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Chapter 2. Formal foundations Notation 2.21 (Initial marking and End marking) Let N = (P, T, F, R) be an RWF-net and i, o be the input and output places of the net. The initial marking of N is denoted as M i and it represents a marking where there is a token in the input place i (i.e., M i = i). Similarly, the end marking of N is denoted as M o and it represents a marking where there is a token in the output place o (i.e., M o = o). We now define two structural properties for an RWF-net: soundness and weak soundness. An RWF-net is sound if and only if it satisfies three criteria: option to complete, proper completion and no dead transitions. Definition 2.22 (Soundness) Let N = (P, T, F, R) be an RWF-net and M i , M o be the initial and end markings. N is sound iff: 1. option to complete: for every marking M reachable from M i , there exists an occurrence sequence leading from M to M o , i.e., for all M ∈ N[M i 〉 : M o ∈ N[M〉, and 2. proper completion: the marking M o is the only marking reachable from M i with at least one token in place o, i.e, for all M ∈ N[M i 〉 : M ≥ M o ⇒ M = M o , and 3. no dead transitions: for every transition t ∈ T , there is a marking M reachable from M i such that M[t〉, i.e, for all t ∈ T there exists an M ∈ N[M i 〉 such that M[t〉. The soundness definition for an RWF-net is based on the previous soundness definition [AH05] for WF-nets. The soundness of WF-nets has been shown to correspond to boundedness and liveness properties of the short-circuited WFnet [Aal97]. The soundness property of a WF-net can be determined from its reachability graph and is decidable. Note that reachability is not decidable for an arbitrary reset net [DJS99]. As the soundness property of an RWF-net relies on reachability results, we will show that the soundness property of an RWF-net is also not decidable 4 . Suppose we have a reset net N with an initial marking I, and we want to know whether marking Q is reachable from I(see Figure 2.4). Note that we assume that every transition in N has at least one input and one output place. We also assume that there are no dead transitions in N. We propose to wrap N in an RWF-net N ′ as shown in Figure 2.5. We will show that N ′ is sound iff Q is not reachable from I. Note that the dotted lines represent the boundary between places and transitions in the original reset net N. Transitions with crosses in them represent global reset transitions. For the sake of clarity, we have not drawn all the reset arcs between all places and the global reset transitions. There is one global reset transition for each place in N which reset all places, including i, p, q, s, and o. An RWF-net also does not support weighted arcs and we assume that weighted arcs are not needed. However, if the marking I and Q require this, it is possible to introduce extra transitions and 4 This proof is due to Eric Verbeek and Marc Voorhoeve from Eindhoven University of Technology. PhD Thesis – c○ 2006 M.T.K Wynn – Page 18
Chapter 2. Formal foundations Figure 2.4: Reachability is undecidable for a reset net. places to support it. Figure 2.5 demonstrates the idea behind the transformation as given in Definition 2.23. We assume that p, q, r, s, t, i, o, u, v, y, t x do not appear in N. i t s N I Any transition in N Q q u v p Any place in I Any place in N Any place in Q r Global resets y o Figure 2.5: Transformation of a reset net into an RWF-net to demonstrate that soundness of an RWF-net is undecidable. Definition 2.23 (transRWF) Let N = (P, T, F, R) be an RWF-net and I, Q ∈ IM(N). The function transRW F (N, I, Q) returns an RWF-net N ′ = (P ′ , T ′ , F ′ , R ′ ) such that P ′ = P ∪ {p, q, r, s, i, o}, PhD Thesis – c○ 2006 M.T.K Wynn – Page 19
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[ADK02] [AH00] [AH02] [AH04] [AH05]
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