snick Outline snack • Prereqs, Learning Goals, and Quiz Notes • Reminder about the Challenge Method CPSC 121: Models of Computation • Generalized De Morgan’s Law 2016W2 • Brief Problems and Discussion Rewriting Predicate Logic Statements • Next Lecture Notes Steve Wolfman, based on notes by Patrice Belleville and others 1 2 Learning Goals: Pre-Class Learning Goals: In-Class By the start of class, you should be able to: By the end of this unit, you should be able – Determine the negation of any quantified statement. to: – Given a quantified statement and an equivalence rule, – Explore alternate forms of predicate logic apply the rule to create an equivalent statement (particularly the De Morgan’s and contrapositive statements using the logical equivalences you rules). have already learned plus negation of – Prove and disprove quantified statements using the quantifiers (a generalized form of De “challenge” method ( Epp, 3 d edition, page 99). Morgan’s Law). – Apply universal instantiation, universal modus ponens, and universal modus tollens to predicate logic statements that correspond to the rules’ premises to infer statements implied by the premises. 3 4 Outline Reminder: Challenge Method A predicate logic statement is like a game with • Prereqs, Learning Goals, and Quiz Notes two players: • Reminder about the Challenge Method • you (trying to prove the statement true) • your adversary (trying to prove it false). • Generalized De Morgan’s Law The two of you pick values for the quantified • Brief Problems and Discussion variables working from the outside (left) in. • Next Lecture Notes Your adversary picks the values of universally quantified variables. You pick the values of existentially quantified variables. 8 9 1
What does it mean to say: Challenge Method Continued “I have a winning strategy at Nim”? If there’s a strategy for you such that no strategy of the adversary’s can beat you, the statement is true. If there’s a strategy for the adversary such that no strategy of yours can beat the adversary, the statement is false. 10 11 Is the same as ? Outline You’re playing on the side of truth * . Which • Prereqs, Learning Goals, and Quiz Notes of these will likely be easier to prove? • Reminder about the Challenge Method x S, y S,Supes(x,y). • Generalized De Morgan’s Law x S, y S,Supes(x,y). • Brief Problems and Discussion a. version • Next Lecture Notes b. version c. Neither, since they’re logically equivalent. d. It’s impossible to tell. the Canadian Way. 12 13 * Also justice and De Morgan’s Law and De Morgan’s Law and Negating Quantifiers Negating Quantifiers Consider the statement: Consider the statement: x Z + , Odd(x) Even(x) x Z + , x*x = x. This is essentially an infinitely big AND: This is essentially an infinitely big OR: (Odd(1) Even(1)) (Odd(2) Even(2)) (1*1 = 1) (2*2 = 2) (3*3 = 3) ... (Odd(3) Even(3)) ... What happens if we negate it? What happens if we negate it? 14 15 2
Generalized De Morgan’s De Morgan’s (for Quantifiers) with Multiple Quantifiers ~ x, P(x) x, ~P(x) Which of the following is equivalent to: ~ n 0 Z 0 , n Z 0 , n > n 0 F(a 1 , a 2 , n). ~ x, P(x) x, ~P(x) a. n 0 Z 0 , ~ n Z 0 , n > n 0 F(a 1 , a 2 , n). b. n 0 Z 0 , n Z 0 , ~(n > n 0 ) F(a 1 , a 2 , n). c. n 0 Z 0 , n Z 0 , ~(n > n 0 F(a 1 , a 2 , n)). (The quantifier changes when a negation d. n 0 Z 0 , n Z 0 , ~(n > n 0 F(a 1 , a 2 , n)). moves across it.) e. n 0 Z 0 , n Z 0 , n > n 0 ~F(a 1 , a 2 , n). P(x) could be anything, of course! 16 17 It can be a “helper predicate”. Which Logical Equivalences Outline Apply? • Prereqs, Learning Goals, and Quiz Notes All of them! • Reminder about the Challenge Method But… we have to be sure to carefully “line up” the • Generalized De Morgan’s Law parts of the logical equivalence with the parts of • Brief Problems and Discussion the logical statement. • Next Lecture Notes 18 19 Problem : Lists (aka Arrays) Outline • Prereqs, Learning Goals, and Quiz Notes Let Length(a, len) mean that list a has the length len. Let A be the set of all arrays. • Reminder about the Challenge Method Problem : Use logical equivalence to show that • Generalized De Morgan’s Law these translations of “an array has exactly one • Brief Problems and Discussion length” are logically equivalent: a A, len N, Length(a,len) • Next Lecture Notes ( len 2 N, Length(a, len 2 ) len = len 2 ). a A, len N, Length(a,len) (~ len 2 N, Length(a, len 2 ) len len 2 ). 20 21 3
