Artificial Intelligence CIS 342 The College of Saint Rose David Goldschmidt, Ph.D.

Uncertainty Uncertainty is the lack of exact knowledge that would enable us to reach a fully reliable solution – Classical logic assumes perfect knowledge exists: IFA is true THENB is true Describing uncertainty: – If A is true, then B is true with probability P

Expert knowledge often uses vague and inexact terms Fuzzy Logic describes fuzziness by specifying degrees – e.g. degrees of height, speed, distance, temperature, beauty, intelligence, etc. Fuzzy Logic

Boolean logic uses sharp distinctions – e.g. temperatures above 85  are “hot”; temperatures less than 85  are “cold” Fuzzy logic attempts to smooth such sharp distinctions between terms – Use real numbers between 0 and 1 to represent the possibility that a given statement is true or false Fuzzy Logic

Concept of a continuum – 1937 paper: “Vagueness: an exercise in logical analysis” (Max Black) – Identify vagueness as a matter of probability Fuzzy Logic

1965 paper: “Fuzzy Sets” (Lotfi Zadeh) – Apply natural language terms to a formal system of mathematical logic Fuzzy Logic is a set of mathematical principles for knowledge representation based on degrees of membership Fuzzy Logic

Unlike Boolean logic, fuzzy logic is multi-valued – Fuzzy logic represents degrees of membership and degrees of truth – Things can be part true and part false at the same time Fuzzy Logic

Fundamental to mathematics, a set is a collection of distinct objects – A fuzzy set is a set whose elements have varying degrees of membership Fuzzy Sets

A comparison of crisp and fuzzy sets depicting height Fuzzy Sets

A crisp (or Boolean ) set is too sharp – Low applicability to real-world knowledge/concepts Fuzzy Sets I’m tall!I’m short?

A fuzzy set provides a natural fit – High applicability to real-world knowledge/concepts Fuzzy Sets

X-axis is the universe of discourse, all possible values Y-axis is the degree of membership Fuzzy Sets

Let X be the universe of discourse – Denote its elements as x – In classical set theory, crisp set A over X is defined by function f A ( x ), the characteristic function of A f A ( x ): X → {0, 1} where f A ( x ) = Fuzzy Sets 1, if x  A 0, if x  A

Let X be the universe of discourse – Denote its elements as x – In fuzzy set theory, fuzzy set A over X is defined by function  A ( x ), the membership function of A  A ( x ): X → [0, 1] where  A ( x ) = 1, if x is entirely in A  A ( x ) = 0, if x is not in A 0 <  A ( x ) < 1, if x is partly in A Fuzzy Sets

Representing Fuzzy Sets

Representing height using three crisp sets: Representing Fuzzy Sets

Representing height using three fuzzy sets: Representing Fuzzy Sets

What’s the degree of membership for Steven and Bob in each fuzzy set? In-Class Exercise write a function or method to calculate degree of membership ( HINT : use analytic geometry)