Logic Puzzles - Knights and Knaves

1. Logic Puzzles - Knights and Knaves AE1APS Algorithmic Problem Solving John Drake

2. The island of Knights and Knaves. • The island of Knights and Knaves is a fictional island to test peoples’ ability to reason logically. • There are two types of people on the island. • Knights who always tell the truth • Knaves who always lie.

3. Knaves?

4. The island of Knights and Knaves. • Logic puzzles are about deducing facts about the island, from statements made by the people on the island. But you do not know whether the statement was made by a knight or a knave.

5. Some examples • What happens if you ask a person if they are a knight or a knave? • There may be gold on the island. What question could you ask to establish this? I’m a Knight I’m a Knight

6. Algebra • At school we learnt to manipulate expressions. The values of expressions are numbers • n2–m2= (n + m)(n - m) • To do this we use laws. • e.g. n + 0 = n and n – n = 0 (for ANY values of n) • Associatively of addition • (m + n) +p = m + (n + p) • I am sure you are all familiar with laws regarding addition, multiplication.

7. 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.

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] • It is transitive: [x ≡ y and y ≡ z implies z ≡ x] • If x ≡ y, then f(x) ≡ f(y) • 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) ]

9. Knights and knaves. • “A” is a native on the island. • Therefore A is either a knight or a knave. • The statement “A is a knight” is either true or false. • The statement “there is gold on the island” is either true or false. • Suppose “A is a knight”, and suppose A makes statement S. • The crucial observation is that the values of these two propositions are the same. • Then “A is a knight” ≡ S

10. More formally… • Suppose A is the proposition “person A is a knight" and suppose A makes a statement S. We can infer that A is true is the same as S is true. That is, • A≡S

11. Examples • If A says “I am a knight” then all we can infer from the statement is A ≡ A. Not much use! • Similarly, it cannot be that a native says “I am a knave” because we would then conclude A≡ ¬A which is always false. • If A says “I am the same type as B". We infer that A ≡ (A ≡ B) which simplifies to B. ¬ Boolean symbol for negation (i.e. NOT A)

12. Going for gold • A native says “there is gold on this island, is the same as, I am a knight”. • Let G denote the proposition “There is gold on the island". A's statement is A ≡ G. So what we are given is: A ≡ A ≡ G • This simplifies to G. So we deduce that there is gold on the island but it is not possible to tell whether A is a knight or a knave.

13. In natural language • If A is a knight, he will always tell the truth so we will know there is gold on the island. • If A is a knave he will always lie so when asked if A ≡ G (i.e. Are the answers to “Are you a knight?” and “Is there gold on the island?” the same), he will answer true when there is gold on the island and false if not. • If not clear will go through case analysis using truth tables

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

15. Getting directions • A tourist comes to a fork in a road, one way leads to a restaurant, the other does not. What question (yes/no) would you ask a native to find out which way it is to the restaurant?

16. What question to ask? • Let Q be the question to be posed. • The response to the question will be A ≡ Q • Let L denote, “the gold can be found by following the left fork” • The requirement is that L is the same as the response to Q. i.e. we require L≡ (A ≡ Q)

17. What question to ask? • {but as equality is associative} • (L ≡A) ≡Q • So the question Q posed is L ≡ A • i.e. “is the value of ‘the restaurant can be found by following the right fork’ equal to the value of ‘you are a knight’”

18. Equals for Equals • There are three natives A, B and C. • C says “A and B are both the same type” • Formulate a question, that when posed to A determines if C is telling the truth.

19. Equals for equals • Let A be the statement “A is a knight”. • Let Q be the unknown question. • The response we want is C (i.e. if C is true then C is a knight). • By the previous example, Q ≡ (A ≡ C) i.e. we replace L by C. • C’s statement is A ≡ B, so now we know C ≡ (A ≡ B) by equality. • So Q ≡ (A ≡ (A ≡ B)) which simplifies to Q ≡ B, Therefore the question to be posed is “is B a knight?”

20. Key points • Never trust a knave • Equality is reflexive, symmetric, transitive, substitutiveand associative – know these! • More knights and knaves on Thursday • Coursework 1, 2 and 3 now online, deadline for Coursework 3 is Thursday 15th November at 1pm • http://www.cs.nott.ac.uk/~jqd/AE1APS-1213/ • John.drake@nottingham.edu.cn • http://en.wikipedia.org/wiki/Knights_and_Knaves