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).
Partial orders are one of the main tools for modelling "true concurrency" semantics of concurrent systems (cf. [26]). They are utilized to develop powerful partial-order based automatic verification techniques, e.g., the partial order reduction technique for model checking of concurrent software (see, e.g., [1,Chapter 10] and [9]). Partial orders are also equipped with traces, their powerful formal language counterpart, introduced by Mazurkiewicz in his seminal paper [25]. In The Book of Traces [8], trace theory has been used to tackle problems from diverse areas including formal language theory, combinatorics, graph theory, algebra, logic, and especially concurrency theory.
However while partial orders and traces can sufficiently model the “earlier than” relationship, Janicki and Koutny argued that it is problematic to use a single partial order to specify both the “earlier than” and the “not later than” relationships [13]. This motivates them to develop the theory of relational structures, where a pair of relations is used to capture concurrent behaviors. The most well-known among the classes of relational structures proposed by Janicki and Koutny is the class of stratified order structures (so-structures) [10,16,17]. A so-structure is a triple (X , ≺, ⊏), where ≺ and ⊏ are binary relations on X . They were invented to model both the “earlier than” (the relation ≺) and “not later than” (the relation ⊏) relationships, under the assumption that system runs can be described using stratified partial orders, i.e., step sequences. So-structures have been successfully used to give semantics of inhibitor and priority systems [15,21,19,20].
The combined trace (comtrace) notion, introduced by Janicki and Koutny [14], generalizes the trace notion by utilizing step sequences instead of words. First the set of all possible steps that generates step sequences are identified by a relation si m, which is called simultaneity. Second a congruence relation is determined by a relation ser , which is called serializability and is in general not symmetric. Then a comtrace is defined to be a congruence class of step sequences. Comtraces were introduced as a formal language representation of so-structures to provide an operational semantics for Petri nets with inhibitor arcs. Unfortunately comtraces have been less often known and applied than so-structures, even though in many cases they appear to be more natural. We believe one reason is that the comtrace notion was too succinctly discussed in [14] without a full treatment dedicated to comtrace theory. Motivated by this, Janicki and the author have devoted our recent effort on the study of comtraces [23,18], yet there are too many different aspects to explore and the truth is that we could barely scratch the surface. In particular, a huge amount of results from trace theory (e.g., from [8,7]) need to be generalized to comtraces. These tasks are often challenging since we are required to develop novel techniques to deal with the complex interactions of the “earlier than” and “not later than” relations.
In the literature of Mazurkiewicz traces, traces are defined using the following three equivalent methods. The first method is to define a trace to be a congruence class of words, where the congruence relation is induced from an independency relation on events. In the second method, a trace can be viewed as a dependence graph (cf. [8,Chapter 2]). A dependence graph is a directed acyclic graph whose vertices are labeled with events, and satisfies the condition that every two distinct vertices with dependent labels must be connected by exactly one directed edge. The third method is to define a trace as a labeled partially ordered set, whose elements are labeled with events, and we also require the partial order to be “compatible” with the independency relation (see Definition 8 for the precise formulation). Although the above three characterizations of traces can be shown to be equivalent, depending on the situation one characterization can be more convenient than the others. When studying graph-theoretics of traces, the dependence graph representation is the most natural. The treatment of traces as congruence classes of words is more convenient in Ochma ński’s characterization of recognizable trace languages [8,Chapter 6] and Zielonka’s theory of asynchronous automata [8,Chapter 7]. On the other hand most results on temporal logics for traces (see, e.g., [28,29,5,4,11]) utilize the labeled poset representation of traces. The reason is that it is more natural to interpret temporal logics on the vertices (for local temporal logics) or finite downward closed subsets (for global temporal logics) of a labeled partially order set. Thus all of these three representations are indispensable in Mazurkiewicz trace theory.
Since our long-term goal is to generalize the results from Mazurkiewicz trace theory to comtraces, there is a strong need for all three analogous representations for com
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