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AE1APS Algorithmic Problem Solving John Drake 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. 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. What happens if you ask a person if they are a knight or a knave? I’m a Knight I’m a Knight There may be gold on the island. What question could you ask to establish this? 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. 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. 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) ] “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 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 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) 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. 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 Given a pair of natives, what question would you ask to discover if the other one is a knight? 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? 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) {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’” 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. 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?” Never trust a knave Equality is reflexive, symmetric, transitive, substitutive and 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/ [email protected] http://en.wikipedia.org/wiki/Knights_and_Knaves