1. AE1APS Algorithmic Problem Solving John Drake

2.  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.  Conjunction: X ∧ Y (AND)  Disjunction: X ∨ Y (OR)  Negation (or complement): ¬X (NOT)  Inequivalence: ≢ (DIFFERENT)  Implication: X ⇒ Y, (X implies Y)

4.  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.  = 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.  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: [¬ x ≡ x ≡ false] p.106  Inequivalance: x ≢ y or ¬(x ≡ y) p.113 [¬(x ≡ y) ≡ x ≡ ¬y ]

8.  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.  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.  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.  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.  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. 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. 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.  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.  In Shakespeare's Merchant of Venice, Portia had three caskets: gold, silver and lead.  Inside one of these caskets Portia had put her portrait and on each was an inscription.

17.  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.  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.  Let pg stand for “the portrait is in the gold casket“  Let ps stand for “the portrait is in the silver casket“  Let ig stand for “the inscription on the gold casket is true"  and let is stand for “the inscription on the silver casket is true“

20.  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.  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.  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. (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.  What can we deduce?  ps is false  pg is true  ig is false  So the portrait is in the Gold casket  Note: we know nothing about the inscription on the Silver casket

25.  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