1. EECS 20 Lecture 4 (January 24, 2001) Tom Henzinger Block Diagrams

2. 1 Systems are functions 2 Signals are functions

3. Systems as Functions InputOutput Domain: set of possible inputs Range: set of possible outputs Graph: set of pairs ( input, output )

4. Factorial System ! Nats

5. Factorial System ! Nats 36

6. Factorial System ! Nats 424

7. Factorial System ! Nats 424 Domain: Nats Range: Nats Graph: { ( 1, 1 ), ( 2, 2 ), ( 3, 6), ( 4, 24 ), … }

8. Inverter System Bools 

9. Inverter System Bools  truefalse

10. Inverter System Bools  falsetrue

11. Inverter System Bools  falsetrue Domain: Bools Range: Bools Graph: { ( true, false ), ( false, true ) }

12. Composition of Systems Bools   true false

13. Composition of Systems Bools   false true

14. This is again a system ! Bools  

15. The Identity System Bools id Domain: Bools Range: Bools Graph: { ( x, y )  Bools 2 | x = y }

16. System Composition is Function Composition    = id because domain (    ) = domain (id) range (    ) = range (id)  x  Bools, (   x ) = id (x)

17. And System Bools 

18. And System Bools 2 Bools 

19. And System Bools  (true, false)false Bools 2

20. And System Bools 

21. And System Bools  true false

22. Exponentiation System Nats exp Nats 2 3 ? - graph (exp) = { ( (x,y), z )  Nats 2  Nats | z = x y }

23. Exponentiation System Nats exp Nats 2 3 8 2 1 graph (exp) = { ( (x,y), z )  Nats 2  Nats | z = x y }

24. Nats exp Nats 2 3 9 1 2 Exponentiation System graph (exp) = { ( (x,y), z )  Nats 2  Nats | z = x y }

25.    Bools Block Diagram This cannot be written easily using .

26.    Bools Block Diagram true false

27.    Bools Block Diagram true false true false

28.    Bools Block Diagram true false true false

29.    Bools Block Diagram true false true false

30.    Bools Or System  domain (  ) = Bools 2 range (  ) = Bools graph (  ) = { ( (x,y), z )  Bools 2  Bools | z = x  y }

31. Bools Or System  domain (  ) = Bools 2 range (  ) = Bools graph (  ) = { ( (x,y), z )  Bools 2  Bools | z = x  y } -

32. Block Diagrams with Forks   

33.    true false

34. Block Diagrams with Forks    true false f

35. Block Diagrams with Forks    true false f f t

36. Block Diagrams with Forks    true false f f t true

37. Joins are illegal   

38.    true false

39. Joins are illegal    true false f t ?

40. Multiple Outputs Nats Nats 0 f f: Nats 2  Nats 0 2 such that  x,y  Nats, f (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y }

41. Division System Nats divide divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } 1 2 1 2 quotient remainder Nats 0

42. Division System Nats divide 7 3 1 2 1 2 divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

43. Division System Nats divide 7 3 1 2 1 2 2 1 divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

44. Division System Nats divide 3 7 1 2 1 2 divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

45. Division System Nats divide 3 7 1 2 1 2 0 3 divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

46. Division System Nats divide 9 3 1 2 1 2 3 0 divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

47. Many possible implementations Nats 1 2 1 2 C program for Euclid’s algorithm divide divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y } Nats 0

48. Many possible implementations Nats Nats 0 1 2 1 2 circuit divide divide: Nats 2  Nats 0 2 such that  x,y  Nats, divide (x,y) = { (q,r)  Nats 0 2 | x = q· y + r  r < y }

49. Block diagrams can hide outputs divide 22 11 zerocheck zerocheck: Nats 0  Bools such that  x  Nats, zerocheck (x)  x = 0. Nats Bools

50. Block Diagrams can hide outputs divide 22 11 zerocheck zerocheck: Nats 0  Bools such that  x  Nats, zerocheck (x)  x = 0. Nats Bools

51. Block Diagrams can hide outputs 2 1 Nats Bools divisible divisible: Nats 2  Bools such that  x,y  Nats, divisible (x,y)  (  q  Nats, x = q· y ).

52. Hidden inputs are illegal, for now 

53.  true ?

54. Constant functions have no inputs  true

55. Constant function  true

56. Cycles are illegal, for now divide

57. Block Diagrams -are nested, directed, acyclic graphs -allow compositional, hierarchical system description

58. Quiz 1.  set x, x  P(x) 2.  function f, { x  domain (f) | x = f(x) } 3.  n  Nats, n = 2  ( n, n+1 )  { 1, 2, 3 } 2 4.  f  [Nats  Nats], f(x) = x 2