Data structures for symbolic multi-valued model-checking

Abstract

Multi-valued logics provide an interesting alternative to classical boolean logic for modeling and reasoning about systems. Such logics can be used for reasoning about partially-specified systems, effectively encode vacuity detection and query-checking problems, help in detecting inconsistencies, and many others.In our earlier work, we identified a useful family of multi-valued logics: those specified over finite distributive lattices where negation preserves involution, i.e., \({{\neg}}{{\neg}} a = a\) for every element a of the logic. Such structures are called quasi-boolean algebras, and model-checking over these not only extends the domain of applicability of automated reasoning to new problems, but can also speed up solutions to some classical verification problems.Symbolic model-checking over quasi-boolean algebras can be cast in terms of operations over multi-valued sets: sets whose membership functions are multi-valued. In this paper, we propose and empirically evaluate several choices for implementing multi-valued sets with decision diagrams. In particular, we describe two major approaches: (1) representing the multi-valued membership function canonically, using MDDs or ADDs; (2) representing multi-valued sets as a collection of classical sets, using a vector of either MBTDDs or BDDs. The naive implementation of (2) includes having a classical set for each value of the algebra. We exploit a result of lattice theory to reduce the number of such sets that need to be represented.The major contribution of this paper is the evaluation of the different implementations of multi-valued sets, done via a series of experiments and using several case studies.