Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. Technical Report Technical Report DAIMI PB 78, Aarhus University, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....theory of free partially commutative monoids generalizes both, the theory of free monoids and the theory of free commutative monoids. In computer science, free partially commutative monoids are commonly called trace monoids and their elements are called traces. Both notions are due to Mazurkiewicz [20], who recognized traces as a model of concurrent processes. 14] gives an extensive overview about current research trends in trace theory. The relevance of trace theory for computer science can be explained as follows. Assume a nite alphabet . An element of the free monoid over , i.e. a ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....Owing to the tight connexion between permutation equivalence and concurrency, permutation equivalence appears in many guises in areas where concurrency is important. For instance, in trace theory the notion of Mazurkiewicz trace as a trace up to the order of independent actions was introduced in [29]. For another example, in transaction processing systems (TPSs: 12] concurrency control protocols must prevent interference by ensuring so called serialisability: a TPS should only accept a schedule if it is equivalent to some serial schedule. Roughly speaking, the notions of conflict and view ....

A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical Report DAIMI PB-78, Aarhus University, 1977.

....[57] This paper investigates the first order theory of the one step rewrite graphs of trace rewriting systems, which generalize semi Thue systems. Cartier and Foata [7] investigated the combinatorics of free partially commutative monoids that became later known as trace monoids. Mazurkiewicz [46] introduced them into computer science. They form a mathematically sound model for the concurrent behaviour of systems of high abstraction. Since trace monoids are a generalization of free monoids, it was tempting to extend the investigation of free monoids to free partially commutative monoids. ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....then we obtain a PSPACE completeness result. Restricting ourselves to trace monoids we still obtain a better complexity result, as it was known previously. 1 Introduction Free partially commutative monoids (or trace monoids) serve as a basic algebraic tool for investigating concurrent systems, [15, 16]. Atomic actions are represented by letters and independency of actions is reflected by an independence relation on the alphabet. The independency defines a partial commutation. If each atomic action a has an inverse a, then, on the algebraic level, we switch from monoids to groups. This means ....

....that D is reflexive and symmetric and that (a; b) 2 D implies (a; b) 2 D for all a; b 2 Gamma . In particular, we have (a; a) 2 D for all a 2 Gamma . The free partially commutative monoid M ( Gamma; I) is defined by the quotient monoid Gamma =fab = ba j (a; b) 2 Ig. According to Mazurkiewicz [15] it is also called a trace monoid and its elements are traces. If the reference to ( Gamma; I) is clear, we also write M instead of M ( Gamma; I) The length jxj of a trace x is the length of any representing word. A letter a 2 Gamma is called minimal (maximal resp. in x, if we can write x = ay ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....Processes (CSP) 22] or the ISO standard Formal Description Technique LOTOS. There were also proposed many different representations of behavioural semantics for these models of various levels of coarseness, e.g. labeled transition systems, Petri nets or event structures. In 1977 Mazurkiewicz [29] used free partially commutative monoids (often called trace monoids) for an intuitive and elegant coarse description of the behaviour of concurrent systems, and he showed that in this way the behaviour of elementary net systems (1 safe Petri nets) can be captured faithfully. In this approach one ....

A. Mazurkiewicz, Concurrent program schemes and their interpretations, DAIMI Rep. PB 78, Aarhus Univ., 1977.

.... 1 and 1 correspond respectively to upward and downward subword closed sets. For example, 1 is precisely the class SRE [2] for which it can be shown that inclusion can be checked in polynomial time. Related work: The notion of semi commutations was proposed in the late 70 s by Mazurkiewicz [13] for describing the semantics of 1 safe Petri nets. Problems related to closure of languages under semi commutations have been studied in the community of Mazurkiewicz trace theory (see e.g. chapter 12 in [8] for a survey) However, the problems we address here and our results are di erent: Our ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....for Petri nets (e.g. 1, 5, 7] except for Meseguer s work [10] not much work on concurrent semantics for objects has been done. We propose a concurrent semantics for actor systems, based on the theory of traces [1, 5, 7] The theory of traces introduced in computer science by Mazurkiewicz [8] to describe the behavior of safe Petri nets views concurrent systems as a 1. a set of atomic actions X, together with 2. a relation I X Theta X specifying which actions can be performed independently or concurrently. This relation independence is assumed to be symmetric and ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretation. DAIMI Rept. PB 78, Aarhus University, Aarhus, 1977.

