Definition

Fuzzy/linguistic IF-THEN rules are structured expressions of natural language having the form

$$ \mathrm{\mathcal{R}}\kern0.5em =\kern0.5em IF\kern0.5em X\kern0.5em is\kern0.5em \mathcal{A}\kern0.5em THEN\kern0.5em Y\kern0.5em is\kern0.5em \mathrm{\mathcal{B}} $$
(1)

where X, Y are variables and \( \mathcal{A},\kern0.5em \mathrm{\mathcal{B}} \) are expressions such as small, very small, medium, roughly medium, more or less big, big, etc. The latter are called evaluative linguistic expressions. Modeling their meaning in fuzzy set theory makes it possible to model the meaning of the whole rule. The part before THEN is called antecedent, the part after it is called consequent.

A linguistic description is a finite set of fuzzy/linguistic IF-THEN rules

$$ \begin{array}{l}{\mathrm{\mathcal{R}}}_1:\kern0.5em \mathrm{IF}\kern0.5em X\kern0.5em \mathrm{is}\kern0.5em {\mathcal{A}}_1\kern0.5em \mathrm{THEN}\kern0.5em Y\kern0.5em \mathrm{is}\kern0.5em {\mathrm{\mathcal{B}}}_1\\ {}\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots.\\ {}{\mathrm{\mathcal{R}}}_m:\kern0.5em \mathrm{IF}\kern0.5em X\kern0.5em \mathrm{is}\kern0.5em {\mathcal{A}}_m\kern0.5em \mathrm{THEN}\kern0.5em Y\kern0.5em \mathrm{is}\kern0.5em {\mathrm{\mathcal{B}}}_m.\end{array} $$
(2)

Linguistic description can be taken as a special structured text in natural language which describes some situation.

Key Points

There are two possible ways how fuzzy/linguistic IF-THEN rules can be interpreted in fuzzy set theory:

  1. (a)

    IF-THEN rule is assigned a fuzzy relation,

  2. (b)

    IF-THEN rule is assigned a function from the set of contexts to the set of fuzzy relations.

Case (b) is more complicated but more realistic as a model of the meaning of linguistic expressions since (1) can be taken in this case as a conditional expression of natural language.

In case (a), the whole linguistic description (2) is assigned a fuzzy relation constructed using one of two possible formulas: let each of the expressions of the form “\( X\kern0.5em \mathrm{is}\kern0.5em {\mathcal{A}}_i \)” be assigned a fuzzy set \( {A}_i\underset{\sim }{\subset }U \) and “Y is ℬi,” i = 1,…m a fuzzy set \( {B}_i\underset{\sim }{\subset }V \). Two possible fuzzy relations can be considered:

$$ {R}^A\left(x,y\right)=\underset{i=1}{\overset{m}{\vee }}\left({A}_i(x)\otimes {B}_i(y)\right), $$
(3)
$$ {R}^I\left(x,y\right)=\underset{i=1}{\overset{m}{\wedge }}\left({A}_i(x)\to {B}_i(y)\right), $$
(4)

where ⊗ is a t-norm and → is a residuation. Then (3) is called disjunctive normal form in which each IF-THEN rule is interpreted as conjunction; (4) is called conjunctive normal form in which each IF-THEN rule is interpreted as implication. Both forms (3) and (4) are two possible interpretations of the linguistic description (2).

The second interpretation of (2) is a set of functions I(ℛi) assigned to the rules ℛi in (2), i = 1, …, m. Each function has the form

$$ I\left({\mathrm{\mathcal{R}}}_i\right):{C}_X\times {C}_Y\to {L}^{U\times V} $$
(5)

where CX, CY are sets of contexts for the variable X and Y, respectively. Each context wX ∈ CX and wY ∈ CY is a certain interval of elements in U and V, respectively. The couple of contexts 〈wX, wY〉 is assigned via (5) a fuzzy relation of the form Ai(x) → Bi(y), x ∈ U, y ∈V. This approach leads to a mathematical model of the meaning of the text (2). Human understanding to such expressions and deriving conclusions on the basis of them can be mimicked. The details can be found in [2].

Cross-References