Complexity Results for Model Checking

The complexity of model checking branching and linear time temporal logics over Kripke structures has been addressed in e.g. [SC85, CES86]. In terms of the size of the Kripke model and the length of the formula, they show that the model checking problem is solvable in polynomial time for CTL and NP-complete for L(F ). The model checking problem can be generalised by allowing more succinct descriptions of systems than Kripke structures. We investigate the complexity of the model checking problem when the instances of the problem consist of a formula and a description of a system whose state space is at most exponentially larger than the description. Based on Turing machines, we define compact systems as a general formalisation of such system descriptions. Examples of such compact systems are K-bounded Petri nets and synchronised automata, and in these cases the well-known algorithms presented in [SC85, CES86] would require exponential space in term of the sizes of the system descriptions and the formulas; we present polynomial space upper bounds for the model checking problem over compact systems and the logics CTL and L(X,U,S). As an example of an application of our general results we show that the model checking problems of both the branching time temporal logic CTL and the linear time temporal logics L(F ) and L(X,U, S) over K-bounded Petri nets are PSPACE-complete.