....transitions of the net can be read on the upper contour of a Tetris like heap of pieces, canonically associated to an (earliest) execution of the system. This representation theorem is best understood by comparison with the following existing approaches. 1. Trace languages. In the landmark paper [20], Mazurkiewicz observed that trace monoids (that is, free partially commutative monoids) and their subsets (trace languages) are the natural model of the untimed behavior of safe Petri nets. Trace monoids, in which certain (independent) letters commute, and others do not, allow us to identify the ....

....ng. This simple example shows that firing sequences, which provide a sequential description of the behavior, are not really adapted in the presence of concurrency. The problem of modeling concurrency in a more efficient way has long been considered. A classical approach, proposed by Mazurkiewicz [20], 21] uses the notion of trace monoid. For a general and recent reference on traces, see [13] DEFINITION III.4. Let T be an alphabet equipped with a reflexive symmetric relation called dependence relation and denoted by D. We denote by I the complement of D, called independence relation. The ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. Research report DAIMI Rep. PB-78, Aarhus Univ., 1977.

....connue pour le calculer. Ici, on introduit quelques nouvelles m#thodes pour une #valuation exacte de ce param#tre dans quelques cas, et une m#thode pour son approximation en g#n#ral. 1 Introduction Trace monoids were introduced by Mazurkiewicz in order to handle an algebraic study of concurrency [10]. the study has been developed and enriched by several authors [1] 3] 8] 9] 11] 12] Arqu#s and Fran#on proposed a measurement of parallelism, for concurrent processes based on the possibility of shuing words representing behaviors of processes. The approach allows some comparisons for ....

A. Mazurkiewicz, Concurrent Program Schemes and their Interpretations, DAIMI, BP78, Aarhus University, (1977).

.... information to allow action refinement within an interleaving model (under some assumptions that effectively require that the refinement of actions respects their dependencies) The idea of action dependencies regulating orderings has been suggested and inten sively studied by Mazurkiewicz [55, 56, 57] and others (see for instance [30] The basic concept in Mazurkiewicz work are traces, which are equivalence classes of sequences of actions, factorised by a permutation equivalence: in a sequence, adjacent independent actions may be commuted. System behaviour is described as a set of traces, ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Report PB-78, Aarhus University, 1977.

....are treated in [146] where more details and pointers to the literature on representative models can be found. Behavioral versus system models. A behavioral model is a model that abstract from repetition of system states. Language models as Hoare languages [63] and Mazurkiewicz trace languages [85, 86], synchronisation trees [141] and event structures [137, 142] are all examples of behavioral models. Models that allow an interpretation of repetition of system states are referred to as system models. The models of labelled transition systems [77] and Petri nets [106] are prime examples of such ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. Technical Report PB-78, DAIMI, Computer Science Department, University of Aarhus, 1977.

....languages over a trace alphabet ( Sigma; I) with I Sigma Theta Sigma being an irreflexive and symmetric independence relation. Here by traces we mean the class of Sigma labelled posets known as Mazurkiewicz traces. These objects play a fundamental role in the theory of distributed systems [2, 11]. It turns out that regular trace languages can be captured by a natural extension of DLTL( Sigma) denoted DLTL( Sigma; I) The main idea is that one must use regular parallel programs instead of regular programs to index the until operator. What is interesting about this characterization is ....

A. Mazurkiewicz, Concurrent program schemes and their interpretations, Technical report DAIMI PB-78, Computer Science Department, University of Aarhus (1977).

