C++ Neural Networks and Fuzzy Logic:Applications of Fuzzy Logic

C++ Neural Networks and Fuzzy Logic
by Valluru B. Rao
M&T Books, IDG Books Worldwide, Inc.
ISBN: 1558515526 Pub Date: 06/01/95

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The result of the operation:

PROJECT (R1 over D1) with LEVEL(D1) = 0.6

is the relation R2 given in Table 16.5.

**Table 16.5** Relation R2, which is the Result of PROJECT Operation

D1
{Georgette, Ernie}
{Georgette, Grace}
Darrell

This projection operation with a condition on the level works as follows. First, the column for D1 is to be picked, of the two columns that R1 shows. Repetitions are removed. The condition says that if two elements of D1 have a similarity of 0.6 or higher, they should be treated as the same. Even though similarity levels of pairs (Georgette, Ernie) and (Georgette, Grace) are both greater than 0.6, the pair (Ernie, Grace) has similarity of 0.4 only so that we do not treat the three as the same.

This type of table is not constructed in the standard database model, so this is part of an extension of the standard model.

Suppose you recast the information in the relation R1 and call it R3, shown in Table 16.6.

**Table 16.6** Example for Relation R3, with Domains D1 and D2

D1D2
Georgette	{Spanish, Italian}
Darrell	{French, Spanish, Japanese}
Grace	{Chinese, French}
Ernie	{Russian, Spanish}

This kind of a table also is not found in standard databases (where there are groups with more than one element used in the relation), and is an example of an extended model.

Possibility Distributions

As an alternative to using similarity relations for introducing fuzziness into a database model, you can, following Umano, et al., use a possibility distribution-relational model. The possibility distributions represent the fuzzy values of attributes in the data. An example of a possibility distribution is the fuzzy set you saw before, nov_rarely = {0.7/1, 0.2/2}, where nov_rarely stands for number of visits considered to be “rarely.”

Example

An example of a database on the lines of this model is shown in Table 16.7:

**Table 16.7** Example of Possibility Distribution Relational Model

Name	Number of Visits Outside the U.S.	Citizenship	Name of Companion on Latest Visit
Peter	3	U.S.	Barbara
Roberto	{10, 11}_p	Spain	Anne
Andre	2	unknown	Carol
Raj	14	{India, U.S.}_p	Uma
Alan	unknown	U.S.	undefined
James	many	U.K.	null

A standard database cannot look like this. Entries like many and {10, 11}_p clearly suggest fuzziness. The entry {10, 11}_p, is a possibility distribution, suggesting that the number of visits outside the United States made by Roberto is either 10 or 11. Similarly, Raj’s citizenship is India or United States, but not dual citizenship in both. Andre’s citizenship and Alan’s number of visits outside the United States are not known, and they can have any values. The possibilities cannot be narrowed down as in the case of Raj’s citizenship and Roberto’s frequency of visits outside the United States The entry undefined is used for Alan because he always traveled alone, he never took a companion.

James’ number of visits is fuzzy. He traveled many times. A fuzzy set for many will provide the possibility distribution. It can be defined, for example, as:

    many = {0.2/6, 0.5/7, 0.8/8, 1/9, 1/10, ...}

The name of the companion on James’ latest visit outside the United States is entered as null because we do not know on the one hand whether he never took a companion, in which case we could have used undefined as in Alan’s case, and on the other whom he took as a companion if he did take one, in which case we could have used unknown. Simply put, we use null when we do not know enough to use either unknown or undefined.

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