1. 1 Undecidable Problems of Decentralized Observation and Control Stavros Tripakis VERIMAG (based on [Puri,Tripakis,Varaiya-SCODES’01], [Tripakis-CDC’01], [Tripakis-WODES’02])

2. 2 Plan of talk Decentralized observation: –Observation architecture and problem definition. –Necessary-sufficient conditions and examples. –Undecidability. –Links to formal language theory. –Open questions. Decentralized control: –A problem from supervisory control theory. –Reducing observation to control. Example: the Alternating Bit Protocol.

3. 3 Observation architecture Plant Observation 1 Decision function Does behavior meet a certain property ? Observation n … observation point 1observation point n

4. 4 Observation architecture Plant: regular lang. L over  plant’s behavior:   L  1 = Proj( ,  1 ) Decision function   K ?  n = Proj( ,  n ) …

5. 5 Definition Given regular languages K  L   * and subalphabets  i   *, i=1,…,n, K is called observable w.r.t. L,  i, if there exists a total function such that for all   L, f   1     n   ** f( P 1(  ), …, P n(  ) ) = 1 iff   K

6. 6 Example of non-observability L={a b,b a} K={a b} Decision function The decision function receives the same input in both cases!  1 ={a}  2 ={b} ab

7. 7 Necessary and sufficient conditions K is observable w.r.t. L,  i, iff  K.  ’  L-K.  i=1,…,n. Pi(  )  Pi(  ’)

8. 8 Example of observability with ``unbounded memory’’ L= (a b)* (1 + b b) K= (a b)* The decision function must count !  1 ={a}  2 ={b}

9. 9 Decidability of centralized observability (n=1) Checking observability is decidable in the centralized case (n=1). In this case nec./suf. condition becomes:  K.  ’  L-K. P1(  )  P1(  ’)  P1(K)  P1(L-K) = 

10. 10 Note Decentralized observability is not equivalent to: This condition is much stronger.  i=1,…,n. Pi(K)  Pi(L-K) = 

11. 11 Illustration of stronger condition L 11 * 22 * K L-K K is observable w.r.t. L,  1,  2 projections x y z

12. 12 Illustration of observability L 11 * 22 * K L-K K is unobservable w.r.t. L,  1,  2 projections

13. 13 Undecidability of decentralized observability (n  2) Checking observability is undecidable: –For two (or more) observers (n=2). –For three (or more) observers, even if K, L are prefix-closed. –Question open for n=2 and K, L prefix- closed.

14. 14 Undecidability proof By reduction of Post’s Correspondence Problem (PCP): –For a given instance of PCP, build K, L,  i, such that K is jointly observable w.r.t. L,  i iff the PCP instance has no solution. the PCP alphabet

15. 15 Links to theory of rational sets (thanks to anonymous reviewer) Given regular language L   * and  i   *, i=1,…,n, define the set which is a rational subset of the monoid Then, observability is equivalent to checking that

16. 16 Links to theory of rational sets Another way of getting at the undecidability result: –Given rational sets A,B, checking if A  B={} is undecidable –For every rational subset A of, there exists regular language L(A)   *, such that A  B={}iff iff

17. 17 Links to theory of traces (working on it) K observable w.r.t.  *,  i  K is a trace language w.r.t. the dependency where

18. 18 Open questions The question n=2 and prefix-closed. Special case of the problem where L=  *. –When  i are a partition of , decidable and equivalent to finite-memory observability (c.f. Zielonka’s theorem). Same problems with finite-memory observers (finite-state automata). –Note that they are not asynchronous automata (they do not synchronize on common events).

19. 19 Decentralized control A problem from supervisory control theory (Ramadge-Wonham et al). Variants of the problem known to be decidable (e.g., [Cieslak et al’88], [Rudie-Wonham’92]). This variant is probably the most interesting: –It captures protocol synthesis. –If the problem was decidable, we could automatically synthesize protocols such as the Alternating Bit Protocol !

20. 20 Centralized control architecture Regular language L   * supervisor

21. 21 Decentralized control architecture (without communication)

22. 22 Property model Responsiveness property over  : a formula of the form A word w over  satisfies a  b if for every a in w there is a b after the a. Specification: a set  of responsiveness properties. A set of words L satisfies  if every word in L satisfies every property in . a  ba, b  

23. 23 Control problem Given: –A plant G over , and subalphabets  Oi,  Ci of  –A specification over  Find: –Supervisors Ci observing  Oi and controlling  Ci Such that: –The language of the closed-loop system satisfies the specification.

24. 24 Control problem (alternative formulation) Given: –A plant G over , and subalphabets  Oi,  Ci of  –A specification over  Find: –Supervisors Ci observing  Oi and controlling  Ci Such that: –The language of the closed-loop system satisfies the specification.

25. 25 Reducing observation to control (for simplicity, n=2) Suppose we want to check whether K is observable w.r.t. L,  i ’, for i=1,…,n. We will reduce this to checking existence of n supervisors: –Each supervisor will initially observe a behavior in L. –Then supervisors 2 to n will “send” their observations to supervisor 1 (how?). –Finally, supervisor 1 will have to decide whether the original behavior was in K or in L-K.

26. 26 Reducing observation to control (for simplicity, n=2) Let Ti be alphabets of “fresh” letters, copies of  i ’, for i=2,…,n. The plant will be:

27. 27 Reducing observation to control (for simplicity, n=2) Let Ti be alphabets of “fresh” letters, copies of  i ’, for i=2,…,n. The plant will be: Everybody observes L (a and b are unobservable)

28. 28 Reducing observation to control (for simplicity, n=2) Let Ti be alphabets of “fresh” letters, copies of  i ’, for i=2,…,n. The plant will be: Supervisor 2 transmits its observation to supervisor 1

29. 29 Reducing observation to control (for simplicity, n=2) Let Ti be alphabets of “fresh” letters, copies of  i ’, for i=2,…,n. The plant will be: Supervisor n transmits its observation to supervisor 1

30. 30 Reducing observation to control (for simplicity, n=2) Let Ti be alphabets of “fresh” letters, copies of  i ’, for i=2,…,n. The plant will be: Supervisor 1 must decide (a’ and b’ are controllable by sup.1)

31. 31 Reducing observation to control (for simplicity, n=2) The specification will be: That is: if behavior was in K, enable a’, otherwise enable b’.

32. 32 Undecidability of control problem Supervisors exist iff K is observable: –If K not observable, supervisors cannot distinguish between an observation in K and an observation in L-K. –If K observable, supervisor 1 gathers the observations of everybody, then applies the f function (note that supervisors are infinite-state). Control problem undecidable for n  2.

33. 33 Example: how to synthesize a reliable transmission protocol over an unreliable channel ? Sending client Receiving client Backward channel Forward channel ? Channels are lossy but FIFO. O,1 donesend deliver

34. 34 The problem is a decentralized control synthesis problem ? Sending client Receiving client Backward channel Forward channel ? O f,1 f O f ’,1 f ’ O b,1 b donesend deliver Plant supervisors observable events controllable events O b ’,1 b ’

35. 35 Other results Decentralized diagnosability is also undecidable. Adding communication with unbounded delays (lossless, FIFO) does not help. Bounded-delay communication helps (but the details have to be worked out). Hierarchy of control problems with communication:

36. 36 Thanks to... Raja Sengupta Anuj Puri Pravin Varaiya David de Frutos Stephane Lafortune Albert Benveniste Karen Rudie John Thistle Anonymous reviewer of SCODES’01 Jean Berstel …

37. 37 Merci !