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EECS 20 Lecture 4 (January 24, 2001) Tom Henzinger

Block Diagrams. EECS 20 Lecture 4 (January 24, 2001) Tom Henzinger . 1 Systems are functions 2 Signals are functions . Systems as Functions. Input. Output. Domain: set of possible inputs Range: set of possible outputs Graph: set of pairs ( input, output ).

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EECS 20 Lecture 4 (January 24, 2001) Tom Henzinger

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

  2. 1 Systems are functions 2 Signals are functions

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

  4. Factorial System Nats Nats !

  5. Factorial System Nats Nats 3 ! 6

  6. Factorial System Nats Nats 4 ! 24

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

  8. Inverter System Bools Bools 

  9. Inverter System Bools Bools  true false

  10. Inverter System Bools Bools  false true

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

  12. Composition of Systems Bools Bools Bools   true true false

  13. Composition of Systems Bools Bools Bools   false false true

  14. This is again a system ! Bools Bools  

  15. The Identity System Bools Bools id Domain: Bools Range: Bools Graph: { ( x, y )  Bools2 | 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 Bools2 Bools 

  19. And System Bools2 Bools  (true, false) false

  20. And System Bools Bools  Bools

  21. And System Bools true Bools  false false Bools

  22. - Exponentiation System Nats 2 Nats exp ? 3 Nats graph (exp) = { ( (x,y), z )  Nats2  Nats | z = xy }

  23. Exponentiation System Nats 2 Nats 1 exp 8 3 2 Nats graph (exp) = { ( (x,y), z )  Nats2  Nats | z = xy }

  24. Exponentiation System Nats 2 Nats 2 exp 9 3 1 Nats graph (exp) = { ( (x,y), z )  Nats2  Nats | z = xy }

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

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

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

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

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

  30. Or System  Bools  Bools    Bools domain () = Bools2 range () = Bools graph () = { ( (x,y), z )  Bools2  Bools | z = x  y }

  31. - Or System Bools Bools  Bools domain () = Bools2 range () = Bools graph () = { ( (x,y), z )  Bools2  Bools | z = x  y }

  32. Block Diagrams with Forks    

  33. Block Diagrams with Forks true     false

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

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

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

  37. Joins are illegal   

  38. Joins are illegal true   false 

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

  40. Multiple Outputs Nats Nats0 f Nats0 Nats f: Nats2 Nats02 such that x,yNats, f (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  41. Division System Nats Nats0 quotient 1 1 divide 2 remainder 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  42. Division System Nats Nats0 7 1 1 divide 3 2 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  43. Division System Nats Nats0 7 2 1 1 divide 3 2 1 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  44. Division System Nats Nats0 3 1 1 divide 7 2 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  45. Division System Nats Nats0 3 0 1 1 divide 7 2 3 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  46. Division System Nats Nats0 9 3 1 1 divide 3 2 0 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  47. Many possible implementations divide Nats Nats0 1 1 C program for Euclid’s algorithm 2 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  48. Many possible implementations divide Nats Nats0 1 1 circuit 2 2 Nats0 Nats divide: Nats2 Nats02 such that x,yNats, divide (x,y) = { (q,r)Nats02 | x = q· y + r  r < y }

  49. Block diagrams can hide outputs Nats 1 1 divide Bools 2 2 zerocheck Nats zerocheck: Nats0 Bools such that xNats, zerocheck (x)  x = 0 .

  50. Block Diagrams can hide outputs Nats 1 1 divide Bools 2 2 zerocheck Nats zerocheck: Nats0 Bools such that xNats, zerocheck (x)  x = 0 .

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