1. EECS 20 Lecture 7 (January 31, 2001) Tom Henzinger Reactive Systems

2. Transducive or Combinational System Input ValueOutput Value transduciveSystem : Values  Values

3. Reactive or Sequential System Input SignalOutput Signal reactiveSystem : [ Time  Values ]  [ Time  Values ]

4. Reactive Systems 1Memory-free systems 2Delays 3Causality 4Finite-memory systems 5Infinite-memory systems

5. A reactive system F : [ Time  Values ]  [ Time  Values ] is memory-free iff there exists a transducive system f : Values  Values such that  x  [ Time  Values ],  y  Time, ( F (x) ) (y) = f ( x (y) ).

6. Memory-free Reactive Systems Normalize Trunc Quantize Negate The identity system Id Every constant system Const Every composition of memory-free systems Every block-diagram composition of memory-free systems Every combinational circuit

7. Id : [ Time  Values ]  [ Time  Values ] such that  x  [ Time  Values ], Id (x) = x. For every constant const  Values, Const : [ Time  Values ]  [ Time  Values ] such that  x  [ Time  Values ],  y  Time, ( Const (x) ) (y) = const. The Identity System The Constant Systems

8. FG Composition of Memory-free Systems fg If F and G are memory-free,

9. FG Composition of Memory-free Systems fg If F and G are memory-free, then G  F is memory-free. G  F g  f

10. Negate And Block-Diagram Composition of Memory-free Systems   Bools

11. Negate And Block-Diagram Composition of Memory-free Systems   again memory-free 1 2

12. The Delay System Delay: [ Time  Values ]  [ Time  Values ] such that  x  [ Time  Values ],  y  Time, ( Delay (x) ) (y) = ? if y < 1 x (y-1) if y  1 {

13. The Delay System Delay c : [ Time  Values ]  [ Time  Values ] such that  x  [ Time  Values ],  y  Time, ( Delay (x) ) (y) = c if y < 1 x (y-1) if y  1 {

14. Delay 0 (sin) : Reals +  Reals such that  y  [0,1), Delay (sin) (y) = 0 and  y  [1,  ), Delay (sin) (y) = sin (y-1). Delay 11 (id) : Reals +  Reals such that  y  [0,1), Delay (id) (y) = 11 and  y  [1,  ), Delay (id) (y) = y - 1. Continuous-Time Delay

15. Delay 13 (id) : Nats 0  Nats 0 such that Delay (id) (0) = 13 and  y  Nats, Delay (id) (y) = y - 1. Delay true (alt) : Nats 0  Bools such that Delay (alt) = { (0, true), (1, false), (2, true), (3, false), … }. Discrete-Time Delay

16. Discrete-time delay over finite set of values : finite memory Continuous-time delay, or infinite set of values: infinite memory

17. Block Diagram with Delay   DtDt DfDf 2 1 1 2 Bools

18. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … DtDt DfDf

19. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … t t f DtDt DfDf

20. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … t, f, t t DtDt DfDf

21. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … t, f f, t f t t DtDt DfDf

22. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … t, f, f, t, f t DtDt DfDf

23. Block Diagram with Delay   2 1 1 2 t, f, t, f, … t, t, t, t, … t, f, t, f, … f, t, f, t, … DtDt DfDf

24. Block Diagram with Cycle  DfDf Bools

25. Block Diagram with Cycle  t, f, t, … DfDf

26. Block Diagram with Cycle  t, f, t, … t fDfDf

27. Block Diagram with Cycle  t, f, t, … t f,DfDf

28. Block Diagram with Cycle  t, f, t, … f, t, t, …DfDf

29. Legal Transducive Block Diagrams -all components are transducive systems -no cycles e.g., combinational circuits

30. Legal Reactive Block Diagrams -all components are memory-free or delay systems -every cycle contains at least one delay e.g., sequential circuits

31. Discrete-Time Moving-Average DiscMovAvg: [ Nats 0  Reals ]  [ Nats 0  Reals ] such that  x  [ Nats 0  Reals ],  y  Nats 0, ( DiscMovAvg (x) ) (y) = ? if y < 2 1/3 · ( x (y) + x (y-1) + x (y-2) ) if y  2 {

32. Discrete-Time Moving-Average DiscMovAvg c : [ Nats 0  Reals ]  [ Nats 0  Reals ] such that  x  [ Nats 0  Reals ],  y  Nats 0, ( DiscMovAvg (x) ) (y) = c if y < 2 1/3 · ( x (y) + x (y-1) + x (y-2) ) if y  2 {

33. Continuous-Time Moving-Average ContMovAvg c : [ Reals +  Reals ]  [ Reals +  Reals ] such that  x  [ Reals +  Reals ],  y  Reals +, ( ContMovAvg (x) ) (y) = c if y < 3 1/3 · x (t) dt if y  3 { y y-3

34. Continuous-Time Moving-Average ContMovAvg c : [ Reals +  Reals ]  [ Reals +  Reals ] such that  x  [ Reals +  Reals ],  y  Reals +, ( ContMovAvg (x) ) (y) = c if y < 3 1/3 · x (t) dt if y  3 { y y-3 Integral is a quantifier that binds the variable t.

35. Discrete-Time Moving-Average DiscMovAvg : [ Nats 0  Reals ]  [ Nats 0  Reals ] such that  x  [ Nats 0  Reals ],  y  Nats 0, ( DiscMovAvg (x) ) (y) = c if y < 2 1/3 ·  x (t) if y  2 { y t = y-2 Sum is a quantifier that binds the variable t.

36. A Noncausal Reactive System Predict: [ Time  Bins ]  [ Time  Bins ] such that  x  [ Time  Bins ],  y  Time, ( Predict (x) ) (y) = 1 if  z  Time, x (z) = 1 0 if  z  Time, x (z) = 0 {

37. A reactive system F : [ Time  Values ]  [ Time  Values ] is causal ( or implementable ) iff  x, y  [ Time  Values ],  z  Time, if (  t  Time, t  z  x (t) = y (t) ) then ( F (x) ) (z) = ( F (y) ) (z).