snick  snack CPSC 121: Models of Computation 2009 Winter Term 1 Rewriting Predicate Logic Statements Steve Wolfman, based on notes by Patrice Belleville and others 1

Outline Prereqs, Learning Goals, and Quiz Notes Reminder about the Challenge Method Generalized De Morgan’s Law Brief Problems and Discussion Next Lecture Notes 2

Lecture Prerequisites Reread Sections 2.1 and 2.3 (including the negation part that we skipped previously). Read Sections 2.2 and 2.4. Solve problems like Exercise Set 2.1 #25-28 and ; Set 2.2 #1-25 and 27-36; Set 2.3 #13-20, , 40-42, and 45-52; and Set 2.4 #2-19, and (You needn’t learn the “diagram” technique, but it may make more sense than other explanations!) Complete the open-book, untimed quiz on Vista that’s due before the next class. 3

Learning Goals: Pre-Class By the start of class, you should be able to: –Determine the negation of any quantified statement. –Given a quantified statement and an equivalence rule, apply the rule to create an equivalent statement (particularly the De Morgan’s and contrapositive rules). –Prove and disprove quantified statements using the “challenge” method (Epp, 3 d edition, page 99). –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. 4

Learning Goals: In-Class By the end of this unit, you should be able to: –Explore alternate forms of predicate logic statements using the logical equivalences you have already learned plus negation of quantifiers (a generalized form of De Morgan’s Law). 5

Quiz 6 Notes 6

Outline Prereqs, Learning Goals, and Quiz Notes Reminder about the Challenge Method Generalized De Morgan’s Law Brief Problems and Discussion Next Lecture Notes 7

Reminder: Challenge Method A predicate logic statement is like a game with two players: you (trying to prove the statement true) your adversary (trying to prove it false). The two of you pick values for the quantified variables working from the outside (left) in. Your adversary picks the values of universally quantified variables. You pick the values of existentially quantified variables. 8

Challenge Method Continued 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. 9

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)  y  P,  x  P, Top(x)  Report(x, y) A.The first one. B.The second one. C.Both D.Neither E.Not enough info 10

Reminder: Challenge Method Your adversary picks universally quantified variables. You pick existentially quantified variables. If you change the turn order (by switching an existential/universal), you may change the way the game works! If you swap two neighbouring existentials, you change what order you announce your choices in but don’t change the strategies. (And similarly for universals.) 11

Outline Prereqs, Learning Goals, and Quiz Notes Reminder about the Challenge Method Generalized De Morgan’s Law Brief Problems and Discussion Next Lecture Notes 12

De Morgan’s Law and Negating Quantifiers Consider the statement:  x  Z +, Odd(x)  Even(x) This is essentially an infinitely big AND: (Odd(1)  Even(1))  (Odd(2)  Even(2))  (Odd(3)  Even(3)) ... What happens if we negate it? 13

De Morgan’s Law and Negating Quantifiers Consider the statement:  x  Z +, x*x = x. This is essentially an infinitely big OR: (1*1 = 1)  (2*2 = 2)  (3*3 = 3) ... What happens if we negate it? 14

Generalized De Morgan’s (for Quantifiers) ~  x, P(x)=  x, ~P(x) ~  x, P(x) =  x, ~P(x) (The quantifier changes when a negation moves across it.) 15

De Morgan’s with Multiple Quantifiers What can we do with the negation on a statement like: ~  n 0  Z 0,  n  Z 0, n > n 0  Faster(a 1, a 2, n) a.The negation cannot be moved inward. b.The negation can only move across one quantifier because the Generalized De Morgan’s rule only handles one quantifier. c.The negation could be moved across the existential or across both of the quantifiers. d.The negation must be moved across both quantifiers because a negation cannot appear between quantifiers. e.None of these. 16

Outline Prereqs, Learning Goals, and Quiz Notes Reminder about the Challenge Method Generalized De Morgan’s Law Brief Problems and Discussion Next Lecture Notes 17

Which Logical Equivalences Apply? Which propositional logic equivalences apply to predicate logic? (Answers taken from your quiz notes.) a.Modus ponens, modus tollens, and De Morgan's (not all equivalences!) b.~(P(x)  Q(x))  P(x)  ~Q(X) c.Commutative, Associative, and “definition of conditional” d.All propositional logic equivalences apply to predicate logic. e.Some other answer is correct. 18

