On Three Alternative Characterizations of Combined Traces

The combined trace (i.e., comtrace) notion was introduced by Janicki and Koutny in 1995 as a generalization of the Mazurkiewicz trace notion. Comtraces are congruence classes of step sequences, where the congruence relation is defined from two relations simultaneity and serializability on events. They also showed that comtraces correspond to some class of labeled stratified order structures, but left open the question of what class of labeled stratified orders represents comtraces. In this work, we proposed a class of labeled stratified order structures that captures exactly the comtrace notion. Our main technical contributions are representation theorems showing that comtrace quotient monoid, combined dependency graph (Kleijn and Koutny 2008) and our labeled stratified order structure characterization are three different and yet equivalent ways to represent comtraces. This paper is a revised and expanded version of L^e (in Proceedings of PETRI NETS 2010, LNCS 6128, pp. 104-124).

💡 Research Summary

The paper revisits the combined trace (comtrace) model, originally introduced by Janicki and Koutny in 1995 as a generalisation of Mazurkiewicz traces, and establishes three mutually equivalent mathematical representations for it. A comtrace is defined over a set of events equipped with two binary relations: simultaneity (which marks pairs of events that may occur together in the same step) and serializability (which marks pairs that may be ordered sequentially). A step‑sequence is a word over subsets of events, and the comtrace equivalence class is the smallest congruence generated by these two relations. While earlier work showed that comtraces correspond to a certain class of labelled stratified order structures (LSOS), it left open exactly which subclass of LSOS captures the comtrace semantics.

The authors answer this by introducing a “structured labeling scheme” – a specific subclass of LSOS characterised by three constraints. First, events are organised by a partial order ≤ that respects the serializability relation. Second, a level function ℓ maps each event to a natural number; events sharing the same level are mutually simultaneous, forming a simultaneity block. Third, each event carries a label from a finite alphabet Σ, preserving the original event types. The level structure thus yields a stratified order where each stratum corresponds to a maximal simultaneity block, and the partial order dictates the permissible inter‑stratum sequencing.

The core technical contribution is a pair of representation theorems. The first theorem proves that for any comtrace class