Decidability Issues for Decentralized Controllability of Open Nets

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Decidability Issues for Decentralized Controllability of Open Nets

  1. 1. Karsten Wolf DecidabilityDecidability IssuesIssues forfor DecentralizedDecentralized ControllabilityControllability of Open Netsof Open Nets
  2. 2. read more: www.service-technology.org Controllability centralized decentralized autonomous
  3. 3. read more: www.service-technology.org Decentralized Controllability Adaptability
  4. 4. read more: www.service-technology.org Decentralized Controllability Adaptability
  5. 5. read more: www.service-technology.org Decentralized Controllability Adaptability
  6. 6. read more: www.service-technology.org Decentralized Controllability Adaptability Realizability
  7. 7. read more: www.service-technology.org Decentralized Controllability Adaptability Realizability
  8. 8. read more: www.service-technology.org So far: decidable for… centralized decentralized autonomous Finite state deadlock freedom weak termination Acyclic deadlock freedom (= weak termination) Finite state deadlock freedom
  9. 9. read more: www.service-technology.org 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
  10. 10. read more: www.service-technology.org 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
  11. 11. read more: www.service-technology.org 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
  12. 12. read more: www.service-technology.org 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
  13. 13. read more: www.service-technology.org 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
  14. 14. read more: www.service-technology.org 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
  15. 15. read more: www.service-technology.org 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, …

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