11th International Conference on Analysis and Optimization by Guy Cohen, Jean-Pierre Quadrat

By Guy Cohen, Jean-Pierre Quadrat

The convention, coorganized via INRIA and Ecole des Mines de Paris, specializes in Discrete occasion structures (DES) and is aimed toward engineers, scientists and mathematicians operating within the fields of computerized keep watch over, Operations examine and facts who're attracted to the modelling, research and optimization of DES. quite a few tools equivalent to Automata conception, Petri nets, and so forth. are proposed to explain and study such structures. comparability of those assorted mathematical ways and the worldwide war of words of theoretical ways with purposes in production, telecommunications, parallel computing, transportation, and so on. are the ambitions of the convention.

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In fact, UI~ o UI~ ... _, ,,-2 o U'i:_~ generates a (necessarily consecutively pairwise dynamically consistent block states) trajectory Xi = Xil, Xi2 . . . Xi, = Xj; this block state trajectory has a set level realization which contains all trajectories starting from all states xi ~ Xi which pass successively through Xil, Xi2 . . . Xi,_~ and which directly terminate in Xj. 82 Furthermore it can be shown that if 7r1 -~ 7l~2,then a Jr2-control event is a collection of dynamically consistent concatenations of some zr~- control events.

Therefore K is not co-closed. On the other hand if we replace K by its closure, that is, K:=Cc+Abc+AaB , then K is co-closed since K=MUM I , where M := Cc + A b c and M r = C + Ab + A a B which are both co sublanguages of K. Next we present the formal results that substantiate the arguments elaborated by this simple example. Proposition 3. Every co-closed sublanguage of L is a wco language. The following proposition implies that the converse statement of the previous proposition holds only for closed languages and ,in general, gives equivalent characterizations of closed wco languages.

The associated theoretical results are summarized by the following theorems. Theorem 5. The language L e×B is the supremal wco sublanguage of K. 45 T h e o r e m 6. A sublanguage M of L is wco if and only if there exists B' E Sn(E~, Eo) such that Lp×B, = if×1. We now proceed to construct the nonblocking supervisor N 6 Sn that synthesizes the supremal co-closed sublanguage of K. First we state the definition of a nonblocking supervisor. Definition 7. A supervisory automaton S 6 S, is said to be nonblocking relative to a language K if for every reachable state x of the closed loop automaton P × S via a string • t s ~ Le×s there exists a string t such that st c Le×s fq K and x -~P/S.

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