Concurrency Theory: Lecture 5, 30 January 2018
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Place invariants:
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- 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.


Petri Net Languages
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- 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.

    Using the theory of regions, it can be shown that the word
    "abbaa" cannot be the language of any unlabelled net.  Such a
    word is called "Petri net unsolvable".

    Recently (2015 and later) there has been some work on
    characterizing solvable and unsolvable words. However, it still
    seems to be open to characterize when a non-trivial regular
    language is "solvable".

- Petri nets can recognize some context-free languages

  - Producer-consumer language = { w | every prefix of w has more
      send actions than receive actions}

  - This language is a variant of a^n b^n

  - Can use labels to actually program a^n b^n

- Petri nets cannot recognize all context-free languages

  - The language of palindromes is not PN recognizable

  - Requires a combinatorial argument about number of reachable
    markings

- Petri nets can recognize some context-sensitive languages

  - Add a third component to producer-consumer: consumer passes a
    message to third component

  - Language is a variant of a^n b^n c^n, which is not
    context-free but is context-sensitive.
  
- Petri net languages are orthogonal to normal Chomsky hierarchy
  of regular, CFL, CSL, ...

Alphabets with independence
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- Sequential languages do not accurately capture concurrent
  behaviour

- Cannot distinguish independent a,b from ab+ba (both a and b are
  enabled but sequentialized by a shared place connected by
  self-loop)

- Augment alphabet with an independence relation

Mazurkiewicz trace theory

- (Sigma,I)

- I is an independence relation on Sigma: binary relation that is
  symmetric and irreflexive

- Two adjacent independent letters can be permuted without
  affecting the computation

  - If (a,b) in I, uabv is equivalent to ubav (write uabv <-> ubav)

- In general, two words are trace-equivalent if they can be
  obtained from one another through a sequence of swaps.

  (a,b) in I:   aabb is equivalent to bbaa

  aabb <-> abab <-> baab <-> baba <-> bbaa

  Write aabb <=> bbaa

- A trace is an equivalence class of equivalent words