Logic Puzzles – Continued

1. Logic Puzzles – Continued AE1APS Algorithmic Problem Solving John Drake

2. Outline of the course • Invariants – Chapter 2 • River Crossing – Chapter 3 • Logic Puzzles – Chapter 5 (13) • Knights and Knaves • Portia’s Casket • Matchstick Games - Chapter 4 • Sum Games – Chapter 4 • Induction – Chapter 6 • Tower of Hanoi – Chapter 8

3. A reminder of Boolean operators • Conjunction: X ∧ Y (AND) • Disjunction: X ∨ Y (OR) • Negation (or complement): ¬X (NOT) • Inequivalence:≢ (DIFFERENT) • Implication: X⇒ Y, (X implies Y)

4. Boolean algebra • Boolean expressions are either true or false. • Boolean valued expressions are called propositions. • “it is sunny” is an atomic Boolean expression. • “it is sunny and warm” is non atomic as it can be broken down into two expressions. • We are concerned with the rules for manipulating Boolean expressions.

5. Equals or equivales? • = or ≡ ? • In which order should x = y = z be processed? • As an example, if x = true, y = false and z = false. • If read continually - true = false = false evaluates to false

6. Equals or equivales? • However if read associatively – (x = y) = z or x = (y = z) it will evaluate to true • To remove this confusion we use ≡ to indicate that an expression should be considered associatively. • The property of associativity is very powerful!

7. Negation and Inequivalence • Negation: [¬ x≡ x ≡ false] p.106 • Inequivalance: x ≢ y or ¬(x ≡ y) p.113 [¬(x≡ y) ≡ x≡ ¬y]

8. Boolean Equality. • Equality is a binary relation. • It is a function with a range of Boolean values true and false. • Equality is reflexive: [x≡ x] • It is symmetric: [x≡ y is the same as y ≡ x]

9. Boolean Equality. • It is transitive: [x ≡ y and y ≡ z implies z ≡ x] • It is associative: • [(x ≡ y) ≡ z is the same as x ≡ (y ≡ z) ] • It is substitutive: [x ≡ (y ≡ z) can be replaced by (y ≡ z) ≡ (y ≡ z) ]

10. Another example to think about… • Given a pair of natives, what question would you ask one to discover if the other is a knight? I’m a Knight I’m a Knight

11. Knights and Knaves • If A is the proposition: “person A is a knight" and suppose the native makes a statement S. • We can infer that A is true is the same as S is true. That is, A ≡ S

12. Knights and Knaves • If native A is asked a yes/no question Q then the response to the question is: A ≡ Q • Let R be the desired response when we ask question Q: i.e. R ≡ (A≡ Q)

13. Knights and Knaves R ≡ (A ≡ Q) • A – The statement “A is a knight” • Q – The response to a given question • R – The desired response from the native

14. Knights and Knaves R ≡ (A ≡ Q) • A – The statement “A is a knight” • We can ask • Q – The response to a given question • What we are trying to find • R – The desired response from the native • “Is B a knight?” – we will call this proposition B

15. Knights and Knaves • So, B ≡ (A ≡ Q) • A and B are statements, • Q is the question to be asked • As equality is associative • B ≡ (A ≡ Q) becomes (B ≡ A)≡ Q • As equality is symmetric • (B ≡ A) ≡ Q becomes Q ≡ (A ≡ B) • So we ask the native is (A ≡ B)? • “Is the statement of your being a knight equivalent to the statement of B being a knight”

16. Portia’s Casket • In Shakespeare's Merchant of Venice, Portia had three caskets: gold, silverand lead. • Inside one of these caskets Portia had put her portrait and on each was an inscription.

17. Portia’s Casket • Portia explained to her suitor that each inscription could be either true or false but on the basis of the inscriptions he was to choose the casket containing the portrait. • If he succeeded he could marry her. • Here we will consider a simpler variant of this problem using only two caskets

18. Portia’s Casket • Suppose there are two caskets, gold and silver, into one of which Portia placed her portrait. • The inscriptions are: • Gold: The portrait is not in here. • Silver: Exactly one of these inscriptions is true. • Which casket contains the portrait?

19. Portia’s Casket • Let pg stand for “the portrait is in the gold casket“ • Let ps stand for “the portrait is in the silver casket“ • Let igstand for “the inscription on the gold casket is true" • and let is stand for “the inscription on the silver casket is true“

20. Portia’s Casket • What do we know? • If the inscription on the gold casket is true then the portrait is in the silver casket • i.e. ig≡ ps • As there is only one portrait pg and ps cannot both be true • i.e. pg≢ ps • The inscriptions are: • Gold: The portrait is not in here. • Silver: Exactly one of these inscriptions is true.

21. Portia’s Casket • What do we know? • For one statement to be true ig≡ ¬is • So for is to be true this condition must hold i.e. is ≡ (ig≡ ¬is) • The inscriptions are: • Gold: The portrait is not in here. • Silver: Exactly one of these inscriptions is true.

22. Portia’s Casket • This leaves us with three statements to help us decide which casket the portrait is in. • (ig≡ ¬pg)∧ (pg≢ ps) ∧ (is ≡ ig≡ ¬is) • The inscriptions are: • Gold: The portrait is not in here. • Silver: Exactly one of these inscriptions is true.

23. Portia’s Casket (ig≡ ps)∧ (pg≢ ps) ∧ (is ≡ ig≡ ¬is) Using the definition of negation and inequivalance (ig≡ ps)∧ (pg≡ ps≡ false) ∧ (is ≡ ig≡ is ≡ false) Using reflexivity of equality [(is ≡ is) ≡ true] (ig≡ ps)∧ (pg≡ ps≡ false) ∧ (ig≡ false) Equality is substitutive, then use reflexivity (false ≡ ps)∧ (pg≡ true) ∧ (ig≡ false)

24. Portia’s Casket • What can we deduce? • psis false • pgis true • igis false • So the portrait is in the Gold casket • Note: we know nothing about the inscription on the Silver casket

25. Finally • Please e-mail me with any questions or problems you are having • Tutorial session this afternoon at 1pm • Hopefully extra slots start next week • John.drake@nottingham.edu.cn