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EECS 20 Lecture 15 (February 21, 2001) Tom Henzinger

Simulation. EECS 20 Lecture 15 (February 21, 2001) Tom Henzinger. Deterministic Reactive System: for every input signal, there is exactly one output signal. Function:. DetSys : [ Time  Inputs ]  [ Time  Outputs ]. Nondeterministic Reactive System:

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EECS 20 Lecture 15 (February 21, 2001) Tom Henzinger

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  1. Simulation EECS 20 Lecture 15 (February 21, 2001) Tom Henzinger

  2. DeterministicReactive System: for every input signal, there is exactly one output signal. Function: DetSys : [ Time  Inputs ]  [ Time  Outputs ]

  3. Nondeterministic Reactive System: for every input signal, there is one or more output signals. Binary relation: NondetSys  [ Time  Inputs ]  [ Time  Outputs ] such that  x  [ Time  Inputs ],  y  [ Time  Outputs ], (x,y)  NondetSys Every pair (x,y)  NondetSys is called a behavior.

  4. S1 is a more detailed description of S2; S2 is an abstraction or property of S1. System S1 refines system S2 iff 1. Time [S1] = Time [S2] , 2. Inputs [S1] = Inputs [S2] , 3. Outputs [S1] = Outputs [S2] , 4. Behaviors [S1]  Behaviors [S2] .

  5. Systems S1 and S2 are equivalent iff 1. Time [S1] = Time [S2] , 2. Inputs [S1] = Inputs [S2] , 3. Outputs [S1] = Outputs [S2] , 4. Behaviors [S1] = Behaviors [S2] .

  6. Nondeterministiccausal discrete-time reactive systems can be implemented by nondeterministicstate machines.

  7. Nondeterministic State Machine Inputs Outputs States possibleInitialStates  States possibleUpdates: States  Inputs  P( States  Outputs ) \ Ø receptiveness (i.e., machine must be prepared to accept every input)

  8. Lossy Channel without Delay Nats0 Bins Nats0 Bins LCwD 0 1 1 0 0 0 1 1 … 0  1 0   1 1 … 0 / 0 0 / 1/ 1 1/ 

  9. Channel that never drops two in a row Nats0 Bins Nats0 Bins NotTwice 0 1 1 0 0 0 1 1 … 0  1 0  0  1 … 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 / 

  10. Typical Exam Questions A. Convert between three (possibly nondeterministic) system representations: 1.Input-output definition Deterministic case: define S: [Nats0Inputs]  [Nats0Outputs] such that for all x  [Nats0Inputs] and all y  Nats0, (S(x))(y) = … Nondeterministic case: define S  [Nats0Inputs]  [Nats0Outputs] such that for all x  [Nats0Inputs] and all y  [Nats0Outputs], (x,y)  S iff … 2.Transition diagram 3.Block diagram B. Equivalence, bisimulation, minimization of deterministic state machines

  11. Refinement, simulation, and minimization of nondeterministic state machines

  12. condition on behaviors Theorem : The nondeterministic state machines M1 refines the nondeterministic state machine M2 if there exists a simulation of M1 by M2 . relation between states

  13. Theorem : The nondeterministic state machines M1 refines the nondeterministic state machine M2 if there exists a simulation of M1 by M2 . Not symmetric ! Not “if-and-only-if” ! (We say that “M2 simulates M1”.)

  14. In the following, assume Inputs [M1] = Inputs [M2] Outputs [M1] = Outputs [M2]

  15. A binary relation S  States [M1]  States [M2] is a simulation of M1 by M2 iff 1.  p  possibleInitialStates [M1] ,  q  possibleInitialStates [M2], ( p, q )  S and 2.  p  States [M1] ,  q  States [M2] , if ( p, q )  S, then  x  Inputs ,  y  Outputs ,  p’  States [M1] , if ( p’, y )  possibleUpdates [M1] ( p, x ) then  q’  States [M2] , ( q’, y )  possibleUpdates [M2] ( q, x ) and ( p’, q’ )  S .

