The Complexity of CTL* + Linear Past

Abstract

We investigate the complexity of satisfiability and finite-state model-checking problems for the branching-time logic CTL\(^*_{lp}\), an extension of CTL* with past-time operators, where past is linear, finite, and cumulative. It is well-known that CTL\(^*_{lp}\) has the same expressiveness as standard CTL*, but the translation of CTL\(^*_{lp}\) into CTL* is of non-elementary complexity, and no elementary upper bounds are known for its satisfiability and finite-state model checking problems. In this paper, we provide an elegant and uniform framework to solve these problems, which non-trivially extends the standard automata-theoretic approach to CTL* model-checking. In particular, we show that the satisfiability problem for CTL\(^*_{lp}\) is 2Exptime-complete, which is the same complexity as that of CTL*, but for the existential fragment of CTL\(^*_{lp}\), the problem is Expspace-complete, hence exponentially harder than that of the existential fragment of CTL*. For the model-checking, the problem is already Expspace-complete for the existential and universal fragments of CTL\(^*_{lp}\). For full CTL\(^*_{lp}\), the proposed algorithm runs in time polynomial in the size of the Kripke structure and doubly exponential in the size of the formula. Thus, the exact complexity of model-checking full CTL\(^*_{lp}\) remains open: it lies somewhere between Expspace and 2Exptime.KeywordsModel CheckTemporal LogicCurrent NodeKripke StructureTree AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.