1. snick  snack CPSC 121: Models of Computation 2013W2 Propositional Logic: A First Model of Computation 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. Introductions, again Steven Wolfman ICICS 239; office hours listed on the website I also have an open door policy: If my door is open, come in and talk! Also, I will usually be available after class. And, you can make appointments with me Additionally, you can use TA office hours 2

3. Outline Prereqs, Learning Goals, and Quiz Notes True, False, and Gates. Why Start Here? Problems and Discussion Next Lecture Notes: Unit Continues... 3

4. 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. –Evaluate the truth of propositional logic statements using truth tables. –Translate back and forth between propositional logic statements and circuits that assess the truth of those statements. 4 How should you achieve pre-class goals? Use the quiz to guide your readings!

5. Quiz 1 Notes What logic statement does this directly implement? a. ~(p  m) b. p  m c. ~(p  m) d. ~p  ~m e. ~(~p  ~m)

6. Quiz 1 Notes What logic statement does this directly implement? a. ~(p  m) b. p  m c. ~(p  m) d. ~p  ~m e. ~(~p  ~m) Morals of the story: 1.Read the supplement! 2.Change NAND into AND then NOT (pull out the bubble) 3.None of these is a real option! 4.Steve uses a “RNG” to order options.

7. Learning Goals: In-Class By the end of this unit, you should be able to: –Build combinational computational systems using propositional logic expressions and equivalent digital logic circuits that solve real problems, e.g., our 7- or 4-segment LED displays (using a “DNF” or any other successful approach). 7

8. Quiz 1 Notes Approaches… Try to understand the “story”: “no matter with switch is flipped, the result will always result in the light turning on” [not quite what we intended] Formalize the problem: “Let a,b,c represent 3 switches from left to right” Solve in propositional logic: “(x ^ y ^ z) v (x ^ ~y ^ ~z) v (~x ^ y ^ ~z) v (~x ^ ~y ^ z)” Try a simpler problem: “I decided to just focus on two parts [switches] of the circuit” Test your answer: “The light switches cannot both be on at the same, but they can both be off. By working out the truth table, it proofs to be correct.” 8

9. Quiz 1 Notes “Marked for Completeness”? Compare these two responses I'm not sure I understand the question...how does a logic circuit diagram connect to an electrical circuit? I understand this question but i dont know how to write it in terms of this circuit and logic stuff 9 Somewhere between these is the “marked for completeness line”. (The left one gets credit; the right does not.) At minimum: give the question a shot or ask a meaningful question in return!

10. Quiz 1 Notes Popular sources of help (especially on the open- ended light-bulb problem): Epp Friends and family members Wikipedia (especially its page of logic gate symbols) Lecture notes Online videos related to logic and circuits 10 But… follow the quiz collaboration guidelines. Closed-ended part: work alone (but open book). Open-ended part: work with anyone (but acknowledge).

11. Where We Are in The Big Stories Theory How do we model computational systems? Now: learning the underpinning of all our models (formal logical reasoning with Boolean values). Hardware How do we build devices to compute? Now: establishing our baseline tool (gates), briefly justifying these as baselines, and designing complex functions from gates. 11

12. Outline Prereqs, Learning Goals, and Quiz Notes True, False, and Gates. Why Start Here? Problems and Discussion Next Lecture Notes: Unit Continues... 12

13. Logic for Reasoning about Truth: Where Should We Start? I will suppose that... some malicious demon of the utmost power and cunning has employed all his energies in order to deceive me. I shall think that the sky, the air, the earth, colours, shapes, sounds and all external things are merely the delusions of dreams which he has devised to ensnare my judgement. I shall consider myself as not having hands or eyes, or flesh, or blood or senses, but as falsely believing that I have all these things. - René Descartes 13

14. Logic as Model for Physical Computations Input a Input b a  b ~aa 5V 0V http://alumni.media.mit.edu/~paulo/courses /howmake/mlfabfinalproject.htm 14

15. “OR” operator and gate a b output Physical System “Truth Table” model ab a  b TTT TFT FTT FFF We think of “flowing water” as true and “no water” as false, and the physical world becomes an effective representation for our ideas! Propositional logic model: a  b means “ a OR b ” Circuit diagram model: the “OR” gate

16. Outline Prereqs, Learning Goals, and Quiz Notes True, False, and Gates. Why Start Here? Problems and Discussion Next Lecture Notes: Unit Continues... 16

17. Problem: Light Switch Problem: Design a circuit to control a light so that the light changes state any time its switch is flipped. ? ? The problem gives the story we have to implement. Be sure you understand the story and always keep it in mind! 17

18. Problem: Light Switch Problem: Design a circuit to control a light so that the light changes state any time its switch is flipped. Identifying inputs/outputs: consider these possible inputs and outputs: Input:the switch flipped or the switch is up Output: the light is shining or the light changed states Which are most useful for this problem? a.flipped and shining b.flipped and changed c.up and shining d.up and changed e.None of these 18 ? ?

