Testing Language Containment for ω-Automata Using BDDs

Abstract

ω-automata provide a well-established basis for the specification and verification of control-intensive systems. To verify that a system satisfies a given property ("task"), one specifies both the system and the task in terms of ω-automata, and then verifies that the ω-regular language of the system automaton is contained in that of the task automaton. This procedure, which is the basis of the COSPAN verification software, has been used in a number of commercial applications. However, its applicability is limited by the computational complexity of the ensuing language containment check, which tends to grow exponentially with the number of components in the system. While reduction techniques such as task decomposition and task-relative homomorphic reduction can greatly extend the complexity of systems which thus may be analyzed, there is a computational cost associated with such reductions as well. Moreover, the system complexity is the ultimate limiting factor. Recent advances in the manipulation of data-structures for binary decision diagrams (BDDs) have suggested that this data-structure may now facilitate checking language containment for far larger system models than has been hitherto possible. We have confirmed this by implementing new BDD-based language containment checks in COSPAN. We exhibit two such algorithms: one with a time advantage and the other with a space advantage. Each has increased significantly the size of system models which can be verified.