1. snick  snack CPSC 121: Models of Computation 2013W2 Propositional Logic: Conditionals and Logical Equivalence Steve Wolfman, based on notes by Patrice Belleville and others 1 This work is licensed under a Creative Commons Attribution 3.0 Unported License.Creative Commons Attribution 3.0 Unported License

2. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Logical equivalence practice –How to write a logical equivalence proof –A flaw in our model Next Lecture Notes 2

3. Quiz 2 Review: “p cannot be true unless q is false” Let’s go one row at a time to check whether the statement is true (not violated). a.Definitely true b.I think it’s true c.I don’t know d.I think it’s false e.Definitely false 3 pqstmt TT TF FT FF

4. Quiz 2 Review: “p cannot be true unless q is false” Compare to the truth table for p  ~q 4 pq p  ~q TT TF FT FF

5. Quiz 2 Review: “Meaning” of  When I say “if it’s raining, I’ll bring my umbrella”, do I mean: In other words, I might bring my umbrella even if it’s not raining? Probably not, in English. BUT, in logic  always means exactly that truth table! It just isn’t a great match for “normal” English. Better for legalistic English… 5 ru r  u TTT TFF FTT FFT

6. Quiz 2 Review: “Meaning” of  Consider the following truth table for p  ~q And, consider the following statement: “p cannot be true unless q is false.” Does this mean the same thing as p  ~q? 6 pq~q p  ~q TTFF TFTT FTFT FFTT

7. Quiz 2 Review: “Meaning” of  Consider the following truth table for p  ~q And, consider the following statement: “p cannot be true unless q is false.” Does this mean the same thing as p  ~q? 7 pq~q p  ~q TTFF TFTT FTFT FFTT True “by default”

8. Quiz 2 Review: “Meaning” of  Consider the following truth table for p  q And, consider the following statement: “Either p and q, or ~p and ~q.” Does this mean the same thing as p  q? 8 pq p  q TTT TFF FTF FFT Turns out many people find this translation much clearer.

9. Quiz 2 Review: “F” vs. “c” Reminder: I think “c” is a terrible symbol to use for a contradiction because “a” and “b” are natural variables to use in propositional logic expressions, and “c” is a natural one to use next. We will use “F” instead and avoid capital letters as names for propositional logic variables. 9

10. Learning Goals: Pre-Class By the start of class, you should be able to: –Translate back and forth between simple natural language statements and propositional logic, now with conditionals and biconditionals. –Evaluate the truth of propositional logical statements that include conditionals and biconditionals using truth tables. –Given a propositional logic statement and an equivalence rule, apply the rule to create an equivalent statement. Example: given (u  s)  s, apply p  q  ~p  q. Note:p maps to (u  s) and q maps to s. Result: ~(u  s)  s 10

11. Learning Goals: In-Class By the end of this unit, you should be able to: –Explore alternate forms of propositional logic statements by application of equivalence rules, especially in order to simplify complex statements or massage statements into a desired form. –Evaluate propositional logic as a “model of computation” for combinational circuits, including at least one explicit shortfall (e.g., referencing gate delays, fan-out, transistor count, wire length, instabilities, shared sub-circuits, etc.). 11

12. Where We Are in The Big Stories Theory How do we model computational systems? Now: practicing our second technique for formally establishing the truth of a statement (logical equivalence proofs). (The first technique was truth tables.) Hardware How do we build devices to compute? Now: learning to modify circuit designs using our logical model, gaining more practice designing circuits, and identifying a flaw in our model for circuits. 12

13. Motivating Problem Simplify this code, proving your simplification correct: 13 List version: ;; ListOfNatural -> Boolean ;; Produces true if all numbers in lon ;; are positive and false otherwise. (define (all-positive? lon) (cond [(empty? lon) true] [else (if (<= (first lon) 0) false (all-positive? (rest lon)))])) Image version: ;; Image -> Boolean ;; Produces true if image is too large to ;; fit on the HEIGHT-by-WIDTH screen. (define (too-large? image) (if (> (image-width image) WIDTH) true (if (> (image-height image) HEIGHT) true false)))

14. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Logical equivalence practice –How to write a logical equivalence proof –A flaw in our model Next Lecture Notes 14

15. Problem: 4-Segment LED Display Problem: Build a circuit that displays the numbers 1 through 9 represented by four Boolean values a, b, c, and d on a 4- segment Boolean display. 12345 6789 15

16. Problem: 4-Segment LED Display Problem: Build a circuit that displays the numbers 1 through 9 represented by four Boolean values a, b, c, and d on a 4- segment Boolean display. 12345 6789 16 321 654 987

17. RECALL: Representing Positive Integers 17 This is the convention we (and computers) use for the positive integers 0-9, which requires 4 variables: #abcd 0FFFF 1FFFT 2FFTF 3FFTT 4FTFF 5FTFT 6FTTF 7FTTT 8TFFF 9TFFT...