  16. S x / y p’ p q

  17. S x / y p’ p x / y q’ q

  18. S x / y p’ p x / y q’ q

  19. 0 / 0 0 / 1/ 1 1/  NotTwice refines Lossy Channel without Delay q 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 / 

  20. 0 / 0 0 / 1/ 1 1/  NotTwice is simulated by Lossy Channel without Delay q Simulation S = { (a, q), (b, q) } 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 / 

  21. 0 / 0 0 / 1/ 1 1/  Lossy Channel without Delay is not simulated by NotTwice q Not a simulation S’ = { (q, b) } 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 / 

  22. 0 / 0 0 / 1/ 1 1/  Lossy Channel without Delay is not simulated by NotTwice q Not a simulation S” = { (q, a), (q, b) } 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 / 

  23. Channel that drops every third zero Nats0 Bins Nats0 Bins ThirdZero 0 1 1 0 0 0 1 1 … 0 1 1 0  0 1 1 …

  24. Channel that drops every third zero Nats0 Bins Nats0 Bins ThirdZero 0 1 1 0 0 0 1 1 … 0 1 1 0  0 1 1 … State between time t-1 and time t: 3 third zero from now will be dropped 2 second zero from now will be dropped 1 next zero will be dropped

  25. Channel that drops every third zero 1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  26. ThirdZero refines NotTwice 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  27. ThirdZero is simulated by NotTwice Simulation S = { (3,b), 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  28. ThirdZero is simulated by NotTwice Simulation S = { (3,b), (2,b), 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  29. ThirdZero is simulated by NotTwice Simulation S = { (3,b), (2,b), (1,b), 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  30. ThirdZero is simulated by NotTwice Simulation S = { (3,b), (2,b), (1,b), (3,a) } 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  31. NotTwice is not simulated by ThirdZero 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 0 / 

  32. NotTwice is not simulated by ThirdZero 0 / 0 1 / 1 a b 0 / 0 0 /  1 / 1 1 /  1 / 1 1 / 1 1 / 1 0 / 0 0 / 0 3 1 2 X 0 /  0 / 

  33. Special cases of nondeterministic state machines • Deterministic • Output-deterministic (“almost deterministic”) For these machines, simulations can be found easily (provided they exist).

  34. A state machine is output-deterministic • iff • there is only one initial state, and • for every state and every input-output pair, there is only one successor state. Deterministic implies output-deterministic, but not vice versa. For example, LCwD and NotTwice are output-deterministic; ThirdZero is deterministic.

  35. Deterministic: for every input signal x, there is exactly one run of the state machine. Output-deterministic: for every behavior (x,y), there is exactly one run.

  36. If M2 is an output-deterministic state machine, then a simulation S of M1 by M2 can be found as follows: 1. If p  possibleInitialStates [M1] and possibleInitialStates [M2] = { q } , then (p,q)  S. 2. If (p,q)  S and (p’,y)  possibleUpdates [M1] (p,x) and possibleUpdates [M2] (q,x) = { q’ } , then (p’,q’)  S.

  37. If M2 is a deterministic state machine, then M1 is simulated by M2 iff M1 is equivalent to M2.

  38. If M2 is a deterministic state machine, then M1 is simulated by M2 iff M1 is equivalent to M2. “if and only if” M1 refines M2, and M2 refines M1

  39. If M2 is a nondeterministic state machine, then M1 is simulated by M2 implies M1 refines M2, but M2 may not refine M1. ( Recall M1 = ThirdZero and M2 = NotTwice .)

  40. If M2 is a nondeterministic state machine, then M1 is simulated by M2 implies M1 refines M2, but M1 refine M2 even if M1 is not simulated by M2.

  41.  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  42. Two behaviors: 00000…, 01000…  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 is equivalent to z  / 0  / 0  / 0 M2 x y  / 1 u  / 0 Same two behaviors: 00000…, 01000…

  43. Simulation S = { (a, x),  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 is simulated by z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  44. Simulation S = { (a, x), (b, y), (c, y),  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 is simulated by z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  45. Simulation S = { (a, x), (b, y), (c, y), (d, z), (e, u) }  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 is simulated by z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  46.  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 does not simulate z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  47.  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 does not simulate z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  48.  / 1 X  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 does not simulate z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

  49.  / 0 b d  / 0  / 0 M1 a  / 1  / 0 c e  / 0 does not simulate X  / 0 z  / 0  / 0  / 0 M2 x y  / 1 u  / 0

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