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Narrowing was introduced, and has traditionally been used, to solve equations in initial and free algebras modulo a set of equations E. This paper proposes a generalization of narrowing which can be used to solve reachability goals in initial and free models of a rewrite theory R. We show that narrowing is sound and weakly complete (i.e., complete for normalized solutions) under reasonable executability assumptions about R. We also show that in general narrowing is not strongly complete, that is, not complete when some solutions can be further rewritten by R. We then identify several large classes of rewrite theories, covering many practical applications, for which narrowing is strongly complete. Finally, we illustrate an application of narrowing to analysis of cryptographic protocols.

We address the issue of efficiently automating assume-guarantee reasoning for simulation conformance between finite state systems and specifications. We focus on a non-circular assume-guarantee proof rule, and show that there is a weakest assumption that can be represented canonically by a deterministic tree automata (DTA). We then present an algorithm L T that learns this DTA automatically in an incremental fashion, in time that is polynomial in the number of states in the equivalent minimal DTA. The algorithm assumes a teacher that can answer membership and candidate queries pertaining to the language of the unknown DTA. We show how the teacher can be implemented using a model checker. We have implemented this framework in the COMFORT toolkit and we report encouraging results (over an order of magnitude improvement in memory consumption) on non-trivial benchmarks. IntroductionFormal verification is an important tool in the hands of software practitioners for ascertaining correctness of safety critical software systems. However, scaling formal techniques like model checking [11] to concurrent software of industrial complexity remains an open challenge. The primary hurdle is the state-space explosion problem whereby the number of reachable states of a concurrent system increases exponentially with the number of components.Two paradigms hold the key to alleviating state-space explosion -abstraction [10,9] and compositional reasoning [23,8]. Both of these techniques have been extensively studied by the formal verification community and there have been significant breakthroughs from time to time. One of the most important advancements in the domain of compositional analysis is the concept of assume-guarantee [23] (AG) reasoning. The essential idea here is to model-check each component independently by making an assumption about its environment, and then discharge the assumption on the collection of the rest of the components. A variety of AG proof-rules are known, of which we will concern ourselves with the following non-circular rule called AG-NC:where M 1 M 2 is the concurrent system to be verified, S is the specification, and an appropriate notion of conformance between the system and the specification. AG-NC is known to be sound and complete for a number of conformance notions, including K.

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