18. Problem: Equivalent Circuits Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table): 1.(~a  b)  (a  ~b  ~c) 2.(~a  b)  a 3.a  b 4.a  b 18 321 654 987

19. Problem: Equivalent Circuits Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table). 1.(~a  b)  (a  ~b  ~c) 2.(~a  b)  a 3.a  b 4.a  b Which of 2, 3, and 4 are equivalent? a.None of them. b.2 and 3 but not 4. c.2 and 4 but not 3. d.3 and 4 but not 2. e.2, 3, and 4. 19 321 654 987

20. Problem: Equivalent Circuits Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table). 1.(~a  b)  (a  ~b  ~c) 2.(~a  b)  a 3.a  b 4.a  b How many of these are correct? a.None of them. b.One of them. c.Two of them. d.Three of them. e.All four of them. 20 321 654 987

21. Problem: Equivalent Circuits Problem: Consider these circuits for the top LED in the 4-segment display (assuming inputs abcd from the previous table). 1.(~a  b)  (a  ~b  ~c) 2.(~a  b)  a 3.a  b 4.a  b Which of these proves that circuit 1 is not equivalent to circuit 3: a.2 and 4 are equivalent b.1 mentions c but 3 does not. c.Their values differ on a=T b=F c=T. d.None of these. 21 321 654 987

22. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Logical equivalence practice –How to write a logical equivalence proof –A flaw in our model Next Lecture Notes 22

23. Writing an Equiv Proof: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b)by distribution...to be filled in...  a  b by identity QED 23

24. Writing an Equiv Proof: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b)by distribution...to be filled in...  a  b by identity QED 24 State your theorem. Explicitly start the proof. Start with one side...

25. Writing an Equiv Proof: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b)by distribution...to be filled in...  a  b by identity QED 25 Start with one side... Each line starts with  to indicate it’s equivalent to the previous line....and work to the other. End with QED!

26. Writing an Equiv Proof: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b)by distribution...  a  b by identity QED 26 Give the next statement. And justify how you got it.

27. Problem: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b) by distribution  ?????? by negation  a  b by identity QED 27 What’s missing? a.(a  b) b.F  (a  b) c.a  (a  b) d.None of these, but I know what it is. e.None of these, and there’s not enough information to tell.

28. PRACTICE Problem: Arbitrary Logic Expression Problem: Aliens hold the Earth hostage, and only you can save it by proving (a  b)  ~(b  a)  ~a  b. Reminder (“IMP” on “Dave’s Formula Sheet”): p  q  ~p  q 28

29. Collected Wisdom So far: Try getting rid of  : p  q  ~p  q Try moving negations inward: ~(p  q)  ~p  ~q Work from the more complex side first, BUT Switch to different strategies/sides when you get stuck In the end, write the proof in clean “one-side-to-the-other” form and double-check steps Also: Try distribution to pull out common terms OR Try distribution to rearrange terms Try working with a simpler part of the statement first Try getting rid of  : p  q  p  q  q  p Try getting rid of  : p  q  ~(p  q) 29

30. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Logical equivalence practice –How to write a logical equivalence proof –A flaw in our model Next Lecture Notes 30

31. “Multiplexer” A circuit that, given three inputs a, b, and c (the “control” signal), outputs a ’s value when c is F and b ’s when c is T. This circuit is called a multiplexer. a b out c 0 1 31 (~a  b  c)  ( a  ~b  ~c)  ( a  b  ~c)  ( a  b  c) abcout FFFF FFTF FTFF FTTT TFFT TFTF TTFT TTTT

32. MUX Design Here’s one implementation of the multiplexer (MUX) circuit. (Is this equivalent to the previous slide’s formula? Good question... prove that it is for practice at home!) 32 a b out c 0 1

33. Truthy MUX What is the intended output if both a and b are T? a.T b.F c.Unknown... but could be answered given a value for c. d.Unknown... and might still be unknown even given a value for c. 33 a b out c 0 1

