String Diagrams for Process Mining

Abstract

Run two process-discovery algorithms on the same event log and they return two different pictures of the same process. Petri nets, causal nets, process trees and BPMN each rely on routing machinery of their own, and their only common ground is the traces they generate. A trace lists activities one after another, so it discards the concurrency the notations exist to express, and two models with identical trace languages can describe genuinely different processes. Whether two discovered models mean the same thing therefore has no notation-independent answer. We show that all four notations admit one canonical presentation, a signature recording a model's activities and the typed interfaces along which they compose, and nothing of the routing machinery. Four construction theorems establish this presentation notation by notation, so two models are compared in a form that each notation determines on its own. Signature equality implies trace equivalence and is strictly finer, separating genuine concurrency from interleaved choice, which a trace comparison cannot, and signature inclusion implies trace inclusion. In a recovery study on one object-centric log, the discovered causal net has exactly the ground-truth signature, while the discovered Petri net's signature strictly contains it and locates every behaviour the Petri net adds in its silent structure. The gain is largest for object-centric data, where a single log supports several notations at once. The four notations become one calculus, in which a translation between them is a claim that can be checked.