Learning Goals: In-Class Learning Goals: Pre-Class Be able for each proof strategy below to: By the start of class, you should be able to: – Identify the form of statement the strategy can prove. – Explore alternate forms of predicate logic – Sketch the structure of a proof that uses the strategy. statements using the logical equivalences you have already learned plus negation of Strategies: constructive/non-constructive proofs of quantifiers (a generalized form of De Morgan’s Law). existence ("witness"), disproof by counterexample, exhaustive proof, generalizing from the generic particular ("WLOG"), direct proof ("antecedent assumption"), proof by contradiction, and proof by cases. 22 23 Alternate names are listed for some techniques. snick Lecture Prerequisites snack In the “Textbook and References” section of the course website: More problems to solve... – Reread the “Rewriting Predicate Logic” sections – Read the “Proof Techniques” sections Complete the open-book, untimed, online (on your own or if we have time) quiz that is due before class. 24 25 Informal Definition: “Independence Why Voting? of Irrelevant Alternatives” (IIA) A voting system is software. It describes how to Philosopher Sidney Morgenbesser is ordering compute a winner from the raw data of marked dessert. The waiter says they have apple and ballots... When [voters, candidates, and blueberry pie. Morgenbesser asks for apple. strategists] are able to use the system to defeat The waiter comes back out and says “Oh, we have the overall will of the voters, blame is properly cherry as well!” laid on the system itself. “In that case,” says Morgenbesser, “I’ll take the - William Poundstone, Gaming the Vote blueberry.” We now play with Arrow’s Impossibility Theorem because > but > > it’s a fascinating proof. But.. Poundstone would remind Huh?? us that there are systems not subject to this theorem! 4
Formal Definition: “Independence General Definition: of Irrelevant Alternatives” (IIA) “Pareto (In)Efficiency” If under a particular set of votes, society If a change in the solution can make prefers A to B, society must still prefer A to everyone better off, then the solution is “Pareto inefficient”. (Used beyond elections!) B if voters rearrange their preferences but maintain their relative rankings of A and B. Question: Which of these is not Pareto Efficient? 1 2 3 4 5 S 1 2 3 4 5 S Candidate (“option”) Key: Voter C B A A D A E C A C D A A A C E E . A E B A B . A B C D E D D D B C . C B C E E . E C E C B . B A D B C . B E B D A B D D E D A B Formal Voting Definition: Formal Definition: “Dictatorship” “Pareto Efficiency” For any two candidates A and B, if everyone In a dictatorship, no matter how everyone votes, society’s preference order precisely prefers candidate A to candidate B, follows one voter’s preference order (even society must prefer A to B. if no one knows who that person is). 1 2 3 4 5 S 1 2 3 4 5 S Candidate (“option”) Key: Voter C B A A D A E C A C D C A A C E E E A E B A B A D D D B C B C B C E E E E C E C B C B A D B C B B E B D A D D D E D A D All hail 4! Arrow’s Impossibility Theorem Arrow’s Impossibility Theorem Arrows Theorem shows, for any voting Which can you prove? system * , if the system exhibits IIA and a. ~ x V, IIA(v) PE(v) ~D(v). Pareto efficiency, then it’s a dictatorship. b. x V, IIA(v) PE(v) ~D(v). Let V be the set of all voting systems and c. x V, D(v) IIA(v) PE(v). IIA(v), PE(v), and D(v) describe the three d. x V, D(v). properties. e. None of these. Problem : Prove using logical equivalences that “there’s no such thing as a fair voting system”. * Technically: ranking-based systems on elections with 2 voters and 3 candidates. 32 33 5
Problem : Bosses Top(x): x is the president Report(x, y): x reports (directly) to y P: the set of all people in the organization Imagine a hierarchical organization in which everyone has exactly one boss except the president. Let the domain of all variables be P. Which of these statements is true in any such organization? x P, y P, Top(x) Report(x, y) A. The first one. y P, x P, Top(x) Report(x, y) B. The second one. C. Both D. Neither 34 E. Not enough info 6
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