19. Problem: Light Switch Problem: Design a circuit to control a light so that the light changes state any time its switch is flipped. ? ? Consider these possible solutions: Which of these solves the problem? a.Only #1 b.Only #2 c.Only #3 d.#1 and #2 e.Some other combination 19

20. Problem: Two-Switch Problem: Design a circuit to control a light so that the light changes state any time either of the two switches that control it is flipped. ? ? 20

21. Problem: Two-Switch Problem: Design a circuit to control a light so that the light changes state any time either of the two switches that control it is flipped. ? ? Getting the Story Right: Is the light on or off when both switches are up? a.On, in every correct solution. b.Off, in every correct solution. c.It depends, but a correct solution should always do the same thing given the same settings for the switches. d.It depends, and a correct solution might do different things at different times with the same switch settings. e.Neither on nor off. 21

22. Problem: Two-Switch Problem: Design a circuit to control a light so that the light changes state any time either of the two switches that control it is flipped. Which of these circuits solves the problem? a.Only #1 b.Only #2 c.#1 and #2 d.#1 and #3 e.All three ? ? 22

23. Problem: Three-Switch Problem: Design a circuit to control a light so that the light changes state any time any of the three switches that control it is flipped. ? ? 23

24. Problem: Three-Switch Problem: Design a circuit to control a light so that the light changes state any time any of the three switches that control it is flipped. ? ? Fill in the circuit’s truth table: s1s1 s2s2 s3s3 TTT TTF TFT TFF FTT FTF FFT FFF a.b.c.d.e. out T F F T F T T F F T T F T F F T F T F T F T F T T F T F T F T F None of these 24

25. Problem: Three-Switch Problem: Design a circuit to control a light so that the light changes state any time any of the three switches that control it is flipped. ? ? Getting the Story Right: Which of these indicates whether the light is on or off in a particular correct solution? a.Whether an odd number of switches is on. b.Whether the majority (two or more) of switches are on. c.Whether all the switches are on. d.Whether a switch has been flipped recently. e.None of these. 25

26. Problem: Three-Switch Problem: Design a circuit to control a light so that the light changes state any time any of the three switches that control it is flipped. ? ? Modelling the Circuit: Which of these describes an incorrect solution? a.(s 1  s 2  s 3 )  (s 1  ~s 2  ~s 3 )  (~s 1  s 2  ~s 3 )  (~s 1  ~s 2  s 3 ) b.s 1  s 2  s 3  (s 1  s 2  s 3 ) c.s 1  (s 2  s 3 ) d.(s 1  ~(s 2  s 3 ))  (~s 1  (s 2  s 3 )) e.None of these is incorrect. 26

27. Problem: n-Switch Problem: Describe an algorithm for designing a circuit to control a light so that the light changes state any time any of its n switches is flipped. ? ?... 27

28. Outline Prereqs, Learning Goals, and Quiz Notes True, False, and Gates. Why Start Here? Problems and Discussion Next Lecture Notes: Unit Continues... 28

29. Learning Goals: In-Class By the end of this unit, you should be able to: –Build combinational computational systems using propositional logic expressions and equivalent digital logic circuits that solve real problems, e.g., our 7- or 4-segment LED displays. 29

30. snick  snack Some Things to Try... (on your own if you have time and interest, not required) 30

31. Some Practice Problems Here are some assignment/exam-like practice problems related to this lecture’s in-class learning goals. Give them a shot! 31 Design a representation (for a digital logic circuit) for the state of a stop light. Design a circuit that takes the current states of a stop light and produces the next state. Design a circuit that takes the current states of two stop lights and produces true if they are “safe in combination” and false otherwise. (By “in combination” we mean that one of the states is the state of the lights along one road at an intersection and the other is the state of the lights along the perpendicular road.

32. Problem: Logicians and Hats Problem: Three logicians are each wearing a black hat or a white hat, but not all white. Nobody can see their own hat. However, A can see the hats of B and C, and B can see the hats of A and C. C is blind. You go and ask them one by one in the order A, B, C, whether they know the color of their own hat. A answers “No”. B answers “No”. Then C answers “Yes”. Explain how this is possible. 32

33. Problem: Criminals and Hats Problem: A warden plans to line up 100 prisoners in order tomorrow. The warden will place a white or black hat on each prisoner’s head so that no prisoner can see the hat on his or her own head, but they can see the hats of the prisoners in front of them. From the back of the line, the warden will ask each prisoner “Is your hat black?” If the prisoner answers correctly, he or she is set free. Incorrect answers lead to immediate and noisy beheading. If they answer anything other than “yes” or “no”, all prisoners are beheaded. They get one hour as a group to plan their strategy. How many prisoners can be saved? The switch problems might give you a hint. 33