The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higher-order processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higher-order processes. For this it is useful to generalise event structures to allow events which “persist.”

Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. Th...

We use input/output automata to define a simple and general model of networks of concurrently executing, nondeterministic processes that communicate through unidirectional, named ports. A notion of the input/output relation computed by a process is defined, and determinate processes are defined to be processes whose input/output relations are single-valued. We show that determinate processes compute continuous functions, and that networks of determinate processes obey Kahn's fixed-point principle. Although these results are already known, our contribution lies in the fact that the input/output automata model yields extremely simple proofs of them (the simplest we have seen), in spite of its generality. 1 Introduction Kahn (1974) describes a simple parallel programming language based on the concept of a network of concurrently executing sequential processes that can communicate by sending values over "channels. " The communication primitives available to processes are sufficiently ...

: Concurrent transition systems (CTS's), are ordinary nondeterministic transition systems that have been equipped with additional concurrency information, specified in terms of a binary residual operation on transitions. Each CTS C freely generates a complete CTS or computation category C , whose arrows are equivalence classes of finite computation sequences, modulo a congruence induced by the concurrency information. The categorical composition on C induces a "prefix" partial order on its arrows, and the computations of C are conveniently defined to be the ideals of this partial order. The definition of computations as ideals has some pleasant properties, one of which 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 ...

We describe systems and their components by functional specification techniques. We define notions of interface and interaction refinement for interactive systems and their components. These notions of refinement allow one to change both the syntactic (the number of channels and sorts of messages at the channels) and the semantic interface (causality flow between messages and interaction granularity) of an interactive system component. We prove that these notions of refinement are compositional with respect to sequential and parallel composition of system components, communication feedback and recursive declarations of system components. According to these proofs refinements of networks can be accomplished in a modular way by refining their components. We generalize the notions of refinement to refining contexts. Finally full abstraction for specifications is defined and compositionality with respect to this abstraction is shown, too. This work was partially sponsored by the German S...

. We recast dataflow in a modern categorical light using profunctors as a generalisation of relations. The well known causal anomalies associated with relational semantics of indeterminate dataflow are avoided, but still we preserve much of the intuitions of a relational model. The development fits with the view of categories of models for concurrency and the general treatment of bisimulation they provide. In particular it fits with the recent categorical formulation of feedback using traced monoidal categories. The payoffs are: (1) explicit relations to existing models and semantics, especially the usual axioms of monotone IO automata are read off from the definition of profunctors, (2) a new definition of bisimulation for dataflow, the proof of the congruence of which benefits from the preservation properties associated with open maps and (3) a treatment of higherorder dataflow as a biproduct, essentially by following the geometry of interaction programme. 1 Introduction A fundament...

) Vineet Gupta Radha Jagadeesan Prakash Panangaden y vgupta@mail.arc.nasa.gov radha@cs.luc.edu prakash@cs.mcgill.ca Caelum Research Corporation Dept. of Math. and Computer Sciences School of Computer Science NASA Ames Research Center Loyola University--Lake Shore Campus McGill University Moffett Field CA 94035, USA Chicago IL 60626, USA Montreal, Quebec, Canada Abstract This paper describes a stochastic concurrent constraint language for the description and programming of concurrent probabilistic systems. The language can be viewed both as a calculus for describing and reasoning about stochastic processes and as an executable language for simulating stochastic processes. In this language programs encode probability distributions over (potentially infinite) sets of objects. We illustrate the subtleties that arise from the interaction of constraints, random choice and recursion. We describe operational semantics of these programs (programs are run by sampling random choices), deno...

Dataflow models of computation have intrigued computer scientists since the 1970s. They were first introduced by Jack Dennis as a basis for parallel programming languages and architectures, and by Gilles Kahn as a model of concurrency. Interest in these models of computation has been recently rekindled by the resurrection of parallel computing, due to the emergence of multicore architectures. However, Dennis and Kahn approached dataflow very differently. Dennis ’ approach was based on an operational notion of atomic firings driven by certain firing rules. Kahn’s approach was based on a denotational notion of processes as continuous functions on infinite streams. This paper bridges the gap between these two points of view, showing that sequences of firings define a continuous Kahn process as the least fixed point of an appropriately constructed functional. The Dennis firing rules are sets of finite prefixes satisfying certain conditions that ensure determinacy. These conditions result in firing rules that are strictly more general than the blocking reads of the Kahn-MacQueen implementation of Kahn process networks, and solve some compositionality problems in the dataflow model. 1

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey.

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 previ...