34. Glitch in MUX Design Imagine the circuit is in steady-state with a, b, and c all T. Trace how changes flow when we change c to F, if each gate takes 10ns to operate. 34 a b out c 0 1

35. Trace:-1 ns 35 T F T T T T F a b out c 0 1

36. Trace:0 ns 36 T F T T T F F OFF * * a b out c 0 1

37. Trace:10 ns 37 T F T F T F T OFF * * a b out c 0 1

38. Tracing a MUX Assume the output starts at 1. We want it to end up at 1. How long does it take before it’s stable? (Choose the best answer.) a.0 ns (it never varies from 1) b.10 ns (it varies before 10ns, but not after) c.20 ns d.30 ns e.40 ns 38 a b out c 0 1

39. Trace:-1 ns 39 T F T T T T F a b out c 0 1

40. Trace:0 ns 40 T F T T T F F OFF * * a b out c 0 1

41. Trace:10 ns 41 T F T F T F T OFF * * a b out c 0 1

42. Trace:20 ns 42 T T F F T F T OFF * a b out c 0 1

43. Trace:30 ns 43 T T T F T F T OFF a b out c 0 1

44. Trace:40 ns 44 T T T F T F T OFF a b out c 0 1

45. Other MUX Glitches? The mux glitches because information from c travels two paths with different delays. While the longer path “catches up” the circuit can be incorrect. PRACTICE: Trace the circuit to show why none of these can cause a glitch: Changing a (keeping b and c constant) Changing b (keeping a and c constant) Changing c (keeping a and b constant if at least one of them is F) 45 a b out c 0 1

46. A Glitchless MUX This circuit uses what we know when a = b = T. PRACTICE: Prove that it’s logically equivalent to the original MUX. Hint: break a  b up into two cases, one where c is true and one where c is false: a  b  c and a  b  ~c. 46 a b out c 0 1

47. Outline Prereqs, Learning Goals, and Quiz Notes Problems and Discussion –Logical equivalence practice –How to write a logical equivalence proof –A flaw in our model Next Lecture Notes 47

48. Learning Goals: In-Class By the end of this unit, you should be able to: –Explore alternate forms of propositional logic statements by application of equivalence rules, especially in order to simplify complex statements or massage statements into a desired form. –Evaluate propositional logic as a “model of computation” for combinational circuits, including at least one explicit shortfall (e.g., referencing gate delays, fan-out, transistor count, wire length, instabilities, shared sub-circuits, etc.). 48

49. Motivating Problem Simplify this code, proving your simplification correct: 49 List version: ;; ListOfNatural -> Boolean ;; Produces true if all numbers in lon ;; are positive and false otherwise. (define (all-positive? lon) (cond [(empty? lon) true] [else (if (<= (first lon) 0) false (all-positive? (rest lon)))])) Image version: ;; Image -> Boolean ;; Produces true if image is too large to ;; fit on the HEIGHT-by-WIDTH screen. (define (too-large? image) (if (> (image-width image) WIDTH) true (if (> (image-height image) HEIGHT) true false))) We’ll focus only on this one. Left as an exercise. But… Note: if you simplify away the cond, you lose the structure that reflects the ListOfNatural data representation. Doing in class if we have time.

50. Motivating Problem Simplify this code, proving your simplification correct: 50 Image version: ;; Image -> Boolean ;; Produces true if image is too large to ;; fit on the HEIGHT-by-WIDTH screen. (define (too-large? image) (if (> (image-width image) WIDTH) true (if (> (image-height image) HEIGHT) true false)))

51. Motivating Problem Simplify the code, proving your simplification correct. 51 Let w = (> (image-width image) WIDTH). Let h = (> (image-height image) HEIGHT). Then, what does the code mean? (if w then true, else (if h then true, else false)) How do we express (if x then y, else z) in propositional logic? (x  y)  (~x  z) Which gives us: (w  T)  (~w  ((h  T)  (~h  F)))

52. Motivating Problem Simplify the code, proving your simplification correct. 52 Simplify: (w  T)  (~w  ((h  T)  (~h  F))) How should we proceed? What’s a promising starting point?

