Fri Sep 30, 2011
60 Evans Hall, 4:10–6 PM
Lotfi A. Zadeh (University of California, Berkeley)
Can Mathematics Deal with Computational Problems Which Are Stated in a Natural Language?
The biweekly LOGIC TEA will be held in the Alfred Tarski Room (727 Evans Hall) immediately following the Colloquium (with support from the Graduate Assembly).
Here are a few very simple examples of computational problems which are stated in a natural language. (a) Most Swedes are tall. What is the average height of Swedes? (b) Probably John is tall. What is the probability that John is short? What is the
probability that John is very short? What is the probability that John is not
very tall? (c) Usually Robert leaves his office at about 5 pm. Usually it takes Robert about an hour to get home from work. At what time does John get home? (d) X is a real-valued random variable. Usually X is much larger than approximately a. Usually X is much smaller than approximately b. What is the probability that X is approximately c, where c is a number between a and b? (e) A and B are boxes, each containing 20 balls of various sizes. Most of the balls in A are large, a few are medium, and a few are small. Most of the balls in B are small, a few are medium, and a few are large. The balls in A and B are put into a box C. What is the number of balls in C which are neither large nor small? For convenience, such problems will be referred to as CNL problems.
It is a long-standing tradition in mathematics to view computational problems which are stated in a natural language as being outside the purview of mathematics. Such problems are dismissed as ill-posed and not worthy of attention. In the instance of CNL problems, mathematics has nothing constructive to say. In my lecture, this tradition is questioned and a system of computation is suggested which opens the door to construction of mathematical solutions of CNL problems. The system draws on the fuzzy-logic-based formalism of computing with words (CW). (Zadeh 2006) A concept which plays a pivotal role in CW is that of precisiation of meaning. More concretely, precisiation involves translation of natural language into a mathematical language in which the objects of computation are well-defined — though not conventional — mathematical constructs.
A key idea involves representation of the meaning of a proposition, p, drawn from a natural language, as a restriction on the values which a variable, X, can take. Generally, X is a variable which is implicit in p. The restriction is represented as an
expression of the form X isr R, where X is the restricted variable, R is the restricting relation and r is an indexical variable which defines the way in which R restricts X. This expression is referred to as the canonical form of p. Canonical forms of propositions in a natural language statement of a computational problem serve as objects of computation in CW.
Construction and use of mathematical solutions of CNL problems is an unexplored domain in mathematics. The importance of this domain derives from the fact that much of human knowledge, and particularly world knowledge, is described in natural language.