1. snick  snack CPSC 121: Models of Computation 2010 Winter Term 2 Sequential Circuits (“Mealy Machines”) Steve Wolfman, based on notes by Patrice Belleville and others 1

2. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 2

3. Learning Goals: Pre-Class We are mostly departing from the readings for next class to talk about a new kind of circuit on our way to a full computer: sequential circuits. The pre-class goals are to be able to: –Trace the operation of a deterministic finite-state automaton (represented as a diagram) on an input, including indicating whether the DFA accepts or rejects the input. –Deduce the language accepted by a simple DFA after working through multiple example inputs. 3

4. Learning Goals: In-Class By the end of this unit, you should be able to: –Translate a DFA into a sequential circuit that implements the DFA. –Explain how and why each part of the resulting circuit works. 4

5. Where We Are in The Big Stories Theory How do we model computational systems? Now: With our powerful modelling language (pred logic), we can prove things like universality of NAND gates for combinational circuits. Our new model (DFAs) are sort of full computers. Hardware How do we build devices to compute? Now: Learning to build a new kind of circuit with memory that will be the key new feature we need to build full- blown computers! 5

6. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 6

7. Problem: Toggle Light Switch Problem: Design a circuit to control a light so that the light changes state any time its “push-button” switch is pressed. (Like the light switches in Dempster: Press and release, and the light changes state. Press and release again, and it changes again.) ? ? 7

8. Concept Q: Toggle Switch How many gates does the circuit take? a.One b.Two c.Three d.None of these (b/c it’s a different number) e.None of these (b/c it can’t be done) ? ? 8

9. A Light Switch “DFA” light off light on toggle This Deterministic Finite Automaton (DFA) isn’t really about accepting/rejecting; its current state is the state of the light. ? ? 9

10. Outline Prereqs, Learning Goals, and !Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 10

11. Departures from Combinational Circuits MEMORY: We need to “remember” the light’s state. EVENTS: We need to act on a button push rather than in response to an input value. 11

12. Worked Problem: “Muxy Memory” Worked Problem: How can we use a mux to store a bit of memory? Let’s have it output the stored bit when the control is 0 and load a new value when the control is 1. 0 1 output ??? new data control 12

13. Worked Problem: Muxy Memory Worked Problem: How can we use a mux to store a bit of memory? Let’s have it output the stored bit when the control is 0 and load a new value when the control is 1. 0 1 output (Q) old output (Q’) new data (D) control (G) This violates our basic combinational constraint: no cycles. 13

14. Problem: Truth Table for Muxy Memory (AKA D Latch) Problem: Write a truth table for the D Latch. GDQ'Q 000 001 010 011 100 101 110 111 14

15. Truth Table for D Latch Fill in the D latch’s truth table: GDQ' 000 001 010 011 100 101 110 111 a.b.c.d.e. Q 0 1 0 1 0 1 0 1 Q 0 1 0 1 0 0 1 1 Q 0 1 0 1 0 F F 1 Q 0 0 0 1 1 1 0 1 None of these 15

16. Worked Problem: Truth Table for D Latch Worked Problem: Write a truth table for the D Latch. GDQ'Q 0000 0011 0100 0111 1000 1010 1101 1111 Like a “normal” mux table, but what happens when Q'  Q? 16

17. Worked Problem: Truth Table for D Latch Worked Problem: Write a truth table for the D Latch. GDQ'Q 0000 0011 0100 0111 1000 1010 1101 1111 Q' “takes on” Q’s value at the “next step”. 17

18. D Latch Symbol + Semantics When G is low, the latch maintains its memory. When G is high, the latch loads a new value from D. 0 1 output (Q) old output (Q’) new data (D) control (G) 18

19. D Latch Symbol + Semantics When G is low, the latch maintains its memory. When G is high, the latch loads a new value from D. 0 1 output (Q) old output (Q’) new data (D) control (G) 19

20. D Latch Symbol + Semantics When G is low, the latch maintains its memory. When G is high, the latch loads a new value from D. output (Q) new data (D) control (G) D G Q 20

21. Hold On!! Why does the D Latch have two inputs and one output when the mux inside has THREE inputs and one output? a.The D Latch is broken as is; it should have three inputs. b.A circuit can always ignore one of its inputs. c.One of the inputs is always true. d.One of the inputs is always false. e.None of these (but the D Latch is not broken as is). 21

22. Using the D Latch for Circuits with Memory Problem: What goes in the cloud? What do we send into G? D G Q Combinational Circuit to calculate next state inputoutput ?? 22

23. Using the D Latch for Our Light Switch Problem: What do we send into G? D G Q inputoutput ?? current light state a.T if the button is down, F if it’s up. b.T if the button is up, F if it’s down. c.Neither of these. 23

