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Structure theory of equal conflict systems

Enrique Teruel, Manuel Silva

1996
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Theoretical Computer Science
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Equal conflict net Systems are a weighted generalisation of the (extended) tiee choice suhclass that keeps the total autonomy of choices. A substantial part of the analytical branch of the structure theory of fiee choice Systems, including decomposition and duality, the rank theorem, characterisations of impartial sequences and prompt interfaces, and the existente of home states, is extended to the equal conflict case. In so doing, several familiar concepts and objects from the structure theory
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... of placekransition net Systems are revisited from a linear algebraic perspective, which is specially adequate to cope with weighted nets. ' Tbis work has been partially supported by tbe projects CICYT TIC-91-0354, Esprit BRA Project 7269 (QMIPS), and Esprit W.G. 6067 (CALIBAN). The Standard definition of (behavioural) conflict between two transitions at a marking M of a P/T System (.N,M,-,) tan be expressed as follows: Two transitions, t, t' E T, are in (behavioural, or egective) conjict at marking M iff M aPre[P, t] and M 2 Pre[P, t'], but M 3 Pre[P, t] + Pre[P, t']. That is, the notion of conflict captures the Situation where two transitions are simultaneously enabled but they cannot be fired concurrently. This notion may be very adequate for the case of safe (i.e. 1-bounded) Systems, but, with Kluge and Lautenbach [19] and Chiola [7], we Claim that in a more general setting, for instance in the case of bounded Systems, it is better to define conflicts in terms of the enabling degree of transitions, instead of using the mere enabling. As an example, consider a place p having two output transitions, t and t', with W(p, t) = 2 and W(p, t') = 3. When M[p] = 5, both transitions are enabled, with degree two and one, respectively. They tan fire concurrently, so with the standard definition they are not in conflict. With the definition in [7] , tt is in conflict with t because the firing of t' reduces in one the enabling degree of t, although t is not in conflict with t' because the firing of t does not modifj the enabling degree of t'. We prefer having a symmetric conflict relation, hence the following definition, which is a particular case of that appearing in [19] , where also place capacities are considered:

doi:10.1016/0304-3975(95)00124-7
fatcat:ed2alvpqvjdx7f2sx3aekhnfmi