CS451/551 Artificial Intelligence Spring 2002 Assignment #4 due: Friday, 4/19/2002 The Assignment: -------------- Represent the following as first order clauses, and prove them using OTTER. 1. There is somebody who loves everybody and is loved by everybody. Love is transitive (i.e., if a person loves somebody else, and that person loves a third person, then the first person loves the third person) Prove that everybody loves everybody. 2. Everybody has a parent. Everybody is male or female. A male parent of anybody is the person's father. A female parent of anybody is the person's mother. Prove that everybody has a mother or a father. 3. Note: This problem deals with integers > 1. It is actually a proof that there are infinitely many primes, because if there are only finitely many primes, then there is a number that is the product of all the primes. When I say that a number divides another number, I mean that it is a smaller number that divides into it evenly. For example, 6 divides 24. Also, the successor of a number is the next number. For example, 15 is the successor of 14. Facts ----- If a number divides another number, then it is smaller than that number. (Example: 6 divides 24, so 6 is smaller than 24) A number is smaller than its successor. (Example: 14 is smaller than 15) No number is less than itself. (Example: 5 is not less than 5) Less than is transitive. (Example: 13 < 14 and 14 < 15. Therefore 13 < 15) If a number divides another number, then it does not divide the successor of the number. (Example: 2 divides 14, therefore 2 does not divide 15) If a number is not prime, then some prime number divides it. (Example: 15 is not prime, so 3 divides 15) Every number has a successor. (This is necessary if successor is viewed as a predicate, but not necessary if successor is viewed as a function.) To prove -------- There is no number that is a product of all the primes.