Formal Verification of Digital Systems by Reduction of Data Paths

Verification of properties (tasks) on a system P containing data paths may require too many resources (memory space and/or computation time) because such systems have very large and deep state spaces. As pointed out by Kurshan, what is needed is a reduced system P' which behaves exactly as P with respect to the properties that must be proved, but more compact than P, so that the verification can be easily performed. The process of finding P' from P is called reduction. P is specified by a network of interacting finite-state machines for data paths and controllers, and tasks are specified by finite-state automate. The verification of a task T on P is performed by the language containment check L(P)⊆L(T), where L(P) is the language generated by P and L(T) is the language accepted by T. It has been shown that, under appropriate conditions, the system P can be reduced to P' and the task T to T' such that L(P')⊆L(T')⇔L(P)⊆L(T). The direct language containment check L(P)⊆L(T) is no longer needed; it is replaced by L(P')⊆L(T'), which is less expensive. More specifically, for the purpose of simplifying the verification of some properties, the system implementation is abstracted locally with respect to the behavior under observation (i.e., bottom-up reduction), in the context of an integrated top-down design/verification technique. The tasks that one may want to verify can express both safety and fairness constraints. In this paper, we prove that the reduction of some data paths to four-state, nondeterministic finite-state machines, and the redundancy removal performed on the controllers is a homomorphic transformation, so that the simplified language containment check can automatically be applied without testing the validity of the homomorphism. This homomorphism correctness verification, required when a formal proof is not available, can be executed using a tool like Cospan, but it may not be completed when the state space to be traversed is too large and deep. The redundancy removal performed on the controllers is important because it eliminates the spurious behaviors introduced in the system by the nondeterminism of the reduced data paths. Redundancy, in fact, may induce a failure in the verification of L(P')⊆L(T'), while L(P)⊆L(T) actually holds. In order to show the effectiveness of the proposed methodology, we verify properties on an extended version of the Mead-Conway Traffic Light Controller, on a modified IRQ communication protocol, and on a relatively prime integers checker and generator