Concurrency Theory: Lecture 7, 08 February 2018
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Constructing a canonical trace from a firing sequence:

- By induction build a net N_w = (P_w,T_w,F_w) and a marking M_w
  for each word w

- Let <=> denote trace equivalence. N_w is identical to N_w' for
  each w'<=> w: each concurrent run generates a canonical net

- Causal net:

  - Each place has at most one incoming and one outgoing edge
  - F* is a acyclic (partial order)
  - Transitions that are not ordered are concurrent

- If we "throw away" the places in N_w, we get a canonical
  partial order representation of the trace [w]

  - Let <= denote trace prefix. [w] <= [w'] iff the N_w is a
    subset, componentwise, of N_{w'}

Given two firing sequences uv and uw, the nets N_uv and N_uw will
have identical prefixes for N_u

Take the union of N_w for all firing sequences w

- Like the computation tree of a sequential system

- 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
    (equivalently, no element is in conflict with itself)

- 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


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