On Recognizable Stable Trace Languages

We relate several models of concurrency introduced in the literature in order to extend classical Mazurkiewicz traces. These are mainly Droste’s concurrent automata and Arnold’s CCI sets of P-traces, studied in the framework of local trace languages. Also, a connection between these models and classical traces is presented in details through a natural notion of projection. These relationships enable us to use efficiently Arnold’s result in two other frameworks. First, we give a finite distributed implementation for regular CCI sets of P-traces (or, equivalently, finite stably concurrent automata) by means of bounded labelled Petri nets. Second, we present a new, simple and constructive method to relate Stark’s trace automata with Bednarczyk’s asynchronous transition systems. This improves a recent result in Scott domain theory. Introduction. Mazurkiewicz trace languages are a well-known and widely studied model of concurrency [4]. They were introduced in [13] to provide a partial order semantics for elementary Petri nets. In the past decade several different generalizations of classical traces have been studied in the literature. First, Droste introduced concurrent automata [5] for which the independence between actions is no longer a global independence relation, but depends on the current state of the system. These automata were shown to extend Bednarczyk’s asynchronous transition systems [2] and Stark’s trace automata [18]. Independently, Arnold introduced an extension of classical traces by means of labelled partial orders called P-traces [1]. In particular, a strong connection between recognizable classical trace languages and regular CCI sets of P-traces was established. More recently, local trace languages were introduced to give a trace semantics for Place/Transition nets [8,14]. There a local independence relation specifies in each configuration which subsets of actions can be executed concurrently. At some point, it seems necessary to classify and relate the different models of concurrency arisen in the literature. For instance, the synthesis problem of Petri nets consists in characterizing which automata (or languages) correspond to the behavior of a Petri net [7,15,8]. More generally, semantical studies bring relationships between models of different levels of abstraction [20,2,16,11]. In this paper, we relate three models of concurrency which are roughly at the same level of abstraction. These are CCI sets of P-traces, stably concurrent automata and a restricted subclass of local trace languages called stable trace languages. The latter are also precisely compared to classical trace languages by ? Supported by the German Research Foundation (DFG/Graduiertenkolleg) J. Tiuryn (Ed.): FOSSACS 2000, LNCS 1784, pp. 177–191, 2000. c © Springer-Verlag Berlin Heidelberg 2000 178 Jean-François Husson and Rémi Morin means of projections. We show that these relationships lead to some improvements for the theories of Petri nets, concurrent automata and dI-domains. After some basic definitions relating recognizable local trace languages and Mukund’s step transition systems [15], we introduce the subclass of stable trace languages with the help of some cube properties. The latter are actually meant to mimic the particular behaviors of stably concurrent automata. In that way, recognizable stable trace languages are easily shown to correspond to the behavior of finite stably concurrent automata. Next we focus on CCI sets of P-traces which are shown to be equivalent to some stable trace languages. Therefore they represent the behavior of stably concurrent automata. Also regular CCI sets of P-traces are associated to recognizable stable trace languages. Thus we obtain precise relationships between these three models. These connections lead us to give a new formulation of a strong result due to Arnold [1, Th. 6.16] showing that these extensions of classical traces are closely related to the original model: any recognizable stable trace language is the projection of a recognizable classical trace language. This relationship holds also for non-recognizable languages over infinite alphabets. However, answering an open problem raised by Arnold, we prove that this relationship fails in the case of non-recognizable stable trace languages over finite alphabets. This relies on a counter-example provided by a Producer-Consumer system. In a seminal paper [21], Zielonka proved that any recognizable classical trace language is described by an asynchronous automaton which provides a finite implementation in the form of distributed processes. In [1], Arnold introduced an extension of Zielonka’s asynchronous automata, called P-asynchronous automata. However these systems failed to describe all regular CCI sets of P-traces. Besides, it is still an open problem to know which regular CCI sets of P-traces are described by P-asynchronous automata (obviously these are not the whole class of regular CCI sets of P-traces, see [10] for a counter-example). In order to avoid this restriction, we present a construction of a finite distributed implementation for any recognizable stable trace language (or any regular CCI set of P-traces) in the form of a labelled Petri net. This construction turns out to complete nicely a somewhat dual approach followed by Droste and Shortt [6]. There the Petri nets whose behavior corresponds to a stably concurrent automaton (or a stable trace language) are characterized by some simple conditions on the weight function. In [17], Schmitt tackles the difficult problem to define a recognizability notion for coherent dI-domains. The basic idea is that a coherent dI-domain should be considered recognizable if it corresponds to the behavior of a finite distributed automaton. However several families of distributed automata might be considered and might give rise to different recognizability notions. The main result of [17] asserts that the coherent dI-domains obtained from either finite trace automata [18] or finite asynchronous transition systems [2] are the same. We present here a new, simple and constructive proof of this result — whereas Schmitt’s approach is not constructive. The proofs of our main results partly rely on technical results borrowed from [1] and [3]. A detailled study is available in [10]. On Recognizable Stable Trace Languages 179