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Frontiers of Computer Science: Selected Publications from Chinese Universities
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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 nonelementary 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.