24. A Timing Problem: We Need EVENTS! Problem: What do we send into G? D G Q output “pulse” when button is pressed current light state button pressed As long as the button is down, D flows to Q flows through the NOT gate and back to D... which is bad! 24

25. A Timing Problem, Metaphor (from MIT 6.004, Fall 2002) What’s wrong with this tollbooth? P.S. Call this a “bar”, not a “gate”, or we’ll tie ourselves in (k)nots. 25

26. A Timing Solution, Metaphor (from MIT 6.004, Fall 2002) Is this one OK? 26

27. A Timing Problem Problem: What do we send into G? D G Q output “pulse” when button is pressed current light state button pressed As long as the button is down, D flows to Q flows through the NOT gate and back to D... which is bad! 27

28. A Timing Solution (Almost) D G Q output button pressed D G Q Never raise both “bars” at the same time. 28

29. A Timing Solution D G Q output D G Q ?? The two latches are never enabled at the same time (except for the moment needed for the NOT gate on the left to compute, which is so short that no “cars” get through). 29

30. A Timing Solution D G Q output D G Q button pressed button press signal 30

31. Master/Slave D Flip-Flop Symbol + Semantics When CLK goes from low to high, the flip-flop loads a new value from D. Otherwise, it maintains its current value. output (Q) new data (D) control or “clock” signal (CLK) D G Q D G Q 31

32. Master/Slave D Flip-Flop Symbol + Semantics When CLK goes from low to high, the flip-flop loads a new value from D. Otherwise, it maintains its current value. output (Q) new data (D) control or “clock” signal (CLK) D G Q D G Q 32

33. Master/Slave D Flip-Flop Symbol + Semantics When CLK goes from low to high, the flip-flop loads a new value from D. Otherwise, it maintains its current value. output (Q) new data (D) control or “clock” signal (CLK) D G Q D G Q 33

34. Master/Slave D Flip-Flop Symbol + Semantics When CLK goes from low to high, the flip-flop loads a new value from D. Otherwise, it maintains its current value. new data clock signal D Q output 34

35. A Pushbutton Light Switch output button press signal D Q What goes here? a.A wirec. An AND gatee. None of these. b.A NOT gated. An XOR gate 35

36. Outline Prereqs, Learning Goals, and !Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 36

37. “Mealy Machines”  DFAs whose state matters A “Mealy Machine” is a DFA that’s not about accepting or rejecting but about outputting values as it transitions. Its output can depend on its state and its input. Let’s work with a specific example... 37

38. Branch Prediction Machine Modern computers “pre-execute” instructions... but to pre-execute an “if” (AKA branch), they have to guess whether to execute the then or else part. Here’s one reasonable way to guess: whatever the last branch did, the next one will, too. (Why? Think loops!) It’s nice to build some “inertia” into this so it won’t flip its guess at the first failure... 38

39. Implementing the Branch Predictor Here’s that algorithm as a DFA (with no accepting state): Can we build a branch prediction circuit? YES! yes? no? NO! not taken taken not taken taken not taken taken not taken 39

40. General Implementation of DFAs as “Mealy Machines” Each time the clock “ticks” move from one state to the next. D Q Combinational circuit to calculate next state/output inputoutput CLK Each of these lines (except the clock) may carry multiple bits; the D flip-flop may be several flip-flops to store several bits. 40

41. How to Implement a DFA as a “Mealy Machine” (1)Number the states, starting with 0, and figure out how many bits you need to store the state number. (2)Number the inputs, starting with 0, and figure out how many bits you need to represent the input. (3)Lay out enough D flip-flops to store the state. (4)For each state, build a combinational circuit that determines the next state and the output given the input. (A separate circuit for each state! These are often very simple!) (5)Send all those “next states” and outputs from the circuits into multiplexers, and use the current state as the control signal. (So, only the appropriate circuit from (4) gets used.) (6)Send the multiplexer output back into your flip-flops. 41

42. Implementing the Predictor Step 1 + 2 YES! 3 yes? 2 no? 1 NO! 0 not taken taken not taken taken not taken taken not taken How many 1-bit flip-flops do we need to represent the state? a.0 (no memory is needed) b.1 c.2 d.3 e.None of these taken = 1 not taken = 0 42

43. Just Truth Tables... Current StatetakenNew state 00000 00101 010 011 100 101 110 111 YES! 3 yes? 2 no? 1 NO! 0 not taken taken not taken taken not taken taken not taken What’s in this row? a.0 0 b.0 1 c.1 0 d.1 1 e.None of these. 43

