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.”

The phenomena of branching time and true or noninterleaving concurrency find their respective homes in automata and schedules. But these two models of computation are formally equivalent via Birkhoff duality, an equivalence we expound on here in tutorial detail. So why should these phenomena prefer one home over the other? We identify dimension as the culprit: 1dimensional automata are skeletons permitting only interleaving concurrency, whereas true n-fold concurrency resides in transitions of dimension n. The truly concurrent automaton dual to a schedule is not a skeletal distributive lattice but a solid one. We introduce true nondeterminism and define it as monoidal homotopy; from this perspective nondeterminism in ordinary automata arises from forking and joining creating nontrivial homotopy. The automaton dual to a poset schedule is simply connected whereas that dual to an event structure schedule need not be, according to monoidal homotopy though not to group homotopy. We conclude...

. We present algorithms for computing similarity relations of labeled graphs. Similarity relations have applications for the refinement and verification of reactive systems. For finite graphs, we present an O(mn) algorithm for computing the similarity relation of a graph with n vertices and m edges (assuming m n). For effectively presented infinite graphs, we present a symbolic similarity-checking procedure that terminates if a finite similarity relation exists. We show that 2D rectangular automata, which model discrete reactive systems with continuous environments, define effectively presented infinite graphs with finite similarity relations. It follows that the refinement problem and the 8CTL model-checking problem are decidable for 2D rectangular automata. 1 Introduction A labeled graph G = (V; E;A; hh\Deltaii) consist of a (possibly infinite) set V of vertices, a set E ` V 2 of edges, a set A of labels, and a function hh\Deltaii : V ! A that maps each vertex v to a label hh...

A general automaton model for timing-based systems is presented and is used as the context for developing a variety of simulation proof techniques for such systems. These techniques include (1) refinements, (2) forward and backward simulations, (3) hybrid forward-backward and backward-forward simulations, and (4) history and prophecy relations. Relationships between the different types of simulations, as well as soundness and completeness results, are stated and proved. These results are (with one exception) analogous to the results for untimed systems in Part I of this paper. In fact, many of the results for the timed case are obtained as consequences of the analogous results for the untimed case.

We define five increasingly comprehensive classes of infinite-state systems, called STS1--STS5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.STS1 These are the systems with finite bisimilarity quotients. They can be analyzed symbolically by iteratively applying predecessor and Boolean operations on state sets, starting from a finite number of observable state sets. Any such iteration is guaranteed to terminate in that only a finite number of state sets can be generated. This enables model checking of the μ-calculus.STS2 These are the systems with finite similarity quotients. They can be analyzed symbolically by iterating the predecessor and positive Boolean operations. This enables model checking of the existential and universal fragments of the μ-calculus.STS3 These are the systems with finite trace-equivalence quotients. They can be analyzed symbolically by iterating the predecessor operation and a restricted form of positive Boolean operations (intersection is restricted to intersection with observables). This enables model checking of all ω-regular properties, including linear temporal logic.STS4 These are the systems with finite distance-equivalence quotients (two states are equivalent if for every distance d, the same observables can be reached in d transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new state sets are generated. This enables model checking of the existential conjunction-free and universal disjunction-free fragments of the μ-calculus.STS5 These are the systems with finite bounded-reachability quotients (two states are equivalent if for every distance d, the same observables can be reached in d or fewer transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new states are encountered (this is a weaker termination condition than above). This enables model checking of reachability properties.

We develop a theory of equivalences for timed systems. Two systems are equivalent iff external observers cannot observe differences in their behavior. The notion of equivalence depends, therefore, on the distinguishing power of the observers. The power of an observer to measure time results in untimed, clock, and timed equivalences: an untimed observer cannot measure the time difference between events; a clock observer uses a clock to measure time differences with finite precision; a timed observer is able to measure time differences with arbitrary precision. We show that the distinguishing power of clock observers grows with the number of observers, and approaches, in the limit, the distinguishing power of a timed observer. More precisely, given any equivalence for untimed systems, two timed systems are k-clock congruent, for a nonnegative integer k, iff their compositions with every environment that uses k clocks are untimed equivalent. Both k-clock bisimulation congruence and k-cloc...

This paper combines and extends the material of [GG-a/c/d/e], except for the part in [GG-c] on refinement of transitions in Petri nets and the discussion of TCSP-like parallel composition in [GG-e]. An informal presentation of some basic ingredients of this paper appeared as [GG-b]. Among others, the treatment of action refinement in stable and non-stable event structures is new. The research reported here was supported by Esprit project 432 (METEOR), Esprit Basic Research Action 3148 (DEMON), Sonderforschungsbereich 342 of the TU Munchen, ONR grant N00014-92-J-1974 and the Human Capital and Mobility Cooperation Network EXPRESS (Expressiveness of Languages for Concurrency). Contents

. There has been considerable controversy in concurrency theory between the `interleaving' and `true concurrency' schools. The former school advocates associating a transition system with a process which captures concurrent execution via the interleaving of occurrences; the latter adopts more complex semantic structures to avoid reducing concurrency to interleaving. In this paper we show that the two approaches are not irreconcilable. We define a timed process algebra where occurrences are associated with intervals of time, and give it a transition system semantics. This semantics has many of the advantages of the interleaving approach; the algebra admits an expansion theorem, and bisimulation semantics can be used as usual. Our transition systems, however, incorporate timing information, and this enables us to express concurrency: merely adding timing appropriately generalises transition systems to asynchronous transition systems, showing that time gives a link between true concurrenc...

Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be motivated by operations on transition systems, but they can also be suggested by investigations of modal formalisms. To show that the equational view of processes from process algebra is closely related to modal logic, we consider various ways of looking at the relation between the calculus of basic process algebra and propositional dynamic logic. More concretely, the paper contains preservation results for various bisimulation notions, a result on the expressive power of propositional dynamic logic, and a definition of bisimulation which is the proper notion of invariance for concurrent propositional dynamic logic. Keywords: modal logic, transition systems, bisimulation, process algebra 1 In...

The logic of norms, called deontic logic, has been used to specify normative constraints for information systems. For example, one can specify in deontic logic the constraints that a book borrowed from a library should be returned within three weeks, and that if it is not returned, the library should send a reminder. Thus, the notion of obligation to perform an action arises naturally in system specification.