On CTL* with Graded Path Modalities

Abstract

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

Benjamin Aminof is supported by the Austrian National Research Network S11403-N23 (RiSE) of the Austrian Science Fund (FWF) and by the Vienna Science and Technology Fund (WWTF) through grant ICT12-059. Aniello Murano is partially supported by the FP7 EU project 600958-SHERPA. Sasha Rubin is a Marie Curie fellow of the Istituto Nazionale di Alta Matematica.