These slides contain my final presentation for my master's thesis, held in May 2013.

1. Strong Distributability Criteria
for Petri Nets
Final Talk
Stephan Mennicke, May 17, 2013
Institut für Programmierung
und Reaktive Systeme

2. Setting: Distributed Systems
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3. Setting: Distributed Systems
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4. Setting: Distributed Systems
Concurrency
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5. Setting: Distributed Systems
Concurrency
Asynchrony
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6. Setting: Distributed Systems
(νz)(x z | z(w).w ) | x(k).k v
Concurrency
Asynchrony
Mobility
e. g., by process calculi
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7. Setting: Distributed Systems
Concurrency
Asynchrony
here: Petri nets
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8. Setting: Distributed Systems
Concurrency
Asynchrony
here: Petri nets
Is this system
distributed?
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9. When are Systems Distributed?
It depends . . .
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10. When are Systems Distributed?
Symmetrically asynchronous systems . . .
. . .
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11. When are Systems Distributed?
Symmetrically asynchronous systems . . .
. . .
τ τ
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12. When are Systems Distributed?
Symmetrically asynchronous systems . . .
. . .
τ τ
. . . are neutral to asynchrony assumptions.
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13. Remember the Initial Question?
Concurrency
Asynchrony
here: Petri nets
Is this system
distributed?
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14. Remember the Initial Question?
Concurrency
Asynchrony
here: Petri nets
Is this system
distributed?
No!
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15. No! But Why?
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16. Symmetric Asynchrony
?
≈
τ τ
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17. Symmetric Asynchrony
≈
τ τ
Symmetrically asynchronous net systems are all
partially reachable N-free net systems.
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18. We can work it out
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19. We can work it out
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20. We can work it out
Distributed: No!
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21. We can work it out
Distributed: No!
Distributed: Yes?
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22. We can work it out
Distributed: No!
Distributed: Yes?
Distributable? YES!
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23. Specifications
“unlabeled”
τ-free
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24. Specifications
x y z
a b c
“unlabeled”
τ-free
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25. Symmetric Distributability
Definition
A specification N is symmetrically distributable iff
there is a symmetrically asynchronous plain-labeled
net system N such that N ≈R N .
1. Which systems are distributable? How?
2. Which systems are not? Why?
Specifications are M-free.
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26. Partially and Fully Reachable N
x
t
u
Result
A partially and fully reachable N is not symmetrically
distributable.
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27. An Implementation Idea
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28. Implementation Idea SYM
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29. Implementation SYM
P1 := P
Pn+1 := Pn ∪ {pt,n+1 | t ∈ O(Nn) ∧ |•t| > 1}
T1 := T
Tn+1 := Tn ∪ {tt,u,n+1 | t ∈ O(Nn) ∧ |•t| > 1 ∧ u ∈ (•t)•}
F1 := F
Fn+1 := {(p, t) ∈ Fn | t ∈ O(Nn) ∨ |•t| 1} ∪
{(p, tt,t,n+1) | t ∈ O(Nn) ∧ |•t| > 1 ∧ (p, t) ∈ Fn} ∪
{(pt,n+1, t) | t ∈ O(Nn) ∧ |•t| > 1} ∪
{(pt,n+1, tt,u,n+1) | t ∈ O(Nn) ∧ |•t| > 1 ∧ u = t} ∪
{(p, tt,u,n+1) | t ∈ O(Nn) ∧ |•t| > 1 ∧ u = t ∧ (p, u) ∈ Fn} ∪
(Fn ∩ (Tn × Pn)) ∪
{(tt,t,n+1, pt,n+1) | t ∈ O(Nn) ∧ |•t| > 1} ∪
{(tt,u,n+1, p) | t ∈ O(Nn) ∧ |•t| > 1 ∧ u = t ∧ (u, p) ∈ Fn} ∪
{(tt,u,n+1, p) | t ∈ O(Nn) ∧ |•t| > 1 ∧ p ∈ •t •u}
l1 := l
ln+1(t) :=



