Concurrency Theory: Lecture 9, 10 February 2017
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We have defined the unfolding of a net as an occurrence net
"axiomatically", by combining N_w for all firing sequences w in a
single net

Can also explicitly construct unfolding for a safe net N =
(S,T,F,Min)

- Assume 1 safe for simplicity

- Create places of the form (p,X) and transitions of the form
  (t,Y) where p and t refer to places and transitions in the
  original net, X and Y refer to "past" of the element.

- Start with (p,emptyset) for each p in Min

- For t in T, let pre(t) = {p1,p2,...,pk} with (p1,X1), ...,
  (pk,Xk) concurrent elements in the unfolding

  Extend the unfolding with a new tranistion (t,Y) where
  Y = {(p1,X1),...,(pk,Xk)} if (t,Y) not already present

  Let post(t) = (p'1,p'2,...,p'm).  Add new output places
  (p'i,{(t,Y)})

Configuration

- Like event structure, downward-closed and conflict free set of
  events
- Mark(C) is marking after C

Cut

- Maximal pairwise concurrent set of places

Extension of C

- All transitions in the unfolding not in C but compatible with C
- Isomorphic to the unfolding of the net with Min = Mark(C)

Lemma:
  If Mark(C) = Mark(C'), extension of C is isomoprhic to extenion
  of C'

Complete prefix

- Prefix of unfolding containing all reachable markings and an
  occurrence of each fireable transition

- For a safe net, only bounded number of reachable markings, so
  finite complete prefix exists.  How to find it efficiently?

Local configuration

- Downarrow(e) for event e = {e' | e' <= e}

Intuition:

- If Mark(Downarrow(e)) = Mark(Downarrow(e')) for event e' added
  earlier, no need to expand below e
- e is a "cutoff" event

Example in the paper shows that this has to be done carefully.

To fix the example, McMillan requires size of Downnarrow(e') to
be strictly less than size of Downarrow(e)

Generalize this to adequate order << on configurations

1. << is well founded
2. C proper subset of C' implies C << C'
3. Preserved by extensions: C << C' impliex C+E << C'+Iso(E) for
   every extension E of C

Note that McMillan's size order is an adequate order

Modified unfolding algorithm:

- At each step of expansion, choose next event to be minimal wrt
  adequate order

- After adding an event, check and mark it if it is a cutoff
  event

- Never expand "below" a cutoff event

Theorem: This algorithm builds a finite complete prefix

Claim 1: The unfolding is finite

  Define depth of a transition and argue that the set of elements
  upto depth k is finite

Claim 2: Every reachable marking is represented

  Go backwards via cutoff events and appeal to well-foundedness
  of <<

Claim 3: Every fireable transition is represented

  Proof similar to that of Claim 2

Why go beyond McMillan's size order?

- Example that requires exponential size unfolding to represent n
  reachable markings

Extend McMillan's order: if sizes equal, compare by lexicographic
  order of Parikh vector

- Fixes the earlier example
- Still a partial order: other bad examples exist

Ideal to find an adequate total order

- One does exist for 1-safe nets
- Unclear how to find such a total order for bounded but not
- 1-safe nets

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