DIEKERT, V. AND M ETIVIER, Y. 1997. Partial commutation and traces. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds., vol. 3, 457--534. Springer-Verlag.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....fields and a self contained theory of traces has also gradually developed. Finally, in 1995 the first monograph on trace theory [18] was published, presenting an overview of various directions of research on partial commutativity, and two years later one chapter of the Handbook of Formal Languages [15] was devoted exclusively to trace monoids. Topics considered in the framework of trace theory belong to many areas of both mathematics and theoretical computer science: algebraic structure of trace monoids as well as their combinatorial properties are studied, the theory of formal languages over ....

.... theories of regular languages, semi )linear sets and rational relations [4] Similarly to the theory of regular languages, the relationships between descriptions of languages using finite monoids, rational operations, standard automata, asynchronous automata and logic were established (see [15]) It is well known that for regular word languages all basic problems are decidable. But this nice property is not shared by the class of rational trace languages, which contains also non recognizable languages and where many problems like deciding universality or recognizability of a ....

[Article contains additional citation context not shown here]

V. Diekert, Y. Metivier, Partial commutation and traces, in: G. Rozenberg, A. Salomaa (Eds.), Handbook of Formal Languages, vol. 3, Springer, Berlin, 1997, pp. 457--533.

....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 while the second author worked at the TU Dresden. Originally, the main interest focused on nite traces. But in order to model the behaviour of non terminating ....

....metric. If the alphabet consists of nitely many letters only, then both metrics are known to be uniformly equivalent. Other metrics that give rise to non homeomorphic topological spaces were introduced in [4] cf. also [5] Plenty of work has been done concerning the monoid of nite traces (cf. [6, 7] for an overview) and the partial order of real traces (see [8, 10] However, apart from Kwiatkowska s results mentioned above, not much is known about the topology of real traces. In this respect, it seems that in nite dependence alphabets have not yet been considered, either. The only ....

V. Diekert and Y. Mtivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 457533. Springer-Verlag, Berlin  Heidelberg, 1997.

....#ft 2 M ; h(t) ng Gamma fl M : Furthermore, the numbers M and fl M are algebraic. Explicit formulas involving the series L and H are given for M and fl M . 2 The Trace Monoid We start by introducing all the necessary notions from the theory of trace monoids. The reader may refer to [11, 12] for further information. In the sequel, a graph is a couple (N; A) where N is a finite non empty set and A ae N Theta N . Hence we consider directed graphs, allowing for selfloops but not multi arcs. Dependence graphs and independence graphs (to be defined below) which are naturally ....

V. Diekert and Y. M'etivier. Partial commutation and traces. In Handbook of formal languages, volume 3, pages 457--533. Springer, 1997.

....reachable i# there is a process execution V 1 . V (k 1) such that s # V (k 1) for some k. Intuitively the process executions are step executions which are in a certain canonical normal form. In fact, this canonical normal form corresponds exactly to the so called Foata normal form [8] from the theory of Mazurkiewicz traces, and also to a partial order semantics for 1 safe Petri nets called processes. For more on this connection, see [9] and further references there. Fig. 1 gives two LTSs, both having the visible actions # 1 = # 2 = a, b . They will be used as a running ....

V. Diekert and Y. Metivier. Partial commutation and traces. In Handbook of formal languages, Vol. 3, pages 457--534. Springer, Berlin, 1997.

....TLC [21] can easily be encoded into MSO(P , Mess) and model checking can be shown to be decidable in this way. However, this approach does not seem to be reasonable for practical issues, due to the non elementary complexity of model checking MSO formulas. In the domain of Mazurkiewicz traces [8], several temporal logics have been studied and sophisticated model checking procedures were developed. These logics usually define trace closed languages. Therefore, as pointed out by Madhusudan and Meenakshi [16] it is desirable to find the existence of temporal logics interpreted over ECMSC ....

....in a natural manner. With respect to #, the dependence relation D(#) # # ) is given by (#, u)D(#) # # , u # ) i# o(#) o(# # ) or (#, # # ) or (# # , #) #= #. For each natural B, # : # # , D(#) # ) turns out to be a Mazurkiewicz trace alphabet [8]. Kuske and Morin investigate relations like this in detail [14, 20] A relation # WB WB provides information about which ECMSC words are seen to be equivalent with respect to D(#) So let # be the least equivalence relation satisfying the following: If w = w 1 (#, u) # # , u # )w 2 and w ....

