An Algebraic Framework for Abstract Model Checking

Abstract

Symbolic forward analysis is a semi-algorithm that in many cases solves the model checking problem for infinite state systems in practice. This semi-algorithm is implemented in many practical model checking tools like UPPAAL [], KRONOS [] and HYTECH []. In most practical experiments, termination of symbolic forward analysis is achieved by employing abstractions resulting in an abstract symbolic forward analysis. This paper presents a unified algebraic framework for deriving and reasoning about abstract symbolic forward analysis procedures for a large class of infinite state systems with variables ranging over a numeric domain. The framework is obtained by lifting notions from classical algebraic theory of automata to constraints representing sets of states. Our framework provides sufficient conditions under which the derived abstract symbolic forward analysis procedure is always terminating or accurate or both. The class of infinite state systems that we consider here are (possibly non-linear) hybrid systems and (possibly non-linear) integer-valued systems. The central notions involved are those of constraint transformer monoids and coverings between constraint transformer monoids. We show concrete applications of our framework in deriving abstract symbolic forward analysis algorithms for timed automata and the two process bakery algorithm that are both terminating and accurate.