Concurrency Theory: Lecture 8, 09 February 2017
-----------------------------------------------

Unfolding of a 1-safe net is an occurrence net

- Each place has at most one incoming edge
- F* is acyclic
- If t and t' share an input place, t and t' are in conflict
- Conflict is "inherited" via F
- Any two elements in conflict have disjoint futures

Occurrence net describes "branching behaviour" of a net

- If we "throw away" the places in the occurrence net, we get an
  event structure

ES = (E, <=, #)

- E is a set of events
- <= is a partial order
- # is the conflict relation, inherited via <=:
      e # e' and e' <= e'' implies e # e"

- Concurrency relation
  - e co e' if not (e <= e' or e' <= e or e # e')
  
- Configuration C is a set of events, describes a computation

  - Set of events that have "happened till now"
  - C is downward closed: e in C and e' <= e implies e' in C as well
  - C is conflict free: only compatible events can happen together

- Configurations correspond to traces

- Configurations ordered by inclusion are isomorphic to traces
  ordered by trace extension

- Algebraic characterization

  - Prime configuration: c is prime if whenever c is included in
    the union of a  set of configurations C, c is included in
    some c' in C 

  - Prime configurations are those of the form downarrow(e), for each
    event e in E

  - Configurations ordered by inclusion form a prime algebraic
    coherent complete partial order

    - Prime configurations form an event structure

      - Events are the prime configurations
      - <= is given by inclusion of configurations
      - c # c' if c and c' do not have a common upper
        bound in the original event structure

    - The configuration space of this event structure is
      isomorphic to the one we started with

  - Hence these event structures are called "prime" event structures

- Difficult to generalize to non-safe nets:

  - Parallel switch:  t or t' can trigger t"

  - Configurations are not prime algebraic coherent cpo

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