Arithmetic + Logic + Geometry = Concurrency

0 Introduction In deening concurrency one dichotomy that has arisen is between reductionism and primitivism. The reductionist viewpoint reduces concurrent behavior to nondeterministic sequential behavior by viewing it as interleaving. The primitive school treats concurrent behavior as a basic form of behavior in its own right, distinct from and indeed more general than sequential behavior. A basic argument for reductionism is that nonsequential behavior is best understood in terms of sequential behavior because the latter is the kind we understand naturally and instinctively, or at least the kind with which we have by far the most experience. At the conscious level each of us appears to be a sequential agent. Our limbs, muscles, and cells may act concurrently, but our control of them is basically sequential and the necessary concurrent control seems to happen below the level of our conscious control, or it would be just as easy to twiddle our thumbs in opposite directions as in the same direction. Primitivists object that reducing concurrent behavior to sequential makes the former harder rather than easier to understand. By analogy, organizing a TV image as the sequence of its scan lines in one dimension, the form in which it arrives at your receiver, makes it much harder to see. It is not so much a question of whether information is lost as the complexity of piecing the information back together. Even if reductionism is the only mode of thought presently available for the perception of concurrent behavior, it is such a poor way of thinking about it that we are better oo learning to deal with concurrency in its own right than persisting with reductionism. Reductionists counter that primitivism is not fully abstract, that is, draws unobservable distinctions. In the context of dataaow nets Kok Kok87], Jonsson Jon89], Rabinovich and Trakhtenbrot RT88], Russell Rus89], and others have shown that reductionism, unlike primitivism, is fully abstract.