V. Diekert and Y. Metivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III. Springer, 1997.

....of partially commutative monoids, in relationship to combinatorial problems. It was later shown to have connections with the theory of formal languages and with concurrent systems. There is a rich corpus of results related to these objects, both from a mathematical and algorithmic point of views [9, 10]. Sets of Optimal Sequences as Traces In this section, we show that the set of optimal sequences sorting a permutation can be described in terms of traces. Recall that a reversal can be identified with the set of displaced elements in the permutation. A sequence of reversals is thus a word over ....

V. Diekert and Y. Mtivier. Partial commutation and traces. In Rozenberg G. and Salomaa A., editors, Handbook of formal languages, Volume 3, pages 457--533. Springer, Berlin, 1997.

....occurrences of causally independent actions that can be naturally grouped together into equivalence classes where two computations are equated in case they are two di#erent interleavings of the same partially ordered stretch of behavior. This observation led to the notion of Mazurkiewicz traces [DM97, DR95] Traces can be understood as partially commutative words that model the behavior of a concurrent system. 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 ....

V. Diekert and Y. Metivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III. Springer, Berlin-Heidelberg-New York, 1997.

....TLC [18] can easily be encoded into MSO(P, Mess) and model checking can be shown to be decidable in this way. However, this approach does not seem to be reasonable for practical issues, due to the non elementary complexity of model checking MSO formulas. In the domain of Mazurkiewicz traces [6], several temporal logics have been studied and sophisticated model checking procedures were developed. These logics usually define trace closed languages. Therefore, as pointed out by Madhusudan and Meenakshi [14] it is desirable to find the existence of temporal logics interpreted over ECMSC ....

....to #, the dependence relation D(#) # # ) is given by (#, u)D(#) # # , u # ) i# 16 o(#) o(# # ) or (#, # # ) or (# # , #) #= #. For each natural B, the pair # : # 1, B, # , D(#)#(# 1, B, # ) turns out to be a Mazurkiewicz trace alphabet [6]. Kuske and Morin investigate relations like this in detail [12, 17] A relation # WB WB provides information about which ECMSC words are seen to be equivalent with respect to D(#) So let be the least equivalence relation satisfying the following: If w = w 1 (#, u) # # , u # )w 2 and w # ....

V. Diekert and Y. Metivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III. Springer, 1997.

....(b) occurs before R (a) or after it. Given #, we define the dependence relation and write (#, #)D(#) # # , # # ) i# P (#) P (# # ) or (#, # # ) Corr and # = # # or (# # , #) Corr and # = # # . It turns out that the pair (# ) is a Mazurkiewicz trace alphabet [6] for every natural B a fact which was already used in [12] providing a direct link between Mazurkiewicz traces and MSCs. We then define the relation to be the least equivalence relation satisfying the following: If # = # 1 (#, #) # # , # # )# 2 and # # = # 1 (# # , # # ) #, #)# 2 for ....

....lexicographic ordering on A # . A word w A # is said to be in lexicographic normal form i# it is minimal in [w] A,D) wrt. Furthermore, for X A # , Min(X) w # w is in lexicographic normal form denotes the set of minimal elements in X. Due to Theorem 4.6 and Corollary 4. 6 in [6], for a regular word language X A # , Min(X # X implies that X # is a regular Mazurkiewicz trace language. Thus, for a regular set X containing at least the representatives in lexicographic normal form, the closure of X wrt. A,D) is a regular (Mazurkiewicz trace) language. We now ....

Volker Diekert and Yves Metivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III. Springer, BerlinHeidelberg -New York, 1997.

....For a process S of sort (A; B) define its IO relation R S , f (ffj A i ; ffj Bo ) j ff 2 Sg L xL B . Actually, this defines a relation R S in (L A = x(L B = where L A = consists of the Mazurkiewicz traces [28] or the elements in the free partially commutative monoid [12]) of the trace language (L ; LA ; I A ) where I A LA xLA is the independence relation defined by ha; ui I A ha ; vi iff a 6= a . Recall that as in [42] the traces are equivalence classes of , the smallest equivalence relation such that ffw wfiffw w if w I A w . For ff 2 L A , ....

