Concurrency Theory: Lecture 4, 23 January 2018
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Coverability tree

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

- Simulate a two counter machine:

  - Counters c and d
  - Instructions l1,l2,...lk
  - Instructions c++, d++, c--, d--, if c==0 goto lj, if d==0 goto lj
  - Subtraction operation blocks if counter is 0

- Use one place for c, one for d

- Each instruction li corresponds to a transition ti

  - c++/d++, tj feeds a token into place c/d
  - c--/d--, tj consumes a token from place c/d
  - if c==0 goto lj
    - transition with inhibitor arc from c takes control to lj
    - transition with self loop from c takes control to l{i+1}

- Reachability of marking where l{k+1} is marked is undecidable

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