K. Schmidt. How to calculate symmetries of petri nets. Acta Informatica, (36):545--590, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... can test in polynomial time whether a permutation belongs to the group [Furst et al. 1980] For permutation group algorithms, see e.g. Butler 1991; Kreher and Stinson 1999] 3 SYMMETRIES OF PLACE TRANSITION NETS The presentation in this section is based on [Starke 1991; Schmidt and Starke 1991; Schmidt 1997; 2000a] 3 SYMMETRIES OF PLACE TRANSITION NETS 3 3.1 P T Nets A Place Transition net (or a P T net) is a tuple N = hP; T; F; V; M 0 i, where 1. P is a nite, non empty set of places, 2. T is a nite, non empty set of transitions such that P T = 3. F (P T ) T P ) is the ow relation ....

....graph of N . For more on these properties and temporal logic model checking under symmetries, see e.g. Starke 1991; Jensen 1995; 1996; Clarke et al. 1996; Emerson and Sistla 1996; Gyuris and Sistla 1999] The integration problem in the (inductive) generation of quotient reachability graphs is [Schmidt 1999; 2000b] Problem 3.2 Given a set Q of already visited markings and a newly generated marking M , nd out whether there is a marking M 0 2 Q such that M M 0 . There are three basic ways to solve the integration problem [Schmidt 1999; 2000b] 1. When Aut(N) is known, loop over all ....

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SCHMIDT, K. 2000a. How to calculate symmetries of Petri nets. Acta Informatica 36, 7, 545590.

....occurring in the reachability analysis of nets. The symmetry reduction method was introduced by Huber et al. 1985; 1991] for colored high level Petri nets. The method was applied to low level nets, the formalism of this paper, by Starke [1991] and further studied in [Schmidt and Starke 1991; Schmidt 1997; 1999; 2000a; 2000b] The main idea of the method is that the symmetries (automorphisms) of a low level net produce corresponding symmetries to the state space of the net. For many veri cation tasks, such as deadlock checking, it is suf cient to inspect only one marking in each set of mutually ....

....of mutually symmetric markings (orbit) Thus a (potentially exponentially smaller) quotient reachability graph can be constructed instead of the normal reachability graph. Schmidt and Starke have presented algorithms for solving many of the problems involved in the method [Schmidt and Starke 1991; Schmidt 1997; 1999; 2000a; 2000b] However, the topic of this paper, the computational complexity issues of the sub tasks appearing in the method, has not been addressed before. 1 The problem of nding the automorphisms of a net is easily proven to be as hard as nding the automorphisms of a graph. This is ....

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SCHMIDT, K. 1997. How to calculate symmetries of Petri nets. Tech. Rep. MATH-AL-81997, Technische Universitt Dresden, Germany. Sept.

....manner. Furthermore, the approach is independent of the syntax of any high level formalism. On the other hand, there is no feasible way to describe low level symmetries symbolically. Thus, calculation is the only way to get the information about symmetries. However, the algorithm proposed in [Sch97] and implemented in INA [RS97] is able to calculate generating sets of maximal symmetry groups in reasonable space and time. The size of the generating set is in worst case quadratic in the net size and its calculation is, though in worst case exponential, polynomial in virtually all cases. For ....

....= m 2 . Sigma is an equivalence relation. We denote the Sigma equivalence class containing some state m by [m] Sigma . A state m of a net is called symmetric with respect to a symmetry group Sigma iff [m] Sigma = fmg. 4 Generating sets The symmetry calculation algorithm as presented in [Sch97] returns a generating set of the symmetry group that enjoys a rather regular structure. This structure is crucial for the methods studied later. As an illustrating example, we consider a distributed algorithm where identical agents interact in a grid like network (see figure 1) For the purpose ....

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K. Schmidt. How to calculate symmetries of Petri nets. Technical Report MATH--AL--8--1997, Dresden University of Technology, 1997. Submitted to a journal.

....N = P; T ; F; V:m 0 ] iff 1. oe(P ) P ; oe(T ) T ; 2. x; y] 2 F iff [oe(x) oe(y) 2 F ; 3. V ( x; y] V ( oe(x) oe(y) For a state m, define oe(m) by oe(m) oe(p) m(p) We denote the set of all symmetries of N by SN . SN forms a group under composition. For several reasons (see [Sch97] for details) it is sometimes reasonable to use a subgroup of SN instead of SN itself. Therefore, we fix an arbitrary subgroup S of SN for the remaining part of the paper. S induces equivalence relations on nodes and on states. Definition 3 (Equivalence) Let x and y be either both states or both ....

....subsets form a partition of S. Definition 4 (Ground set) A subset G of S containing exactly one element of every non empty S jk is a ground set of S. When referring to a ground set G, we denote the unique element of S jk by oe jk . For all j, let oe jj be the identity mapping on P [ T . In [Sch97], we show that Proposition 1 For every oe 2 S there are k 1 ; Delta Delta Delta ; k card(P[T ) such that oe = fl card(P[T ) j=1 oe jk j . This means, that every ground set is a generating set of S. Ground sets are not necessarily minimal generating sets of S. For instance, if oe is an ....

K. Schmidt. How to calculate symmetries of Petri nets. Technical Report MATH--AL--8--1997, Dresden University of Technology, 1997.

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K. Schmidt. How to calculate symmetries of petri nets. Acta Informatica, (36):545--590, 2000.

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