# From *pre*-historic to *post*-modern symbolic model checking

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## Abstract

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 states, returns the set of all predecessor states. This is because specifiers usally 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 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 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 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.

## Keywords

Model Check Linear Temporal Logic Kripke Structure Query Logic Symbolic Model Check## References

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