Concurrency Theory: Lecture 5, 28 August 2018
----------------------------------------------

Petri nets with inhibitor arcs
------------------------------
  - Destroys monotonicity
  - 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

Place and transition invariants:
-------------------------------

- Weighted sum is the same for all reachable markings
  - A marking that does not satisfy the invariant is unreachable
  - Non-negative invariants and bounded places.

- Transition invariants and loops in the marking graph

Petri Net Languages
-------------------

- A bounded net generates a regular language over T
  - Prefix closed, no final markings

- Can add a labelling function l: T -> Sigma
  - Petri net language is now a language over Sigma
  - Again bounded nets generate regular languages

- Converse?

  - From DFA/NFA to labelled net is trivial: for each edge
    s-a->s', add a transition labelled a between places s and s'

  - What if we interpret DFA/NFA as the language of an unlabelled
    net?  Not so simple!

    A single word language is trivially regular.  If L = {w}, then
    the DFA for L is a straight line and the corresponding net will
    have reachable markings with the same structure.  This amounts to
    asking a stronger question: whether there is a net whose
    reachable markings are isomorphic to a straight line DFA.

    There is a well-developed theory of "regions" that characterizes
    whena a given transition system can be isomorphic to the
    reachable markings of a net.

======================================================================