Parameterised boolean equation systems

Boolean equation system are a useful tool for verifying formulas from modal μ-calculus on transition systems (see [Mader, Lecture Notes in Computer Science, Vol. 1019, 1995, pp. 72–88] for an excellent treatment). We are interested in an extension of boolean equation systems with data. This allows to formulate and prove a substantially wider range of properties on much larger and even infinite state systems. In previous works [Groote and Mateescu, Lecture Notes in Computer Science, Vol. 1548, 1999, pp. 74–90; Groote and Willemse, Sci. Comput. Program., 2005] it has been outlined how to transform a modal formula and a process, both containing data, to a so-called parameterised boolean equation system, or equation system for short. In this article we focus on techniques to solve such equation systems. We introduce a new equivalence between equation systems, because existing equivalences are not compositional. We present techniques similar to Gauß elimination as outlined in [Mader, Lecture Notes in Computer Science, Vol. 1019, 1995, pp. 72–88] that allow to solve each equation system provided a single equation can be solved. We give several techniques for solving single equations, such as approximation (known), patterns (new) and invariants (new). Finally, we provide several small but illustrative examples of verifications of modal μ-calculus formulas on concrete processes to show the use of the techniques.