The Density of Linear-Time Properties

Finding models for linear-time properties is a central problem in verification and planning. We study the distribution of lineartime models by investigating the density of linear-time properties over the space of ultimately periodic words. The density of a property over a bound n is the ratio of the number of lasso-shaped words of length n, that satisfy the property, to the total number of lasso-shaped words of length n. We investigate the problem of computing the density for both linear-time properties in general and for the important special case of ωregular properties. For general linear-time properties, the density is not necessarily convergent and can oscillates indefinitely for certain properties. However, we show that the oscillation is bounded by the growth of the sets of badand good-prefix of the property. For ω-regular properties, we show that the density is always convergent and provide a general algorithm for computing the density of ω-regular properties as well as more specialized algorithms for certain sub-classes and their combinations.