M. Droste. Concurrent automata and domains. International Journal of Foundations of Computer Science, 3:389418, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 exception that we are aware of is a result by Majster Cederbaum and Baier ....

M. Droste. Concurrent automata and domains. International Journal of Foundations of Computer Science, 3:389418, 1992.

....dynamic behavior is due to investigations of Panangaden and Stark [80] who introduced the notion of trace automata. Close to this is the notion of concurrent automata being transition systems where the dependence relation may vary with the state and their monoids of computations due to Droste [35]. Many important theorems of trace theory have been generalized to these monoids, see [38] 70 ....

Manfred Droste. Concurrent automata and domains. International Journal of Foundations of Computer Science, 3:389--418, 1992.

.... with some auxiliary structure to capture causality have been proposed as models of concurrency, cf. the asynchronous transition systems of Shields [23] and Bednarczyk [2] the behaviour structures of Rabinovich Trakhtenbrot [21] the concurrent transition systems of Stark [24] and Droste, [5] and the transition systems with independence of Nielsen Winskel [26] In each of these cases the added structure does not fundamentally increase their expressiveness: after a suitable behaviourpreserving unfolding, the causalities expressed by this added structure are completely determined by ....

M. Droste (1992): Concurrent automata and domains. International Journal of Foundations of Computer Science 3(4), pp. 389--418.

....of a given net can be fired simultaneously, without any change to the terminal states of the net and to the existence of non termination. The possibility of firing a transition simultaneously with others reduces the number of states by half. In another example of such an approach, Droste [2] also proposes models of parallel systems where interactions between components take place simultaneously. For CCS, the same idea can be expressed as a possibility of simultaneous actions performed by different components of a given composition (cf. e.g. 4, 5] However, the main aim of the cited ....

Droste M.: Concurrent Automata and Domains, Intern. J. of Foundations of Comp. Sc. 3, No. 4, pp. 389-417, 1992.

....of actions within the system. Because of their single locum of control, they necessarily represent so called interleaving model of concurrency. Petri nets may be used to model concurrent actions within the system by simultaneous firing of more than one transition at the same time (see for instance [5, 6]) Hence, they may represent a noninterleaving model of concurrency. If a set of actions that may happen together replaces all of their possible interleavings, the system state space may be reduced considerably, thus leading to more efficient analysis tools. For instance, replacement of 2 possible ....

Droste M.: Concurrent Automata and Domains, Intern. J. of Foundations of Comp. Sc. 3, No. 4, pp. 389-417, 1992.

....of actions within the system. Because of their single locum of control, they necessarily represent so called interleaving model of concurrency. Petri nets may be used to model concurrent actions within the system by simultaneous firing of more than one transition at the same time (see for instance [13, 2]) Hence, they may represent a noninterleaving model of concurrency. If a set of actions that may happen together replaces all of their possible interleavings, the system state space may be reduced considerably, thus leading to more efficient analysis tools. For instance, replacement of 2 possible ....

Droste M.: Concurrent Automata and Domains, Intern. J. of Foundations of Comp. Sc. 3, No. 4, pp. 389-417, 1992.

....produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R N ThetaN ....

M. Droste. Concurrent automata and domains. Internat. J. Found. Comput. Sci., 3(4):389--418, 1992.

....systems with a notion of concurrency or independence , in the same way as LES may be seen as adding concurrency to ST. Moreover, such enriched transition systems should also represent the system model version of event structures. Several such models have appeared in the literature [17, 1, 18, 3]. However, the asynchronous automata of [17] are not suited to our programme, since they are inherently deterministic. Also the transition systems introduced in [1, 18, 3] do not fit directly the frame, as they are unlabelled. Nevertheless, we could use them profitably provided a layer of labels ....

....should also represent the system model version of event structures. Several such models have appeared in the literature [17, 1, 18, 3] However, the asynchronous automata of [17] are not suited to our programme, since they are inherently deterministic. Also the transition systems introduced in [1, 18, 3] do not fit directly the frame, as they are unlabelled. Nevertheless, we could use them profitably provided a layer of labels is added on top of the events which decorate their transitions. However, since such a double decoration of transitions would not be mathematically very pleasant, here we ....

