Computing with Words Principal Concepts and Ideas
In essence, Computing with Words (CWW) is a system of computation in which the objects of computation are predominantly words, phrases and propositions drawn from a natural language. CWW is based on fuzzy logic. In science there is a deep-seated tradition
- PDF / 204,226 Bytes
- 17 Pages / 429.442 x 659.895 pts Page_size
- 83 Downloads / 255 Views
COMPUTATION WITH RESTRICTIONS z
In CWW, through representation of the meaning of a proposition as a restriction, the problem of computation with information described in natural language reduces to the problem of computation with restrictions. As was noted earlier, in the realm of natural languages restrictions are for the most part possibilistic.
137/263
L.A. Zadeh: Computing with Words, STUDFUZZ 277, p. 73–89. springerlink.com © Springer-Verlag Berlin Heidelberg 2012
74
Phase 2—Computation
For this reason, in the following attention is focused on computation with possibilistic restrictions.
138/263
FROM PRECISIATION TO COMPUTATION Phase 1 p1 I
p1* : X1 isr1 R1
. .
pn-1 Pn q
. .
precisiation
I* pn-1*: X isr R n-1 n-1 n-1 pn*: X isr R n n n q*
Phase 2 X1 isr1 R1 I*
. .
Xn-1 isrn-1 Rn-1 Xn isrn Rn q*
139/263
computation with restrictions
ans(q/I)
Phase 2—Computation
z
In large measure, computation with restrictions (generalized constraints) involves the use of rules which govern propagation and counterpropagation of restrictions (Calculus of fuzzy restrictions, Zadeh 1974). Among such rules, the principal rule is the extension principle (Zadeh 1965, 1975a, 2011).
140/263
z
There are many versions of the extension principle. Let Y be a function of X1, …, Xn, Y=f(X1, …, Xn). Basically, an extension principle is a rule which governs computation of the restriction on Y, R(Y), given restrictions on X1, …, Xn.
141/263
75
76
Phase 2—Computation
NOTE z
The calculi of fuzzy if-then rules in Level 1 CWW may be viewed as special cases of propagation of possibilistic restrictions.
142/263
EXTENSION PRINCIPLE (POSSIBILISTIC) z
z
X is a variable which takes values in U, and f is a function from U to V. The point of departure is a possibilistic restriction on f(X) expressed as f(X) is A, where A is a fuzzy set in V which is defined by its membership function μA(v), vεV. g is a function from U to W. The possibilistic restriction on f(X) induces a possibilistic restriction on g(X) which may be expressed as g(X) is ?B, where B is a fuzzy set in W. The question is: What is B? In symbols,
143/263
Phase 2—Computation
f(X) is A g(X) is ?B The answer to this question is the solution of a variational problem expressed as:
μ B ( w ) = supu μ A ( f ( u )) subject to
w = g( u ) where µA and µB are the membership functions of A and B, respectively. 144/263
z
Equivalently, the possibilistic extension principle may be expressed as: ?Y=g(X) f(X) is A
μY ( v ) = supu μ A ( f ( u )) subject to
v = g( u ) 145/263
77
78
Phase 2—Computation
NOTE z
In a more general setting, the extension principle is concerned with propagation of restrictions—restrictions on both functions and their arguments. Schematically,
Z = g(X,Y) z
There are many different ways in which X, Y and g may be restricted. Only a few have been explored.
146/263
A SUMMARY OF COMPUTATION OF ans(q/I) z
For convenience, in the following the information set, I, is represented as a composite proposition, p=(p1, …, pn).
147/263
Phas
Data Loading...
