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

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

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.Alan Turing,

Turing Test n Turing asked the question "Can a Machine Think?" He argued "yes", but "If a computer could think, how could we tell?" n 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.

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

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

Sample Spaces n We will use the concept of sample space to introduce the concept of multiple but well defined events in multivariate logic. n 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.

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

Venn Diagram Example n 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.

Universe and Null Set n 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 "  ”or “{}”.

Complementation n 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. n The complement of A is designated as A' or an overscored A.

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  B. The intersection (or product) of events A and B is designated as AB or A  B. It has all elementary events in both A and B.

Axioms of Event Algebra n 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' =  7.AI = A

Theorems of Event Algebra n 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 +  = A 7.A  =  8.A + I = I 9.A + A = A 10. AA = A 11. A + AB = A 12. A + A'B = A + B

Mutually Exclusive and Collectively Exhaustive Events n 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. n Events are collectively exhaustive if every elementary element is contained in at least one event set. n 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

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

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

Axioms of Event Algebra n 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' =  7.AI = A

Theorem 6: Prove A +  = A n A I = AAxiom 7 n A' + I' = A'Axiom 4 A' +  = A'I'=  definition A +  = AChange names

Prove (A'B')' = A + B n (AB)' = A' + B'Axiom 4 n (A'B')' = A + BChange names