44. Just Truth Tables... Current StatetakenNew state 00000 00101 01000 01111 10000 10111 11010 11111 YES! 3 yes? 2 no? 1 NO! 0 not taken taken not taken taken not taken taken not taken 44

45. Renaming to Make a Pattern Clearer... predlasttakenpred'last' 00000 00101 01000 01111 10000 10111 11010 11111 YES! 3 yes? 2 no? 1 NO! 0 not taken taken not taken taken not taken taken not taken 45

46. Implementing the Predictor: Step 3 We always use this pattern. In this case, we need two flip-flops. D Q Combinational circuit to calculate next state/output inputoutput CLK D Q Let’s switch to TkGate schematics... 46

47. ??? Implementing the Predictor: Step 3 D Q ?? input output CLK D Q 47

48. Implementing the Predictor: Step 3 48 ???

49. Implementing the Predictor: Step 4 + 5 + 6 49

50. Implementing the Predictor: Step 4 + 5 + 6 What should the next “prediction” be in state 0? a.same value as “taken” switch b.negation of “taken” switch (“~taken”) c.T d.F e.None of these. 50

51. Implementing the Predictor: Step 4 + 5 + 6 What should the next prediction be in state 1? a.taken b.~taken c.T d.F e.None of these. 51

52. Implementing the Predictor: Step 4 + 5 + 6 What should the next prediction be in state 2? a.taken b.~taken c.T d.F e.None of these. 52

53. Implementing the Predictor: Step 4 + 5 + 6 What should the next prediction be in state 3? a.taken b.~taken c.T d.F e.None of these. 53

54. Just Truth Tables... predlasttakenpred'last' 00000 00101 01000 01111 10000 10111 11010 11111 54

55. Implementing the Predictor: Step 4 + 5 + 6 55

56. Implementing the Predictor: Step 4 + 5 + 6 What should the next “last” value be in state 0? a.taken b.~taken c.T d.F e.None of these. 56

57. Implementing the Predictor: Step 4 + 5 + 6 TADA!! 57

58. You can often simplify, but that’s not the point. 58

59. Outline Prereqs, Learning Goals, and !Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 59

60. Steps to Build a Powerful Machine (1)Number the states, starting with 0, and figure out how many bits you need to store the state number. (2)Number the inputs, starting with 0, and figure out how many bits you need to represent the input. (3)Lay out enough D flip-flops to store the state. (4)For each state, build a combinational circuit that determines the next state and the output given the input. (A separate circuit for each state!) (5)Send all those “next states” and outputs from the circuits into multiplexers, and use the current state as the control signal. (6)Send the multiplexer output back into your flip-flops. 60

61. How Powerful Is a DFA? How does a DFA compare to a modern computer? a.Modern computer is more powerful. b.DFA is more powerful. c.They’re the same. 61

62. Where We’ll Go From Here... We’ll come back to DFAs again later in lecture. In two upcoming labs, you’ll: Build a circuit to perform a practical task using techniques like these. Explore how to build something like a DFA that can run other DFAs... a general purpose computer. 62

63. Outline Prereqs, Learning Goals, and !Quiz Notes Problems and Discussion –A Pushbutton Light Switch –Memory and Events: D Latches & Flip-Flops –General Implementation of DFAs (with a more complex DFA as an example) –How Powerful are DFAs? Next Lecture Notes 63

64. Learning Goals: In-Class By the end of this unit, you should be able to: –Translate a DFA into a sequential circuit that implements the DFA. –Explain how and why each part of the resulting circuit works. 64

65. Next Lecture Learning Goals: Pre-Class By the start of class, you should be able to: –Convert sequences to and from explicit formulas that describe the sequence. –Convert sums to and from summation/”sigma” notation. –Convert products to and from product/”pi” notation. –Manipulate formulas in summation/product notation by adjusting their bounds, merging or splitting summations/products, and factoring out values. 65

66. Next Lecture Prerequisites See the Mathematical Induction Textbook Sections at the bottom of the “Textbook and References” page.Mathematical Induction Textbook Sections Complete the open-book, untimed quiz on Vista that’s due before the next class. 66

67. snick  snack More problems to solve... (on your own or if we have time) 67

68. Problem: Traffic Light Problem: Design a DFA with outputs to control a set of traffic lights. 68

69. Problem: Traffic Light Problem: Design a DFA with outputs to control a set of traffic lights. Thought: try allowing an output that sets a timer which in turn causes an input like our toggle when it goes off. Variants to try: - Pedestrian cross-walks - Turn signals - Inductive sensors to indicate presence of cars - Left-turn signals 69

70. 1 23 aa a,b,c b b,c c PROBLEM: Does this accept the empty string? 70