.... the computional resources needed for the veri cation task can be substantially reduced by means of the so called partial order methods for veri cation [8, 12, 18] Such equivalence classes can be canonically represented by restricted labelled partial orders known as Mazurkiewicz traces [5, 10]. These objects apart from alleviating the state explosion problems of veri cation also allow direct formulations of properties expressing concurrency and causality. A number of linear time temporal logics to be interpreted directly over Mazurkiewicz traces (e.g. 1, 4, 11, 14, 15, 16, 17] ....

Mazurkiewicz, A.: Concurrent program schemes and their interpretations. Technical report PB-78, Department of Computer Science, University of Aarhus, Denmark (1977)

....[7] Two semantics have been already considered for template graphs, or graphs resp. and graphs, 7] Here we focuss on specifications (templates) given as and or MSC graphs, which can be seen as alternating transition systems with an associated causal order related to Mazurkiewicz traces, [5]. Like alternating automata on words [10] it is more convenient to translate specifications into and or MSC graphs than just or graphs. The matching problem for and or template graphs is shown to be PSPACE complete. However, our proof shows somewhat surprisingly that and or templates are not ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

.... Sigma) nD, called independence relation, induces an equivalence relation j I on Sigma , generated by the pairs (uabv; ubav) with u; v 2 Sigma , a; b) 2 I. The relation j I turns out to be a congruence. The quotient monoid M ( Sigma; D) Sigma = j I was called trace monoid by Mazurkiewicz [13] and it was first used in combinatorics for rearrangement problems [1] The canonical surjective homomorphism associated to j I will be denoted by : Sigma M ( Sigma; D) For a 2 Sigma, A Sigma, let D(a) fb 2 Sigma j (a; b) 2 Dg and D(A) S a2A D(a) By definition, a trace t is a ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....one may 1 apply to the model include static concurrency analysis [22] path finding [21] and symbolic interpretation [9] ffl Finite state machines . Ordering of actions in the model is described by simple automata in a quite standard way, based on the classic notion of a global system state [13]. We have eliminated the problem of state explosion by utilizing the concept of sum machines and by encapsulating interprocess communication in objects that model communications events. Techniques that can be applied to these objects deal with test case design for parallel programs [18] and their ....

....to communication statements that depend on one another. In order to capture that it is required to extend the notion of a flowgraph. One possible extension is the Petri net model [16] with a number of tokens representing control flow of component processes as they progress through the system [13]. Unfortunately, analysis of a Petri net seems to be more abstract to programmers than the intuitively simpler flowgraph, and requires a great deal of formalism. Another extension is the model based on finite state machines [12] A single flowgraph may be viewed as a finite state machine (FSM) ....

Mazurkiewicz, A.: Concurrent program schemes and their interpretations. DAIMI Rep. PB-78, Aarhus University, 1977.

....are shown to have low polynomial time complexity. 1. Introduction Free partially commutative semigroups (often called trace semigroups) are a natural generalization of free semigroups. Introduced in a combinatorial context by Cartier and Foata [1] later they were proposed by Mazurkiewicz [4] as a model for concurrent systems. Afterwards, the trace theory developed rapidly and many classical results on free semigroups were generalized to this context (see [2] for an introduction and an overview of the theory) We consider free partially commutative semigroups in semigroup varieties. ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Report PB 78, Aarhus University, Aarhus, 1977.

....with the property that every APC expression for pn q n is of exponential size. 3 Semi Commutation Rewriting and APC Semi commutations are a natural way of expressing causal independence in concurrent systems in an algebraic way. The original notion was proposed in the late 70 s by Mazurkiewicz [12] for the semantics of Petri nets. Mazurkiewicz traces and semi traces are a model of true concurrency with nice algorithmical properties, which can be exploited for automatic verification methods. A semi commutation relation R defined over an alphabet Sigma of actions is an irreflexive binary ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....operationalsemanticsforhighlevelprocesslanguageswithvalue passinglike OCCAM (seee.g. 18] Many equivalenceresultsbetweenoperationaland denotationalsemantics existforthelineartimeversusbranchingtimespectrummentionedabove,seee.g. 15,16,35 ] An earlyreferenceforMazurkiewicztracesis[61],thoughthematerialcanalso be foundin[62] Mazurkiewicztracesaregenerallydeflneda littledifierently. In particularitisnotusualtoinsistonthecoherence axiom(referredtoasproperness by Mazurkiewicz[62] intheirdeflnition. As remarked,Mazurkiewicztracesmay be viewedasspecialkindsoflabelled ....

