Concurrent transition systems

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Concurrent transition systems (CTS's), are ordinary nondeterministic transition systems that have been equipped with additional concurrency information. This concurrency information is specified in terms of a binary residual operation on transitions, which describes how certain pairs of transitions “commute.” The defining axioms for a CTS generate a rich algebraic theory, which we develop in detail. Each CTS C freely generates a complete CTS or computational category C ∗, whose arrows are equivalence classes of finite computation sequences, modulo a congruence induced by the residual operation. The notion “computation tree” for ordinary transition systems generalizes to computation diagram for CTS's, leading to the convenient definition of computations of a CTS as the ideals of its computation diagram. A pleasant property of this definition is that the notion of a maximal ideal in certain circumstances can serve as a replacement for the more troublesome notion of a fair computation sequence To illustrate the utility of CTS's, we use them to define and investigate a dataflow-like model of concurrent computation. The model consists of machines, which generalize the sequential machines of classical automata theory, and various operations (parallel product, input and output relabeling, and feedback) on machines that correspond to ways of combining machines into networks. Using our definition of computations as ideals, we define a natural notion of observable equivalence of machines, and show that it is the largest congruence, respecting parallel product and feedback, that does not relate two machines with distinct input/output relations. In an attempt to obtain information about the algebra of observable equivalence classes, we investigate a series of abstractions of the machine model, show that these abstractions respect the feedback operation, and characterize the homomorphic image of this operation in each case. A byproduct of our analysis is a structural characterization of a large class of processes with functional input/output behavior, and a proof that the feedback operation on such processes obeys Kahn's fixed-point principle.