snick  snack CPSC 121: Models of Computation 2008/9 Winter Term 2 Rewriting Predicate Logic Statements Steve Wolfman, based on notes by Patrice Belleville and others

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.

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.

Learning Goals: In-Class By the end of this unit, you should be able to: –Reinforce a couple of pre-class goals. –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).

Quiz 6 Notes Generally good. Where there were problems, there’s no particular pattern I can address. Please review the problems and post any questions on WebCT! Problems with lowest marks: Universal Instantiation (the “curiosity” question), Determining Negation 3 (the “pigeon” question), and Equivalence Rules and Predicate Logic (the “sock” question).

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.

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.

Problem: Bosses Top(x): x is the president Report(x, y): x reports 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 every such organization?  x,  y, Top(x)  Report(x, y)  y,  x, Top(x)  Report(x, y) A.The first one. B.The second one. C.Both D.Neither E.Insufficient information

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.)

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?

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?

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.)

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

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.

Learning Goals: In-Class By the start of class, you should be able to: –Reinforce a couple of pre-class goals. –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).

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”), disproof by counterexample, exhaustive proof, generalizing from the generic particular (“WLOG”), direct proof (“antecedent assumption”), proof by contradiction, and proof by cases. Note: most names listed are Epp’s, although I list my names for the same or closely related techniques in parentheses.

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! Note: these are not exactly “prerequisites” this week. Please complete the readings after the first day of the nextlecture.