Graded Computation Tree Logic

In modal logics, graded (world) modalities have been deeply investigated as a useful framework for generalizing standard existential and universal modalities in such a way that they can express statements about a given number of immediately accessible worlds. These modalities have been recently investigated with respect to the μCALCULUS, which have provided succinctness, without affecting the satisfiability of the extended logic, that is, it remains solvable in EXPTIME. A natural question that arises is how logics that allow reasoning about paths could be affected by considering graded path modalities. In this article, we investigate this question in the case of the branching-time temporal logic CTL (GCTL, for short). We prove that, although GCTL is more expressive than CTL, the satisfiability problem for GCTL remains solvable in EXPTIME, even in the case that the graded numbers are coded in binary. This result is obtained by exploiting an automata-theoretic approach, which involves a model of alternating automata with satellites. The satisfiability result turns out to be even more interesting as we show that GCTL is at least exponentially more succinct than graded μCALCULUS.