Graded Computation Tree Logic

Abstract

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 $\mu$Calculus, which have provided succinctness, without affecting the satisfiability of the extended logic, i.e., 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 work, we investigate this question in the case of the branching-time temporal logic Ctl (G Ctl, for short). We prove that, although G Ctl is more expressive than Ctl, the satisfiability problem for G Ctl 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 G Ctl is at least exponentially more succinct than graded $\mu$Calculus.