By Kai-Uwe Sattler

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**Extra info for Algorithmen und Datenstrukturen**

**Example text**

We define the mapping iz: B' u E' ---t 8 x {1,2, ... } u T x {1,2, ... } in the following manner: (Vb E B')iz(b) = (rp'(b), k) where k = card( {b' E B'I(b', b) E (F')* /\ rp'(b') = rp'(b)}) (Ve E E')iz(e) = (rp'(e),k) where k = card({e' E E'I(e', e) E (F')* /\ rp'(e') = rp'(e)}). Now, let B = iz(B'),E = iz(E') and F ~ (B (iz(e),iz(b)) E F B',e E E'. {? x E) u (E x B) be defined as: (e,b) E F' and (iz(b),iz(e)) E F {? (b,e) E F', for b E We leave to the reader the straightforward proof that iz : B' U E' ---t B U E is one-to-one, ON = (B,E,F) is isomorphic to ON' = (B',E',F'), and that ON can be derived by the above algorithm.

Similarly for a E E,b E B. 1. Elements of Net Theory (d,1) I I l ___ J ___ I I ~~~~:::~~~ __ _ The marked net N l ______________ _ ~--------------------An occurrence net ON _ _ _ _ _.. 9. Marked net and one of its processes Chapter 2. Formal Theory of Basic COSY 48 (3) From (1) we know that either a, bE B or a, bEE. If a, bE B then, from (3) of the process definition it follows that there is no B-cut containing both a and b, thus either (a,b) E (F')· or (b,a) E (F'Y. Let a,b E E. The condition is clearly true if a = b, so assume a =f:.

0. Hence (3i)Ot n Sj =f:. 0 and (to \ °t) n Si =f:. 0, which means that card(M n Si) = 2, a contradiction. 0 A net which is contact-free but not state machine decomposable is shown in Fig. 12. Let N = (S, T, F, Mo) be a smd-net, fixed for the rest of this section, with SMD(N) = {N1, ... ,Nn } where Ni = (Si,Ti,Fi,M~) for i = 1, ... ,n. In Sect. 2 we defined the set V ev* ~ Ev~ x ... x Ev~ on the basis of the sets EVll ... , Evn • The only assumption about the Ev/s was that they were finite, so we can repeat this construction using the sets T1 , ••• , Tn.