decidability issues for decentralized controllability of open nets

Karsten Wolf

DecidabilityDecidability IssuesIssues forforDecentralizedDecentralized ControllabilityControllability of Open Netsof Open Nets

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Controllability

centralizeddecentralized

autonomous

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Decentralized ControllabilityAdaptability

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Decentralized ControllabilityAdaptability

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Decentralized ControllabilityAdaptability

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Decentralized ControllabilityAdaptability Realizability

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Decentralized ControllabilityAdaptability Realizability

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So far: decidable for…

centralizeddecentralized

autonomous

Finite statedeadlock freedomweak termination

Acyclicdeadlock freedom(= weak termination)

Finite statedeadlock 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 timedecide internally, check one

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Service (Sketch)

b b

1,2,3

1,2,3

# #

1 12 2 3 3

(a,aba) , (ab,bb), (baa,aa)

a abaab bb baa aa

# #a a

c c

Weak terminating controller must be bothvalid 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 ofservice 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 = languagePCP 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, …