From pre-historic to post-modern symbolic model checking

Symbolic model checking, which enables the automatic verification of large systems, proceeds by calculating expressions that represent state sets. Traditionally, symbolic model-checking tools are based on back- ward state traversal; their basic operation is the function pre, which, given a set of states, 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 shown experimentally that symbolic model checking can perform significantly better 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 state traversal can ensure that only parts of the state space that are reachable from an initial state and relevant for the satisfaction or violation of the specification are explored; 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, and the post-μ calculus is based on the post operation. These two μ-calculi induce query logics, which augment fixpoint expressions with a boolean emptiness query. Using query logics, we are 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, we show that there are simple branching-time specifications that cannot be checked in this way.