DIEKERT, V., AND M ETIVIER, Y. Handbook of Formal Languages. Springer, 1997, ch. Partial Commutation and Traces.

....#. It is easy to prove that there are markings M 1 , M 2 , Mn such that in the initial state M 0 of # the step execution M 0 [S 0 #M 1 [S 1 #M 2 Mn 1 [S n 1 #Mn can occur. This normal form is actually the Foata normal form from the theory of Mazurkiewicz traces, see e.g. [5]. It is only (quite trivially) adapted to processes of 1 safe net systems. To our knowledge it was rst applied to processes of 1 safe net systems in the veri cation algorithm setting in [7] The fact that the technique used is a Foata normal form is discussed in more detail in an extended ....

....a Foata normal form of a process. We will in our implementation use a di erent de nition, which is equivalent but more suitable for the implementation techniques we use. We have not found this version in the literature. However, it is just a simple adaptation of the version for traces, see e.g. [5]. De nition 2. The sequence of steps # = S 0 #[S 1 # [S n 1 # is a step execution of a 1 safe net system # in Foata normal form if: a) # = # (i.e. # is the empty step sequence) or (b) There are markings M 1 , M 2 , Mn such that in the initial state M 0 of # the step execution ....

V. Diekert and Y. Mtivier. Partial commutation and traces. In Handbook of formal languages, Vol. 3, pages 457534. Springer, Berlin, 1997.

....assume a finiteness condition for the commutation behavior of the monoid elements, and then we can prove that any c rational language is recognizable. This uses an extension of the notion of the rank of a language, which was already shown to be very useful in trace theory by Hashigushi [13] cf. [9, 8]. To deal with the iteration, differently from trace theory, we need Ramsey s theorem. Recall that an equation ab = cd with irreducible generators a; b; c; d of M states that the different sequential executions ab and cd give rise to the same effect. If now a 6= c, the effect of a in the ....

....easily show that s i does not have a proper divisor. In addition, t i 1 = s i 1 (x [s 1 s 2 : s i ] and therefore d x (s 1 s 2 : s n ) t 1 t 2 : t n . The uniqueness is immediate by the proof. 4 Commutation grids and the rank In trace theory, the generalized Levi Lemma (cf. [8]) plays an important role. It was extended to concurrency monoids in [11] Here, we develop a further generalization to divisibility monoids using commutation grids. This enables us to obtain a concept of rank of a language in these monoids, similar to the one given by Hashigushi [13] for trace ....

[Article contains additional citation context not shown here]

V. Diekert and Y. M'etivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3. Springer, 1997.

....for its census function CL . If further CL is polynomially bounded, then there exists a polynomial time r.e.c. for L. 6 Rational Trace Languages Another application of our method concerns the uniform random generation and the census function of trace languages. To study this case we refer to [10] for basic de nitions. We only recall that, given a trace monoid M ( I) over the independence alphabet ( I) a trace language, i.e. a subset of M ( I) is usually speci ed by considering a string language L and taking the closure [L] ft 2 M ( I) t = x] for some x 2 Lg. In ....

V. Diekert and Y. Mtivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III, pages 457527. Springer, Berlin-Heidelberg, 1997.

....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 focused on nite traces. But in order to model the behaviour of non terminating concurrent systems, one is also interested in in nite traces (cf. Gastin and Petit ....

....Mauri, and Pighizzini [1] by taking the distance of Foata normal forms. We call this metric the Foata normal form metric. If the alphabet consists of nitely many letters only, then both metrics are uniformly equivalent. Plenty of work has been done concerning the monoid of nite traces (cf. [5, 6] for an overview) and the partial order of real traces (see [9, 7, 2] However, apart from Kwiatkowska s results mentioned above, not much is known about the topology of real traces. In this respect, it seems that in nite dependence alphabets have not yet been considered, either. In the ....

V. Diekert and Y. Mtivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 457533. Springer-Verlag, Berlin  Heidelberg, 1997.

