A Characterization of Combined Traces Using Labeled Stratified Order Structures

This paper defines a class of labeled stratified order structures that characterizes exactly the notion of combined traces (i.e., comtraces) proposed by Janicki and Koutny in 1995. Our main technical contributions are the 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.

💡 Research Summary

The paper presents a new characterization of combined traces (comtraces) by means of labeled stratified order structures (so‑structures). After recalling basic notions of partial orders, step sequences, and Mazurkiewicz trace theory, the authors introduce the Janicki‑Koutny notion of a stratified order structure, a relational triple (X, ≺, ⊏) satisfying four axioms (S1‑S4). Here ≺ models “earlier‑than” while ⊏ models “not‑later‑than”. They define stratified extensions of a so‑structure and prove a Szpilrajn‑type theorem: any so‑structure is the intersection of all its stratified extensions.

Next, the paper reviews the definition of a comtrace alphabet θ = (E, sim, ser), where sim is an irreflexive symmetric simultaneity relation and ser ⊆ sim is a (generally non‑symmetric) serializability relation. Steps are cliques of the sim‑graph, and step sequences form the monoid Sθ*. The comtrace congruence ≈θ identifies two step sequences when a step A can be split into B·C with B × C ⊆ ser, yielding the quotient monoid (Sθ*/≡θ, ⊛) of comtraces.

The core technical contribution is the construction, for any step sequence u, of a relational structure Sₜ = (Σₜ, ≺ₜ, ⊏ₜ) where Σₜ is the set of event occurrences, ≺ₜ captures “earlier‑than” pairs not in ser, and ⊏ₜ captures “not‑later‑than” pairs not in ser. To turn the locally defined relations into a proper so‑structure, the authors introduce the ♦‑closure operator: for a relational structure (X,R₁,R₂), S♦ = (X,(R₁∪R₂)⁎∘R₁∘(R₁∪R₂)⁎,(R₁∪R₂)⁎\id_X). They prove that Sₜ = (Σₜ,≺ₜ,⊏ₜ)♦ satisfies the so‑structure axioms, and that every step sequence belonging to the comtrace