Concurrency Theory: Lecture 22, 19 April 2018
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Well-structured transition systems
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Abstracts the essential properties that make some problems
decidable for Petri nets and lossy channel systems.

1. Well quasi ordering (wqo) on states
2. The transition relation is compatible with the wqo.

   If s --> s' and s <= s1, then there exists s'1, s1-->s'1 and
   s' <= s'1

Two main techniques:

Set saturation:

Backward reachability (a la lossy channel systems) solves
"coverability" --- can we reach q' >= q?

Tree saturation:

Computing the forward tree (a la Karp-Milller covering tree for
Petri nets) solves "termination" --- is an infinite execution
possible?

Need some additional properties of wqo and transition relation:

1. wqo is effectively computable (given q and q', can check if
   q <= q')

2. Effective pred-basis

   - Given a state q, we compute its upward closure Up(q).

   - We then compute the set of states going backwards one step
     Pred(Up(q)).

   - We then look at the upward closure of this set,
     Up(Pred(Up(q)))

   - Since we have a wqo, this upward closed set has a finite set 
     of minimal elements (all minimal elements are incomparable
     and any set of incomparable elements in a wqo is finite)

   - We need to be able to effectively compute this finite set

See "Well-structured transition systems everywhere!" by Finkel
and Schnoebelen, Sections 1-4.

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