Problem: Lists (aka Arrays) Let Length(a, len) mean that list a has the length len. Let A be the set of all arrays. Problem: Use logical equivalence to show that these translations of “an array has exactly one length” are logically equivalent:  a  A,  len  Z 0, Length(a,len)  (  len 2  Z 0, Length(a, len 2 )  len = len 2 ).  a  A,  len  Z 0, Length(a,len)  (~  len 2  Z 0, Length(a, len 2 )  len  len 2 ). 19

Outline Prereqs, Learning Goals, and Quiz Notes Reminder about the Challenge Method Generalized De Morgan’s Law Brief Problems and Discussion Next Lecture Notes 20

Learning Goals: In-Class By the start of class, you should be able to: –Explore alternate forms of predicate logic statements using the logical equivalences you have already learned plus negation of quantifiers (a generalized form of De Morgan’s Law). 21

Learning Goals: “Pre-Class” After the Ch 3 readings and lecture through slide “A New Proof Strategy ‘Antecedent Assumption’” of the next slide set, you should be able for each proof strategy below to: (1) identify the form of statement the strategy can prove and (2) sketch the structure of a proof that uses the strategy. Strategies: constructive/non-constructive proofs of existence (“witness” proofs), disproof by counter- example, exhaustive proof, generalizing from the generic particular (“WLOG”), direct proof (“antecedent assumption”), proof by contradiction, and proof by cases. Alternate names are listed for some techniques. 22

Lecture “Prerequisites” Reread Ch 2 and be able to solve any of the problems in that chapter. Read Sections 3.1, Theorem and pages (“Representations of Integers”), 3.6, and 3.7. Be able to solve (or at least outline the proof structure of) problems like: , (20-23 are particularly relevant!), 3.4 #24-27, 29-30, and 49-53, and 3.6 #1-27 (19-20 are particularly relevant!). Complete the open-book, untimed quiz on Vista that’s due after some lecture! 23

snick  snack More problems to solve... (on your own or if we have time) 24

Why Voting? A voting system is software. It describes how to compute a winner from the raw data of marked ballots... When [voters, candidates, and strategists] are able to use the system to defeat the overall will of the voters, blame is properly laid on the system itself. - William Poundstone, Gaming the Vote We now play with Arrow’s Impossibility Theorem because it’s a fascinating proof. But.. Poundstone would remind us that there are systems not subject to this theorem!

Informal Definition: “Independence of Irrelevant Alternatives” (IIA) Philosopher Sidney Morgenbesser is ordering dessert. The waiter says they have apple and blueberry pie. Morgenbesser asks for apple. The waiter comes back out and says “Oh, we have cherry as well!” “In that case,” says Morgenbesser, “I’ll take the blueberry.” > but > > Huh??

Formal Definition: “Independence of Irrelevant Alternatives” (IIA) If under a particular set of votes, society prefers A to B, society must still prefer A to B if voters rearrange their preferences but maintain their relative rankings of A and B.

General Definition: “Pareto (In)Efficiency” If a change in the solution can make everyone better off, then the solution is “Pareto inefficient”. (Used beyond elections!) EDCBA Candidate (“option”)VoterKey: Question: Which of these is not Pareto Efficient?

Formal Voting Definition: “Pareto Efficiency” For any two candidates A and B, if everyone prefers candidate A to candidate B, society must prefer A to B. Candidate (“option”)VoterKey:

Formal Definition: “Dictatorship” In a dictatorship, no matter how everyone votes, society’s preference order precisely follows one voter’s preference order (even if no one knows who that person is). All hail 4!

Arrow’s Impossibility Theorem Arrows Theorem shows, for any voting system *, if the system exhibits IIA and Pareto efficiency, then it’s a dictatorship. Let V be the set of all voting systems and IIA(v), PE(v), and D(v) describe the three properties. Problem: Prove using logical equivalences that “there’s no such thing as a fair voting system”. 31 * Technically: ranking-based systems on elections with  2 voters and  3 candidates.

Arrow’s Impossibility Theorem Which can you prove? a.~  x  V, IIA(v)  PE(v)  ~D(v). b.  x  V, IIA(v)  PE(v)  ~D(v). c.  x  V, D(v)  IIA(v)  PE(v). d.  x  V, D(v). e.None of these. 32