Concurrency Theory: Lecture 4, 23 August 2019
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omega-markings and coverability tree

- Generalize markings to allow components to have omega
  (infinite number)  of tokens.

  - omega > n for any finite integer n
  - firing rule is amended so that omega+1 =  omega-1 = omega

- Build a tree of reachable markings starting with Min as root

  - To each marking M, add as its children the new markings
    reachable in one step by firing a transition at M

  - If the new node M' has M'(p) > M"(p) for some ancestor M",
    set M'(p) = omega

  - If the new node M' = M" for some ancestor M", mark M' and do
    not expand the tree further below M'

Claim: The coverability tree of a net is always finite

Proof: The tree is finitely branching.  An infinite tree must
  have an infinite branch.  This branch must contain an infinite
  set of incomparable markings.  By Dickson's Lemma, this is
  impossible.

Theorem: M is coverable iff we can find a marking M' >= M in the
  coverability tree.

This follows from the following claim.

Claim: Let M be an omega marking in the coverability tree with
  omega-marked places p1,p2,...,pk.  Then given a simultaneous
  set of bounds n1,n2,...,nk, we can reach a marking M' from Min
  such that M' agrees with M on all places with a finite marking
  and for each pj with a omega marking, M'(pj) >= nj, where nj is
  the lower bound specified for pj.

Proof: There is a path from Min to M in the coverability tree.
Along this path, mark the positions where places in M are first
promoted to omega. Segments between marked positions can be
"pumped". Pumping a later segment could remove tokens from a
place that was promoted earlier. Hence, pump the segments from
last to first by appropriate amounts to ensure that all places
are boosted sufficiently high to account for loss of tokens in
the later part of the run.

Corollary: Net is bounded iff no reachable marking has omega in
coverability tree.

Extensions to the model:

- Arc weights: consider vector addition system model where
  transitions are arbitrary change vectors.

- Place capacities:

  - Can eliminate using complementary places

  - Contact-free elementary net systems are a special case

- Inhibitor arcs

  - Change the model, reachability becomes undecidable

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