Mazurkiewicz,A.,Concurrent programschemes and theirinterpretations, DAIM PB-78,ComputerScience Department,Universityof Aarhus,1977.

....constraints on shuffle the reader may consult [14] 24] 25] 29] However, these semantic constraints on shuffle are not investigated in this paper but they will be the subject of a forthcoming paper. Also, we do not consider the shuffle operation between trace languages, introduced in [32], although trace languages are widely investigated in connection to parallel computation and the theory of concurrency, see [1] 7] 8] 33] The syntactic constraints that we consider here are based on the notion of a trajectory. Roughly speaking, a trajectory is a segment of a line in plane, ....

A. Mazurkiewicz, Concurrent program schemes and their interpretations, DAIMI Rep. PB 78, Aarhus, 1977.

....of such transition systems. Asynchronous cellular automata (ACA) form another fundamental class of transition systems with built in concurrency. They were introduced for traces by Zielonka [Zie87, Zie89] Mazurkiewicz introduced traces in order to describe the behaviors of one safe Petri nets [Maz77, Maz86]. A trace is a pomset where the partial order is dictated by a static dependence relation over the actions of the system. The primary aim of this work is to generalize the notion of ACA so that they can work on pomsets without auto concurrency. In Section 3, we define our notion of ACAs. There are ....

A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.

....specification of concurrent programs. Institut fur Algebra, Technische Universitat Dresden, D 01062 Dresden, Germany E mail: fdroste,kuskeg math.tu dresden.de Research supported by the German Research Foundation (DFG) A possible framework for the modeling of a concurrent system are traces [Ma77, Di90, DR95] whose formulation was inspired by the behaviour of Petri nets. A trace alphabet consists of a finite set E of actions and a binary independence relation I. Then I is the smallest congruence on E that identifies ab and ba whenever a and b are independent. The quotient of the free monoid E ....

A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.

....class satisfy a certain property or none do. For such properties the veri cation task can be substantially improved by the partial order methods for veri cation [6, 10] Such equivalence classes can be canonically represented by restricted labelled partial orders known as Mazurkiewicz traces [4, 8]. These objects allow direct Part of this work was done at Lehrstuhl f ur Informatik VII, RWTH Aachen, Germany Basic Research in Computer Science, Centre of the Danish National Research Foundation. formulations of properties expressing concurrency and causality. A number of linear time ....

Mazurkiewicz, A.: Concurrent program schemes and their interpretations. Report PB-78, Department of Computer Science, University of Aarhus, Denmark (1977)

....of nite non trivial and of in nite connected components and on the number of isolated letters of the dependence alphabet. 1 Introduction Trace monoids were introduced by Cartier and Foata [3] who investigated combinatorial problems concerning the rearrangement of words, and by Mazurkiewicz [14], who was motivated to provide a mathematical model for concurrent systems. Since then trace theory has become a very popular topic, see the recent surveys [5, 6] Corresponding author. This work was written while the second author worked at the TU Dresden. 1 Originally, the main interest ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Report PB-78, Aarhus University, Aarhus, 1977.