l(t) if t ∈ T1
lk(t) if t ∈ Tk and k n
lk(u) if t = tt,u,n+1 and u = t
τ otherwise
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30. Results on SYM
1. Construction is finite.
2. τ-steps do not obstruct potential behavior.
3. Construction is not plain-labeled.
4. Construction respects weak step bisimulation.
5. Construction also works for partially and fully
reachable Ns.
6. SYM(N) is weak symmetrically distributed.
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31. Weak Symmetric Distributability
x τ
a
a
b
Result
Weak symmetric distributability iff asymmetric
distributability.
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32. Tackling Symmetric Distr.
Impossible to implement: partially and fully
reachable Ns.
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33. Tackling Symmetric Distr.
Impossible to implement: partially and fully
reachable Ns.
Remaining cases
1. Neither partially nor fully
2. fully but not partially
3. partially but not fully
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34. Tackling Symmetric Distr.
Impossible to implement: partially and fully
reachable Ns.
Remaining cases
1. Neither partially nor fully
2. fully but not partially
3. partially but not fully
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35. Fully but not Partially
Nfull(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •t ⊆ M ⇒ •u ⊆ M}.
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36. Fully but not Partially
Nfull(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •t ⊆ M ⇒ •u ⊆ M}.
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37. Fully but not Partially
Nfull(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •t ⊆ M ⇒ •u ⊆ M}.
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38. Fully but not Partially
Nfull(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •t ⊆ M ⇒ •u ⊆ M}.
Cool Implemenation Name
Symmetric Extended Free Choice (EFCSYM)
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39. EFCSYM
Algorithmic Idea
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40. EFCSYM
Algorithmic Idea
Iterate over all pairs (t, u) ∈ T × T
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41. EFCSYM
Algorithmic Idea
Iterate over all pairs (t, u) ∈ T × T
If (t, u) ∈ Nfull(N), make preplaces dependent
(EFC-construction)
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42. EFCSYM
Algorithmic Idea
Iterate over all pairs (t, u) ∈ T × T
If (t, u) ∈ Nfull(N), make preplaces dependent
(EFC-construction)
Result
N ≈B EFCSYM(N)
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43. Why is EFCSYM correct? I
t u
Lemma
Let (t, u) ∈ Nfull(N) and there is a reachable marking
M such that •
u ⊆ M. If (u, t) ∈ Nfull(N), then :-(
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44. Why is EFCSYM correct? II
t u v
Lemma
If (t, u), (u, v) ∈ Nfull(N) and t = v, then
(t, v) ∈ Nfull(N).
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45. Why is EFCSYM correct? III
t u v
Lemma
If (t, u), (v, u) ∈ Nfull(N), t = v, but
(t, v), (v, t) ∈ Nfull(N), then :-(
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46. Why is EFCSYM correct? IV
t uv
Lemma
If (t, u), (t, v) ∈ Nfull(N), u = v, but
(u, v), (v, u) ∈ Nfull(N), then :-(
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47. Nfull(EFCSYM(N)) is a Beauty!
Nfull(EFCSYM(N)) := {(t, u), (u, t) | (t, u) ∈ Nfull(N)}
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48. Nfull(EFCSYM(N)) is a Beauty!
Nfull(EFCSYM(N)) := {(t, u), (u, t) | (t, u) ∈ Nfull(N)}
It is sort of transitive
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49. Nfull(EFCSYM(N)) is a Beauty!
Nfull(EFCSYM(N)) := {(t, u), (u, t) | (t, u) ∈ Nfull(N)}
It is sort of transitive
It is symmetrical
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50. Nfull(EFCSYM(N)) is a Beauty!
Nfull(EFCSYM(N)) := {(t, u), (u, t) | (t, u) ∈ Nfull(N)}
It is sort of transitive
It is symmetrical
It is almost an equivalence relation
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51. Nfull(EFCSYM(N)) is a Beauty!
Nfull(EFCSYM(N)) := {(t, u), (u, t) | (t, u) ∈ Nfull(N)}
It is sort of transitive
It is symmetrical
It is almost an equivalence relation
It has at least as many elements as Nfull(N)
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52. SYMfull
1. Apply EFCSYM(N)
2. Make Nfull(EFCSYM(N)) an equivalence relation
3. Interpret an equivalence class as a cluster
4. Apply FC to all clusters
τ
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53. Results on SYMfull
N ≈B SYMfull(N)
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54. Results on SYMfull
N ≈B SYMfull(N)
Nfull(SYMfull(N)) = ∅
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55. Results on SYMfull
N ≈B SYMfull(N)
Nfull(SYMfull(N)) = ∅
Result
If N is an M-free specification that has no partially and
fully reachable Ns as well as no partially but not fully
reachable Ns, then N is symmetrically distributable.
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56. Partially but not Fully I
Npart(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •u ⊆ M ⇒ •t ⊆ M}.
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57. Partially but not Fully I
Npart(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •u ⊆ M ⇒ •t ⊆ M}.
Lemma
If (t, u) ∈ Npart(N) and u may be enabled, then
(u, t) ∈ Npart(N).
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58. Partially but not Fully I
Npart(N) := {(t, u) ∈ T × T | •t ∩ •u = ∅ ∧ t = u ∧ |•u| > 1∧
∃M ∈ [N : •t ⊆ M∧
∀M ∈ [N : •u ⊆ M ⇒ •t ⊆ M}.
Lemma
If (t, u) ∈ Npart(N) and u may be enabled, then
(u, t) ∈ Npart(N).
Lemma
If (t, u) ∈ Npart(N) and (u, t) ∈ Npart(N), then u is
dead.
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59. Partially but not Full II
at bu at bu
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60. Making Npart(N) symmetric
Remove non-symmetric partially but not fully
reachable Ns
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61. Making Npart(N) symmetric
Remove non-symmetric partially but not fully
reachable Ns
Remaining and still Open Question
What about the symmetric partially but not fully
reachable Ns?
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62. Examples
at bu a b
Idea
Divide and Conquer!
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63. x y
a b
x y
a b

64. x y
a b
x y
a b

65. x y
a b

66. x y
a b

67. Promising Idea
Partial Unfoldings
x y
a b
x y
a b
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68. x y
a b
z

69. x y
a b
z

70. x y
a b
z

71. Conclusions
We investigated three forms of symmetric
distributability
Free Choice Distributability (not in this talk, closed)
Weak Symmetric Distributability (closed)
Strong Symmetric Distributability (open)
Implementation algorithms are either standard or
generalizations of the standard ones
We identified two critical structures
a partially and fully reachable N (proven)
the evil W (sketched)
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72. Further things we have done
We
motivated a framework of distributability for net
based systems
instantiated distributability notions from the
literature
motivated another benefit of our research
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73. Future Work
Close the gap in symmetric distributability (next
few months, paper on symmetric distributability)
Complete the characterization of the W
Is there any more?
What about the non-plain-labeled case?
What about causality-respecting implementations?
Applications in the field of asynchronous circuits
(paper on free choice distributability)
Find an appropriate gap-filler for our
hierarchy/hierarchies of distributability
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74. x y
a b
z
Thank you for your
attention!