an:04060692 Zbl 0649.68019 Panangaden, Prakash; Stark, Eugene W. Computations, residuals, and the power of indeterminacy EN Automata, languages and programming, Proc. 15th Int. Colloq., Tampere/Finn. 1988, Lect. Notes Comput. Sci. 317, 439-454 (1988). 1988
a
68N25 68Q60 68Q45 networks of communicating processes; dataflow networks; indeterminate behavior; monotone networks; indeterminate branching; fair merge; reasoning about concurrent systems; monotone port automata [For the entire collection see Zbl 0639.00042.] We investigate the power of Kahn-style dataflow networks, with processes that may exhibit indeterminate behavior. Our main result is a theorem about networks of monotone'' processes, which shows: (1) that the input/$$output$$ relation of such a network is a total and monotone relation; and (2) every relation that is total, monotone, and continuous in a certain sense, is the input/$$output$$ relation of such a network. Now, the class of monotone networks includes networks that compute arbitrary continuous input/$$output$$ functions, an angelic merge'' network, and an infinity-fair merge'' network that exhibits countably indeterminate branching. Since the fair merge'' relation is neither monotone nor continuous, a corollary of our main result is the impossibility of implementing fair merge in terms of continuous functions, angelic merge, and infinity-fair merge. Our results are established by applying the powerful technique of residuals'' to the computations of a network. Residuals, which have previously been used to investigate optimal reduction strategies for the $$\lambda$$-calculus, have recently been demonstrated by one of the authors (Stark) also to be of use in reasoning about concurrent systems. Here, we define the general notion of a residual operation'' on an automaton, and show how residual operations defined on the components of a network induce a certain preorder $$\sqsubset \underset \tilde{} {\;}$$ on the set of computations of the network. For networks of monotone port automata,'' we show that the fair'' computations coincide with $$\sqsubset \underset \tilde{} {\;}$$-maximal computations. Our results follow from this extremely convenient property. 0639.00042