Transition and cancellation in concurrency and branching time

Abstract

We review the conceptual development of (true) concurrency and branching time starting from Petri nets and proceeding via Mazurkiewicz traces, pomsets, bisimulation, and event structures up to higher dimensional automata (HDAs), whose acyclic case may be identified with triadic event structures and triadic Chu spaces. Acyclic HDAs may be understood as extending the two truth values of Boolean logic with a third value expressing transition. We prove the necessity of such a third value under mild assumptions about the nature of observable events, and show that the expansion of any complete Boolean basis L to L with a third literal a expressing a = forms an expressively complete basis for the representation of acyclic HDAs. The main contribution is a new value × of cancellation, which is a sibling of , serving to distinguish a(b + c) from ab + ac while simplifying the extensional definitions of termination XA and sequence A B. We show that every HDAX (acyclic HDA with ×) is representable in the expansion of L to L × with a fourth literal a expressing a = ×.

Comparing Process Behaviors with Finite Chu Spaces

“…Theoretical researches on Chu spaces have been fruitful (e.g. [5]- [7]). Applications of Chu spaces include the modeling of RTL (Register Transfer Level) [8], Verilog [9], general concurrent programs [4], WS-BPEL [10], and physical systems [11].…”

“…In this paper, we use K = {0, ∠, 1, ×}, which is proposed in [5]. This K supports both true concurrency and branching time semantics, which means it is suitable for the refined modeling of concurrent behavior [12].…”

Linear Process Algebra

“…In particular aligning the two intervals at both ends corresponds to the one two-dimensional state 1 0. Rodriguez and Anger [69] have studied extensions of Allen's configurations to handle richer notions of time such as relativistic time, accomplished by suitably enlarging the local event set K. With one extension they characterize as branching time the 13 states extend to 29, with their relativistic one it extends to 82 states, see [67] for further details.…”

“…The last were so interesting in their own right as to distract me from process algebra in order to study their applications to mathematics for a few years before returning to their process algebra applications [63][64][65]67]. [67] in particular made the observation that the usual two values 0 and 1 of Chu spaces, denoting event states of ready (not yet started) and done, could be extended with either or both of the intermediate value of transition and the alternative value × of cancelled. These permit respectively higher-dimensional automata and what we now call cancellation automata to be represented as Chu spaces.…”

“…The Chu representation of concurrent processes is not as well known in concurrency circles as I feel it should be, despite having been in the literature for two decades [8,9,30,31], justifying the tutorial part. The emphasis here is more on rationale, intuition, definitions, and perspective and less on a formal theorem-and-proof development of the theory; for more in-depth technical details see [62] for the relevant theory of Chu spaces and [67] for more on process algebra based on transition and cancellation.…”