AWPN 2010

1. Karsten Wolf
DecidabilityDecidability IssuesIssues forfor
DecentralizedDecentralized ControllabilityControllability of Open Netsof Open Nets

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Controllability
centralized
decentralized
autonomous

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Decentralized Controllability
Adaptability

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Decentralized Controllability
Adaptability

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Decentralized Controllability
Adaptability

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Decentralized Controllability
Adaptability Realizability

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Decentralized Controllability
Adaptability Realizability

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So far: decidable for…
centralized
decentralized
autonomous
Finite state
deadlock freedom
weak termination
Acyclic
deadlock freedom
(= weak termination)
Finite state
deadlock freedom

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Decentralized (acyclic)
-Start with centralized most permissive partner, unrolled to tree
-Remove nodes (subtrees) where actions of different ports do not
commute
?a ?b
!a !b
!a disables !b
!a !b
Does not work on graphs

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Post‘s Correspondance Problem
Given: finitely many pairs of words
(a,aba) , (ab,bb), (baa,aa).
Problem: Is there a (nonempty) sequence of pairs such that left
words concatenate to the same result as right words?
yes, 1 3 2 3:
a baa ab baa = aba aa bb aa.
Problem undecidable. Proof by reduction from halting problem of
TM – difference between left and right corresponds to TM
configuration

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Reduce Decentralized
Controllability to PCP
Idea: Controller corresponds to solution of PCP
(a,aba) , (ab,bb), (baa,aa).
Left: 1 a 3 baa 2 ab 3 baa #
Right: 1 aba 3 aa 2 bb 3 aa #
Service
-Checks whether input is solution
-Checks whether input is valid
-Goes to deadlock if anything goes wrong
Impossible at the same time
decide internally, check one

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Service (Sketch)
b b
1,2,3
1,2,3
# #
1 1
2 2
3 3
(a,aba) , (ab,bb), (baa,aa)
a aba
ab bb baa aa
# #
a a
c c
Weak terminating controller must be both
valid and solution

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What about deadlock freedom?
Can avoid deadlock by sending infinite sequence
What about centralized
controllability?
Centralized controller can detect internal decision of
service according to different progress at left and right ports
after detection proceed with either none-solution or invalid
sequence

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What about the others?
[Tripakis]
Control problem = language
PCP is encoded in language to be enforced
[Bontemps & Schobbens]
-Internal decision: send either left or right sequence to A,
indices to B
-A may talk to B
-In the end, A and B must return ‚left‘ or ‚right‘
(claim: impossible if and only if PCP has solution)
implicitly requires unbounded memory
In our setting: „may talk to each other“ = centralized setting,
decidable

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Conclusion
-Decentralized controllability for weak termination is
undecidable
-This result apparently not covered by previous approaches
-Realizability, adaptability most likely undecidable
-Decentralized controllability for deadlock freedom still open
-Centralized and autonomous settings provide useful lower
and upper bounds
-Floor open for approximations, heuristics, subclasses, …