....for every graph G, there exists a real number , 0 1, such that PG ( 0 and j j for every root of PG (z) di erent from . This result allows to give the asymptotic expression of some sequences arising from enumeration problems and analysis of algorithms de ned on trace monoids [11, 10]. Indeed, every undirected graph G de nes a trace monoid M , where the alphabet and the concurrent relation are simply given by the vertices and the edges of G, respectively. Moreover, it is well known that 1 PG (z) is the generating function of the sequence ft n g, where t n is the number of ....

.... time complexity on input of size n turns out to be proportional to E( n ) The interest for clique polynomials is also related to the M obius function of trace monoids, a natural extension of the classical M obius function on positive integers, introduced by Cartier and Foata in [6] see also [10]) and more recently studied in [8, 7, 9] This is de ned as the function M mapping any element t 2 M in ( 1) jtj or 0, according whether t forms a clique of G or not. Hence, PG (z) is the commutative image of M obtained through the morphism mapping every generator of M into the unique symbol ....

V. Diekert and Y Metivier. Partial commutation and traces, in Handbook of formal languages: beyond words, Rozenberg G. and Salomaa A. editors, Springer, Berlin - Heidelberg - New York, p. 457-534, 1997.

....congruence classes is called trace monoid. It is determined uniquely by the generating alphabet and the relation I of independent letters. The free monoid is obtained as a special case, when all letters are dependent. Trace monoids have been studied in many publications. Good starting points are [8], 9] and [7] The factor problem in a trace monoid M is to decide for two words, whether the trace l, represented by the first word, is a factor of the trace t, represented by the second word (where l is called factor of t, when t = pls for some traces p, s) A linear time algorithm for the ....

Volker Diekert and Yves M'etivier. Partial commutation and traces. In G. Rozenberg and A. Salomaa, editors, Handbook on Formal Languages, volume III. Springer, Berlin-Heidelberg-New York. To appear. 26

....such as traces, Petri nets, or process algebra [Aalbersberg and Rozenberg 88, Hennessy 89, Sassone et al. 96] An important feature of the semantics defined by synchronization languages is that it splits each action into two parts, the start and the termination. For instance, in trace semantics [Diekert and M etivier 97] parallel execution is interpreted as a k b = ab ba, but in order to control the execution of parallel processes it is necessary to distinguish sequences like ab and ba from real parallel execution. By having the start and termination of an action as separate instantaneous parts such ....

Diekert, V., M'etivier, Y.: "Partial commutation and traces"; in: Handbook of Formal Languages, Vol. III. (G. Rozenberg, A. Salomaa, eds.) pp. 457--533, Springer-Verlag, 1997.

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DIEKERT, V. AND M ETIVIER, Y. 1997. Partial commutation and traces. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds., vol. 3, 457--534. Springer-Verlag.

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DIEKERT, V. AND M ETIVIER, Y. 1997. Partial commutation and traces. In Handbook of formal languages, G. Rozenberg and A. Salomaa, Eds., vol. 3, 457--534. Springer-Verlag.

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V. Diekert and Y. Metivier. Partial commutation and traces. In Handbook of formal languages, Vol. 3, pages 457--534. Springer, Berlin, 1997.

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V. Diekert and Y. M'etivier, Partial commutation and traces. In: Handbook of Formal Languages, Vol. 3 (G. Rozenberg and A. Salomaa, Eds.), Springer-Verlag, 1997, pp. 457--533.

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V. Diekert and Y. Metivier. Partial commutation and traces. In Rozenberg G. and Salomaa A., editors, Handbook of formal languages, Volume 3, pages 457-533. Springer, Berlin, 1997.

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V. Diekert and Y. Metivier. Partial commutation and traces. In Handbook of Formal Languages: Beyond Words, pages 457-534. Springer, 1997.

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V. Diekert and Y. Metivier. Partial commutation and traces. In Handbook of Formal Languages: Beyond Words, pages 457-534. Springer, 1997.

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V. Diekert and Y. Metivier. Partial commutation and traces. In Handbook of formal languages, Vol. 3, pages 457--534. Springer, Berlin, 1997.

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