一个公式f是否有对偶模型的判定问题(记为DM)和在一个Kripke-结构中确定是否存在从两个给定状态出发的对偶模型满足给定公式f的判定问题(记为KDM)的复杂性进行了研究.证明了以下结果:对于只含有F("Future")算子的命题线性时序逻辑,DM和KDM都是NP完全的;而对于以下命题线性时序逻辑,DM和KDM都是PSPACE完全的:含有F,X ("Next")算子的逻辑、含有U("Until")算子的逻辑、含有U,S,X算子的逻辑以及由Wolper给出的含有正规语言算子的逻辑(一般称为扩展时序逻辑,简称ETL).