Petri Nets, Discrete Physics, and Distributed Quantum Computation

Abstract

The genius, the success, and the limitation of process calculi is their linguistic character. This provides an ingenious way of studying processes, information flow, etc. without quite knowing, independently of the particular linguistic setting, what any of these notions are. One could try to say that they are implicitly defined by the calculus. But then the fact that there are so many calculi, potential and actual, does not leave us on very firm ground.

An important quality of Petri’s conception of concurrency is that it does seek to determine fundamental concepts: causality, concurrency, process, etc. in a syntax-independent fashion. Another important point, which may originally have seemed merely eccentric, but now looks rather ahead of its time, is the extent to which Petri’s thinking was explicitly influenced by physics (see e.g. [7] . As one example, note that K-density comes from one of Carnap’s axiomatizations of relativity). To a large extent, and by design, Net Theory can be seen as a kind of discrete physics: lines are time-like causal flows, cuts are space-like regions, process unfoldings of a marked net are like the solution trajectories of a differential equation.

This acquires new significance today, when the consequences of the idea that “Information is physical” are being explored in the rapidly developing field of quantum informatics. (One feature conspicuously lacking in Petri Net theory is an account of the non-local information flows arising from entangled states, which play a key role in quantum informatics. Locality is so plausible to us — and yet, at a fundamental physical level, apparently so wrong!). Meanwhile, there are now some matching developments on the physics side, and a greatly increased interest in discrete models. As one example, the causal sets approach to discrete spacetime of Sorkin et al. [8] is very close in spirit to event structures.

My own recent work with Bob Coecke on a categorical axiomatics for Quantum Mechanics [4,5], adequate for modelling and reasoning about quantum information and computation, is strikingly close in the formal structures used to my earlier work on Interaction Categories [6] — which represented an attempt to find a more intrinsic, syntax-free formulation of concurrency theory; and on Geometry of Interaction [1], which can be seen as capturing a notion of interactive behaviour, in a mathematically rather robust form, which can be used to model the dynamics of logical proof theory and functional computation.

The categorical formulation of Quantum Mechanics admits a striking (and very useful) diagrammatic presentation, which suggests a link to geometry — and indeed there are solid connections with some of the central ideas relating geometry and physics which have been so prominent in the mathematics of the past 20 years [3].