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Reasoning about CTL ∗ with Graded Path Modalities

Graded path modalities count the number of paths satisfying a property, and generalize the existential (E) and universal (A) path modalities of CTL∗. The resulting logic is denoted GCTL∗, and is a very powerful logic since (as we show) it is equivalent to monadic path logic. We settle the complexity of the satisfiability problem of GCTL∗, i.e., 2EXPTIME-COMPLETE, and the complexity of the model checking problem of GCTL∗, i.e., PSPACE-COMPLETE. The lower bounds already hold for CTL∗, and so we supply the upper bounds. The significance of this work is two-fold: GCTL∗ is much more expressive than CTL∗ as it adds to it a form of quantitative reasoning, and this is done at no extra cost in computational complexity.