A Framework for Ranking Vacuity Results

Vacuity detection is a method for finding errors in the modelchecking process when the specification is found to hold in the model. Most vacuity algorithms are based on checking the effect of applying mutations on the specification. It has been recognized that vacuity results differ in their significance. While in many cases vacuity results are valued as highly informative, there are also cases in which the results are viewed as meaningless by users. As of today, there is no study about ranking vacuity results according to their level of importance, and there is no formal framework or algorithms for defining and finding such ranks. The lack of framework often causes designers to ignore vacuity information altogether, potentially causing real problems to be overlooked. We suggest and study such a framework, based on the probability of the mutated specification to hold in a random computation. For example, two natural mutations of the specification G(req → F ready) are G(¬req) and GF ready . It is agreed that vacuity information about satisfying the first mutation is more alarming than information about satisfying the second. Our methodology formally explains this, as the probability of G(¬req) to hold in a random computation is 0, whereas the probability of GF ready is 1. From a theoretical point of view, our contribution includes a study of the problem of finding the probability of LTL formulas to be satisfied in a random computation and the existence and use of 0/1-laws for fragments of LTL. From a practical point of view, we propose an efficient algorithm for estimating the probability of LTL formulas, and argue that ranking vacuity results according to our probability-based criteria corresponds to our intuition about their level of importance.