Transition and cancellation in concurrency and branching time

• Authors: Vaughan R. Pratt
• Venue: Mathematical Structures in Computer Science
• Year: 2003

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 $\T$ 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 $L_{\T}$ with a third literal $\that a$ expressing $a=\T$ forms an expressively complete basis for the representation of acyclic HDAs. The main contribution is a new value $\cancel$ of cancellation, which is a sibling of $\T$, serving to distinguish $a(b+c)$ from $ab+ac$ while simplifying the extensional definitions of termination $\tick\A$ and sequence $\A\B$. We show that every HDAX (acyclic HDA with $\cancel$) is representable in the expansion of $L_{\T}$ to $L_{\T\cancel}$ with a fourth literal $\can a$ expressing $a=\cancel$.