Concurrency Theory: Lecture 6, 18 January 2017
----------------------------------------------

Traces
------

Independence alphabet:

(Sigma, I): I irreflexive and symmetric

Dependence relation: complement of I, reflexive and symmetric

One step equivalence: uabv <-> ubav for (a,b) in I

Trace equivalence: transitive closure of one step equivalence

  w <=> w' if w = w1 <-> w2 <-> ... <-> wk = w'

Example:

  (a,b) in I:   aabb <=> bbaa  because

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

- A trace is an equivalence class with respect to <=>
  - [w] is the trace containing w: [w] = { w' | w <=> w' }

- L is trace consistent if w in L implies [w] subset of L
  - Every trace is either completely in L or completely out of L 

- A trace consistent language L is called a "trace language"

- A regular trace language is a trace language who underlying
  word language is regular in the usual sense

Traces as labelled partial orders:

- Better way of thinking of a trace

- (E,<=,l) where

  - (E,<=) is a partial order, E is a finite set of events

  - l: E -> Sigma is the labelling function
    - (l(e),l(e')) in D iff e <= e' or e' <= e
    - e immediately less than e' => (l(e),l(e')) in D
    
- The words in [w] are all the valid linearizations of the
  partial order corresponding to [w]

Trace prefix:

- For traces [w], [w'], [w] <= [w'] if [w'] "extends" [w].

  - For some u in [w] and u' in [w'], u is a prefix of u'

  - We call [w] a trace prefix of [w']

  - Example: With (a,b) in I, [b] <= [ab] because b in [b]
    and ba in [ab] such that b is a prefix of ba

- In the partial order representation, [w] <= [w'] if the partial
  order [w'] extends that of [w].  In other words [w] is
  downward closed subset of [w'].

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

- For each firing sequence wt extending wt, N_w is included in
  N_wt

- N_w is identical to N_w' for each w'<=>w : each concurrent run
  generates a canonical net

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

- Now [w] <= [w'] iff the N_w is a prefix of N_{w'}

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