Thurs., March 4, 1999

1. Thurs., March 4, 1999 • Homework 2 due • Handout Homework 3 (Due Thurs., March 18) • Turing Test • Multivariate Logic and Event Spaces • Week 8: Subjective Probabilities • Week 9: Bayes Nets/Influence Diagrams

2. Multivariate Logic • Venn Diagrams • Axioms of Event Algebra • Theorems of Event Algebra • Mutually Exclusive and Collectively Exhaustive Events • Exercises

3. Alan Turing(the Enigma) • In 1950, Alan Turing, the brilliant British mathematician wrote an article "Computing Machinery and Intelligence" which appeared in the philosophical journal Mind.

4. Turing asked the question "Can a Machine Think?" He argued "yes", but "If a computer could think, how could we tell?" The Turing Test states that a computer is intelligent if a person cannot tell whether he or she is interacting with a computer or another person. The person testing the program sits at a terminal asking and answering questions of the person or computer at the other end. Turing Test

5. Loebner Prize • The Loebner Prize of $100,000 is still available for anyone who can design a program that successfully passes the Turing Test.

6. Introduction to Multivariate Logic • In multivariate logic, the atomic expressions no longer evaluate to either "true" or "false” as they do in classical logic. • In fact, they may evaluate to many events each having a measure of "trueness" associated with it.

7. Sample Spaces • We will use the concept of sample space to introduce the concept of multiple but well defined events in multivariate logic. • A sample space is a generalized collection of points representing elementary events. These elementary events may be conveniently combined into collections or sets of events.

8. Venn Diagrams • Venn diagrams are graphical representations of sample spaces, elementary events, and event sets. • Each event is assigned an area in the sample space, corresponding to its proportion of elementary events.

9. Venn Diagram Example • Because the area of an event in a Venn diagram corresponds to the number of elementary events it contains, this implies that event B is a larger set than event A, which is in turn a larger set than event C. We also observe that events A and B share some elementary events in common as do events A and C. In this diagram the event sets are A, B, and C.

10. Universe and Null Set • The set of all elementary events is called the universe and is designated as " I". • A set which contains no elementary events is called the null set and is designated as "Æ”or “{}”.

11. Complementation • The complement of an event A is defined as the set of all events in the sample space that does not include any of the elementary events in A. • The complement of A is designated as A' or an overscored A.

12. Intersection and Union • The union of two events A and B is the collection of all elementary events contained in A or B or both. The union is designated as A + B or A È B. • The intersection (or product) of events A and B is designated as AB or AÇ B. It has all elementary events in both A and B .

13. Axioms of Event Algebra • The following list of seven axioms provides a logical foundation for this event algebra. 1. A + B = B + A 2. (A')' = A 3. A + (B + C) = (A + B) + C 4. AB = (A' + B')' or (AB)' = A' + B' 5. A + (BC) = (A + B)(A +C) 6. AA' = Æ 7. AI= A

14. All of these theorems can be derived from the seven axioms: 1. AB = BA 2. A(BC) = (AB)C 3. (A'B')' = A + B or A'B' = (A+B)' 4. A(B+C) = AB + AC 5. A + A' = I 6. A + Æ = A 7. AÆ = Æ 8. A + I = I 9. A + A = A 10. AA = A 11. A + AB = A 12. A + A'B = A + B Theorems of Event Algebra

15. Mutually Exclusive and Collectively Exhaustive Events • Events in a sample space are mutually exclusive if none of the events intersect one another – are no elementary events that are contained in more than one event. • Events are collectively exhaustive if every elementary element is contained in at least one event set. • A sample space may consist of events that are both mutually exclusive and collectively exhaustive. Mutually Exclusive Collectively Exhaustive Both Mutually Exclusive and Collectively Exhaustive

16. For example, the union of each of the following mutually exclusive event sets is equal to the set A + B. Although two events may not be mutually exclusive, we can always represent some combination of these events in mutually exclusive form. Mutually Exclusive and Collectively Exhaustive Form

17. Exercises • Prove theorem 6 of event algebra using only the seven axioms given in section 2 and substitutions or name changes. • Prove A + Æ = A

18. Axioms of Event Algebra • The following list of seven axioms provides a logical foundation for this event algebra. 1. A + B = B + A 2. (A')' = A 3. A + (B + C) = (A + B) + C 4. AB = (A' + B')' or (AB)' = A' + B' 5. A + (BC) = (A + B)(A +C) 6. AA' = Æ 7. AI= A

19. Theorem 6:Prove A + Æ = A • A I = A Axiom 7 • A' + I' = A' Axiom 4 • A' + Æ = A' I'=Æ definition • A + Æ = A Change names

20. Prove (A'B')' = A + B • (AB)' = A' + B' Axiom 4 • (A'B')' = A + B Change names