....associated to an alphabet Sigma and a symmetric, irreflexive commutation relation I Sigma Theta Sigma as the quotient monoid M ( Sigma; I) Sigma =fab = ba j (a; b) 2 Ig. Trace monoids became meaningful for computer science in concurrency theory, where they were introduced by Mazurkiewicz [12] in connection with the semantic of labelled Petri nets. As a mathematical model traces are simple enough in order to be able to generalize many interesting results from free monoids in areas as automata theory or logic. However, traces are complex enough in order to require in most cases ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....theory of free partially commutative monoids generalizes both, the theory of free monoids and the theory of free commutative monoids. In computer science, free partially commutative monoids are commonly called trace monoids and their elements are called traces. Both notions are due to Mazurkiewicz [Maz77], who recognized trace monoids as a model of concurrent processes. DR95] gives an extensive overview about current research trends in trace theory. The relevance of trace theory for computer science can be explained as follows. Assume a nite alphabet . An element of the free monoid over , ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....order reduction is only applicable for speci cations that respect this independence. However, to decide whether a formula is so called trace consistent is complete in PSPACE [Pel98] An approach on the same line is to describe the runs of a system by partial orders which are Mazurkiewicz traces [Maz77, DR95] and to formulate speci cations directly for traces. This guarantees that every formula of a corresponding logic is automatically trace consistent. Furthermore, given a decision procedure in terms of an automaton accepting all linearisations of the partial orders satisfying the speci cation, ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....second thought. Marek Bednarczyk has told me how he thinks the approach might be generalized to Place Transition Petri nets with multiple dots in states. These nets may be embedded in (a generalisation of) the category of asynchronous transition systems, and, as argued in [1] Mazurkiewicz traces [10] may be regarded as their computations. Marek s idea is that the path homotopy classes generalize trace equivalence of paths as defined by Mazurkiewicz. After some (not too many) technical discussions, we both agreed that this direction should be pursued. No effort has yet been undertaken to ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. Report DAIMI PB-78, Dept. of Computer Science, University of Arhus, 1977.

....2300 RA Leiden, The Netherlands Anca Muscholl Universitat Stuttgart, Institut fur Informatik, Breitwiesenstr. 20 22, 70565 Stuttgart, Germany 1 Introduction First studied in combinatorics [5] the theory of traces was introduced in computer science through the initiating work of Mazurkiewicz [14]. The initial idea was to provide a convenient model for the behaviour of labelled 1 safe Petri nets. Trace theory now represents a well investigated mathematical framework for studying concurrent systems (for a detailed overview see [10] The basic underlying concept is to consider a system as a ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....commutative monoids has its origins in combinatorics, 13] The existence of the Foata normal form and the characterization of the Mobius function as the polynomial over the independence cliques is shown there. In computer science they became popular mainly by the work of Mazurkiewicz, see [68, 69]. Early ideas can also be found in the paper by Keller (1973) where the Projection Lemma is (implicitly) stated in [64, Lem. 2.5] The Projection Lemma in the more general context of semi commutations is due to Clerbout [15] Levi s Lemma and the Projection Lemma, as stated here, can be found in ....

....a star free expression of dot depth m 1. Since u 2 U , v 2 V , we have uav 2 T , hence L( Psi ) T . The converse follows by known techniques by the Congruence Lemma (Lem. 76) Lem. 75, and Lem. 74. 6. 4 Bibliographical Remarks Dependence graphs for traces were already used by Mazurkiewicz in [68]. In Anisimov and Knuth we find the description as labeled partial orders [4] The characterization of recognizable trace languages by monadic second order logic over traces is due to Thomas [89] Our proof follows Ebinger [41] see also [34, Chapt.10] This allows to include the first order ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....of nite non trivial and of in nite connected components and on the number of isolated letters of the dependence alphabet. 1 Introduction Trace monoids were introduced by Cartier and Foata [3] who investigated combinatorial problems concerning the rearrangement of words, and by Mazurkiewicz [16], who was motivated to provide a mathematical model for concurrent systems. Since then trace theory has become a very popular topic, see the recent surveys [6, 7] Current a liation: 3SOFT GmbH, Frauenweiherstrae 14, D 91058 Erlangen, Germany. Corresponding author. This work was written ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Report PB-78, Aarhus University, Aarhus, 1977.

....While the study of partially commutative monoids traces back to implementation. For example, we also call a protocol definition in some formal language an implementation rather than only its implementation in some low level programming language. Cartier and Foata [CF69] it was Mazurkiewicz [Maz77] who applied these structures for modeling concurrent systems. The exciting point on Mazurkiewicz traces is that only a single representative for each equivalence class might be employed for verifying a desired property. This is the insight underlying many of the partial order based verification ....