53. Motivating Problem Simplify the code, proving your simplification correct. 53 Simplify: (w  T)  (~w  ((h  T)  (~h  F))) How should we proceed? What’s a promising starting point? Which of these is relevant? Try getting rid of  : p  q  ~p  q Try moving negations inward: ~(p  q)  ~p  ~q Work from the simpler side first, BUT Switch to different strategies/sides when you get stuck In the end, write the proof in clean “one-side-to-the-other” form and double- check steps Try distribution to pull out common terms OR Try distribution to rearrange terms Try working with a simpler part of the statement first Try getting rid of  : p  q  p  q  q  p Try getting rid of  : p  q  ~(p  q)

54. Motivating Problem Simplify the code, proving your simplification correct. 54 Simplify: (w  T)  (~w  ((h  T)  (~h  F))) How should we proceed? What’s a promising starting point? “Lemma 1”, proven below: (h  T)  (~h  F)  h (h  T)  (~h  F)  (~h  T)  (~h  F)(by IMP)  (~h  T)  (~~h  F)(by IMP)  (~h  T)  (h  F)(by DNEG) skippable  T  (h  F)(by UB)  T  h(by I)  h(by I)

55. Motivating Problem Simplify the code, proving your simplification correct. 55 Simplify: (w  T)  (~w  ((h  T)  (~h  F))) How should we proceed? What’s a promising starting point? (w  T)  (~w  ((h  T)  (~h  F)))  (w  T)  (~w  h)(by Lemma 1)  (~w  T)  (~w  h)(by IMP)  (~w  T)  (w  h)(by IMP)  T  (w  h)(by UB)  w  h(by I)

56. Motivating Problem Simplify the code, proving your simplification correct. 56 What can we conclude from: (w  T)  (~w  ((h  T)  (~h  F)))  w  h This code: (if (> (image-width image) WIDTH) true (if (> (image-height image) HEIGHT) true false))) Is equivalent to this code: (or (> (image-width image) WIDTH) (> (image-height image) HEIGHT)) WARNING: Be careful as you use proofs like this if evaluating part of the code matters for more than its result (e.g., for printing something to the screen).

57. Motivating Problem We’ve proven that this: (if (> (image-width image) WIDTH) true (if (> (image-height image) HEIGHT) true false))) Is equivalent to: (or (> (image-width image) WIDTH) (> (image-height image) HEIGHT)) Should you “even bother” running your tests, since we proved it? 57 Of course! Your tests give you confidence that your proof was correct, and that your model was accurate. By the way, notice something interesting: our proof prvoved the code transformation correct. Do passing tests prove the code they test is correct? Tests and proofs serve different purposes.

58. Next Lecture Learning Goals: Pre-Class By the start of class, you should be able to: –Convert positive numbers from decimal to binary and back. –Convert positive numbers from hexadecimal to binary and back. –Take the two’s complement of a binary number. –Convert signed (either positive or negative) numbers to binary and back. –Add binary numbers. 58

59. Next Lecture Prerequisites Read Section 2.5 and the supplemental handout: http://www.ugrad.cs.ubc.ca/~cs121/current/ handouts/signed-binary-decimal-conversions.html http://www.ugrad.cs.ubc.ca/~cs121/current/ handouts/signed-binary-decimal-conversions.html Complete the open-book, untimed quiz on Connect that’s due before the next class. 59

60. snick  snack Some Things to Try... (on your own if you have time, not required) 60

61. Problem: String equals Consider the following from the Java 6 API documentation for the String class’s equals method:...The [method returns] true if and only if the argument is not null and is a String object that represents the same sequence of characters as this object. Let n1 mean “this string is null”, n2 mean “the argument is null”, r mean “the method returns true”, and s mean “the two strings are objects that represent the same sequence of characters”. Presumably any two null strings are equal to each other. Then, equality would become something like “the method returns true if and only if the two strings both null or are objects that represent the same sequence of characters”. Problem: Is that logically equivalent to the statement from the API? Why or why not? 61

62. Problem: Decisions in Code Consider the following code, part of a “binary bounds search”: if target equals value then if lean-left-mode is true call the go-left routine otherwise call the go-right routine otherwise if target is less than value then call the go-left routine otherwise call the go-right routine 62

63. Problem: Decisions in Code Let gl mean “the go-left routine is called” Problem: complete the logical expression gl  _____________________ Problem: Use your logic to simplify the pseudocode so it requires just one “if/otherwise”. 63

64. Problem: Prove (~a  b)  a  a  b Theorem: (~a  b)  a  a  b Proof: (~a  b)  a  a  (~a  b) by commutativity  (a  ~a)  (a  b)by distribution...  a  b by identity QED 64 State your theorem. Explicitly start the proof. Start with one side... Each line starts with  to indicate it’s equivalent to the previous line. Give the next statement. And justify how you got it. End with QED!...and work to the other.