From pre-historic to post-modern symbolic model checking

Symbolic model checking, which enables the automatic verification of large systems, proceeds by calculating with expressions that represent state sets. Traditionally, symbolic model-checking tools arc based on backward state traversal; their basic operation is the function pre, which given a set of slates, returns the set of all predecessor states. This is because specifiers usually employ formalisms with future-time modalities, which are naturally evaluated by iterating applications of pre. It has been recently shown experimentally that symbolic model checking can perform significantly belter if it is based, instead, on forward state traversal; in this case, the basic operation is the function post, which given a set of states, returns the set of all successor states. This is because forward stale traversal can ensure that only those parts of the state space are explored which are reachable from an initial state and relevant for satisfaction or violation of the specification; that is, errors can be detected as soon as possible. In this paper, we investigate which specifications can be checked by symbolic forward state traversal. We formulate the problems of symbolic backward and forward model checking by means of two μ-calculi. The pre-μ calculus is based on the pre operation; the post-μ calculus, on the post operation. These two μ-calculi induce query logics, which augment fixpoint expressions with a boolean emptiness query. Using query logics, we arc able to relate and compare the symbolic backward and forward approaches. In particular, we prove that all Ω-regular (linear-time) specifications can be expressed as post-μ queries, and therefore checked using symbolic forward state traversal. On the other hand, wc show that there are simple branching-time specifications that cannot be checked in this way.