....all executions which we consider to be equivalent. Each of its paths represents a single observation. Every (maximal) path of the configuration graph is isomorphic to a linearization of the trace T , which will be defined in the next section. 3. 4 Traces and Words In its original formulation [Maz77] Mazurkiewicz introduced traces as certain equivalence classes of words, and this correspondence turns out to be essential for our developments here. While considering traces as partial orders reveals their correspondence to partial order executions within the application domain of concurrency, ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....for example that the universality problem for this class is EXPSPACE complete. The same complexity bound follows for both MSC problems. Furthermore, we show that the connectivity property for automata is co NP complete. 2 Preliminaries We first recall the notion of Mazurkiewicz (semi ) traces, [6, 10]. An independence alphabet is a pair (A; I) where A is an alphabet endowed by an irreflexive relation I A Theta A, called independence relation (or commutation relation) Note that we do not assume that I is symmetric. With a given independence alphabet (A; I) we associate a rewriting relation ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

....Danish National Research Foundation, Computer Science Department, Aarhus University, Denmark. 2 The author was partially supported by Polish KBN grant No. 8 T11C 002 11. 1 Introduction We propose a linear time temporal logic called LTrL whose models are Mazurkiewicz traces. From its inception [16] the class of labelled partial orders known as (Mazurkiewicz) traces has played a fundamental role in the theory of distributed systems. In particular, traces constitute the natural tool for capturing the non interleaved linear time behaviour of models such as Petri nets, event structures and ....

A. Mazurkiewicz. Concurrent program schemes and their interpretations. Report DAIMI-PB-78, Computer Science Department, _ Arhus University, _ Arhus, Denmark, 1978.

....these methods lies in the fact that via these methods the computational resources required for the veri cation task can often be dramatically reduced. It is often the case that the equivalence classes of computations generated by a distributed system constitute objects called Mazurkiewicz traces [DR95,Maz77]. They can be canonically represented as restricted labelled partial orders. This opens up an alternative way of exploiting the non sequential nature of the computations of a distributed system and the attendant partial order based methods. It consists of developing linear time temporal logics ....

....0 i there exists an e 2 E such that (e) a, e 62 c and c 0 = c [ feg. Con gurations of C T are the trace theoretic analogues of nite pre xes of strings. As will become apparent in Section 3, the formulas of LTL are to be interpreted at con gurations of traces. In its original formulation [Maz77], Mazurkiewicz introduced traces as certain equivalence classes of strings, and this correspondence turns out to be essential to our developments here. To bring this out, let be the set of nite strings over and be the set of (countably) in nite strings generated by with = f0; 1; ....

Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

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Antoni Mazurkiewicz. Concurrent program schemes and their interpretations. Technical Report Technical Report DAIMI PB 78, Aarhus University, 1977.

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A. Mazurkiewicz. Concurrent Program Schemes and their Interpretation. Technical Report DAIMI PB-78, Arhus University, Denmark, 1977.

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. Technical Report DAIMI Rep. PB-78, Aarhus University, 1977.

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. Technical Report DAIMI PB 78, Aarhus University, 1977.

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Mazurkiewicz, A (1977) Concurrent program schemes and their interpretations. Technical Report DAIMI Pb-78, Aarhus University, Denmark.

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

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A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI PB--78, Arhus University, 1977.

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A. Mazurkiewicz, Concurrent program schemes and their interpretations, DAIMI Rep. PB 78, Aarhus Univ., 1977.

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

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A. Mazurkiewicz, Concurrent program schemes and their interpretations, DAIMI Rep. PB 78, Aarhus University, 1977. 12

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. DAIMI Rep. PB 78, Aarhus University, Aarhus, 1977.

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Mazurkiewicz, A.: Concurrent program schemes and their interpretations. Technical report DAIMI PB-78, Department of Computer

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Mazurkiewicz, A.: Concurrent program schemes and their interpretations. Technical report DAIMI PB-78, Department of Computer Science, University of Aarhus, Denmark (1977) 46

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A. Mazurkiewicz. Concurrent program schemes and their interpretations. Report DAIMI PB-78, Dept. of Computer Science, University of Arhus, 1977.

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