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M. Droste. Concurrent Automata and Domains. Int. Journal of Foundations of Computer Science, n. 3, pp. 389--418, 1992.

....computations. The preorder . on CS(A) induces a partial order on M 1 (A) by [fl] ffi] fl . ffi for any fl; ffi 2 CS(A) It can be shown that the finite prefixes of a computation [fl] approximate [fl] precisely as in domain theory arbitrary elements are approximated by compact elements (cf. [Dr92]) Let 2 Q be an distinguished initial state of A. A computation sequence fl is initial if either fl = ffl or dom(fl) Let D(A) fx 2 M 1 (A) j dom(x) or x = 1g. Then, in particular, D(A) is a domain (algebraic pointed cpo, cf. Dr92] In [Dr90, Dr92, Ku94a, Ku94b] the relation ....

....are approximated by compact elements (cf. Dr92] Let 2 Q be an distinguished initial state of A. A computation sequence fl is initial if either fl = ffl or dom(fl) Let D(A) fx 2 M 1 (A) j dom(x) or x = 1g. Then, in particular, D(A) is a domain (algebraic pointed cpo, cf. [Dr92]) In [Dr90, Dr92, Ku94a, Ku94b] the relation between the order theoretic properties of this domain and the algebraic properties of the automaton A have been considered. Now we define a (partial) transition function on A that describes the change of the state of an automaton in response to a ....

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M. Droste. Concurrent automata and domains. Intern. J. of Found. of Comp. Science, 3:389--418, 1992.

....the implication ( is called the inverse cube axiom. Theorem 7.3 Let A be a stably concurrent automaton. Then (M (A) Delta; is a labeled divisibility monoid with finite commutation behavior. Cancellation from the left and the existence of infima for any two monoid elements follows from [10], cancellation from the right has been shown in [3] and the distributivity of the lattice (# m; was proved in [14, 15] It is obvious that the monoid M (A) is generated by its irreducible elements T and that this set is finite. Hence the monoid is a divisibility monoid (alternatively, one ....

....It is obvious that the monoid M (A) is generated by its irreducible elements T and that this set is finite. Hence the monoid is a divisibility monoid (alternatively, one can use Theorem 3. 5 to show this) The finite commutation behavior was discovered in [11] The monoid is even labeled by [10]. Now it is immediate by Theorem 6.8 that recognizable, c rational and mc rational languages coincide in M (A) 26 Corollary 7.4 Let A be a stably concurrent automaton and let L M (A) Then L is recognizable iff it is c rational iff it is mc rational. In [11] a similar equivalence ....

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M. Droste. Concurrent automata and domains. Intern. J. of Found. of Comp. Science, 3:389--418, 1992.

....A has only one state, i.e. Q = fqg, the canonical bijection between CS(A) and E induces an isomorphism between the quotient monoids CS(A) and E = kq ; hence concurrency monoids provide a generalization of trace monoids. Automata with concurrency relations were introduced and studied in [Dr90, Dr92, BD93, BD94, Ku94b], where their domains of computation sequences were investigated and shown to be closely related with event domains and dI domains arising in denotational semantics of programming languages. Similar structures, in a slightly different form, and their applications were investigated independently in ....

....monoids we wish to generalize here to concurrency monoids M(A) For this, a useful (and almost necessary) assumption is that A is stably concurrent, i.e. satisfies the cube and the inverse cube axiom (see Def. 2. 6 for a precise formulation) These axioms arose several times in the literature, see [PS88, S89a, PSS90, Dr90, Dr92, Ku94b]. Intuitively, they mean that the concurrency relations of A depend locally (but not globally) on each other and they ensure that the monoid M(A) has nice structural properties. As shown in [Ku94b] stably concurrent automata generate precisely the class of dI domains, and these distributivity ....

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Droste, M.: Concurrent automata and domains. Intern. J. of Found. of Comp. Science 3 (1992), 389-418.

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