8 Simulatability of Net Types

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Osborn Rice
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1 8 Simulatability of Net Types In chapter 7 was presented different higher order net types. Now we discuss the relations between these net types, place transition nets, and condition event nets. Especially it would be considered the mutual simulatability of different net types. We have seen that modelling with higher order net types is easier but on the other side some properties like boundedness or liveness are not decidable. Therefore exist innocuous and dangerous modifications. Because the introduction of the firing rule with capacities can be reduced to standard concepts this is innocuously. But introducing the firing rule with priorities it leads to a higher order net class than the class of place transition nets. Definition 8.1 A net type NT 1 is called simulatable by net type NT 2 2 if exists an algorithm which for every net N 1 of type NT 1 constructs a net N 2 of type NT 2 and unique mappings ζ (translation of states) and τ (translation of transitions) such that: (a) Mapping ζ maps the reachability set R 2 of N 2 to the reachability set R 1 of N 1 where the initial marking m 0 (2) of N 2 is mapped to the initial marking m 0 (1) of N 1. (b) Mapping τ maps the set of transition T 2 of N 2 to the set of transitions T 1 of N 1 whereas additionally is included the empty word. (c) If m 2, m 2 R 2, q W(T 2 ) and m 2 [q> m 2 (in N 2 ) then holds ζ (m 2 ) [τ(q) > ζ (m 2 ) (in N 1 ). (d) If m 1, m 1 R 1, q W(T 1 ) and m 1 [q> m 1 (in N 1 ) then exist markings m 2, m 2 R 2, q W(T 2 ) with m 1 = ζ (m 2 ), q = τ(q ), m 2 [q > m 2 (in N 2 ), and m 1 = ζ (m 2 ). If furthermore the mapping ζ is bijective we speak about simulatability of the state behaviour. If the construction of net N 2, ζ, and τ is done node by node and arc by arc whereas only information about the node and its environment are used but not about the whole net or its reachability set, then we are speaking about local simulatability. 120

2 If a net type NT 1 is simulatable by net type NT 2 and vice versa then we say that the net types are equivalent. Theorem 8.1 A place transition net is locally simulatable by a loop free ordinary place transition net. Proof idea: Every place p of N which input arcs have the maximal multiplicity in(p) and which output arcs have maximal the multiplicity out(p) is substituted by a loop of places and transitions with in(p) + out(p)-places. The multiple arcs are distributed to the places of the loop. Using mapping τ the empty word ε is allocated to the transitions of the loop. For a marking m of the constructed net the marking ζ(m) has exactly the same number of tokens in place p like in m in the places of the corresponding loop. Figure 8.1 shows an example for this construction. Fig. 8.1: Transformation of a PT-net into an ordinary PT-net Theorem 8.2 Every bounded place transition net is locally simulatable by a safe Petri net (CE-net). 121

3 Proof idea: Every k-bounded place p (k 1) is splitted into k+1 places p 0,...,p n and the adjacent transitions are splitted in k transitions t 1,..., t k (where holds τ(t 1 ) =... =τ(t k ) = t) such that in every reachable marking m exactly one place contains a token. It is marked the place pi iff place p contains exactly i tokens in the corresponding marking ζ (m). Figure 8.2 shows an example for theorem 8.2. Fig. 8.2: Transformation of a PT-net into a safe PT-net Theorem 8.3 Every PT-net with capacities is locally simulatable with a PT-net (without capacities). The next figure demonstrates how the introducing of co-places realises the transformation of theorem 8.3. A co-place gets as initial marking a number of tokens which equals to the difference between the capacity of the corresponding place and its initial marking. If a transition puts tokens to a place then it consumes the same number of tokens from the coplace and vice versa if a transition consumes tokens from a place then it puts the same number of tokens to the corresponding co-place. 122

4 Fig. 8.3: Transformation of a PT-net with capacities into a PT-net without capacities In definition was introduced the firing rule with maximal steps. If we use this firing rule for normal place transitions nets without time then it is also possible to implement the null test for unbounded places. How to test if a place test is clean or not shows the following figure. Fig. 8.4: Null test for unbounded places in a PT-net using the firing rule with maximal steps If an unbounded place test is clean then fires the sequence of sets [t 1 ], [t 2 ] and place p 0 gets a token. But if place test is marked then the fire sequence are the sets [t 1, t 3 ], [t 4 ] and place p 1 would be marked. 123

5 9. Model Checking - see INA-handbook, p

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