Linguistic variables represent crisp information in a form and precision appropriate for the problem. For example, to answer the question "How are you?" one may say "I am fine." the linguistic variables like "fine", so common in everyday speech. In this paper an alternative interpretation of linguistic variables is introduced with the notion of a linguistic description of a value or set of values. The use of linguistic variables in many applications reduces the overall computation complexity of the application. Linguistic variables have been shown to be particularly useful in complex non-linear applications. Here we are applying the concept of reasoning with Linguistic Quantifiers to define the Linguistic Finite Automata along with the expansion of \delta^{\box} and \lambda^{\box} over \delta and \lambda.
The concept of a linguistic variable was first introduced by Zadeh [6] as a model of how words or labels can represent vague concepts in natural language. Some Formal Definitions we are discussing here as: Definition 1.1(Linguistic variable): A linguistic variable is a quadruple [L,T(L),Ω, M] in which L is the name of the variable, T(L) is a countable term set of labels or words (i.e. the linguistic values), Ω is a universe of discourse and M is a semantic rule.
The semantic rule M is defined as a function that associates a normalized fuzzy subset of X with each word in T (L). In other words the fuzzy set M (w) can be viewed as encoding the meaning of w so that for u ∈ U the membership value µ M (w)(u) quantifies the suitability or applicability of the word w as a label for the value u. We can regard the semantic function M as being determined by a group voting model [7] across a population of voters as follows. Each voter is asked to provide the subset of words from the finite set T (L) which are appropriate as labels for the value u. The membership value µ M (w)(u) is then taken to be the proportion of voters who include w in their set of labels.
In [6] Zadeh originally defined a linguistic variable as a quintuple by including a syntactic rule according to which new terms (i.e., linguistic values) could be formed by applying hedges to existing words. However, the semantics of such hedges seem far from clear and the rather arbitrary definitions given in [6] appear inadequate. Indeed, in our view, it is far from apparent that there should be a simple functional relationship between the meanings of a word and the meaning of a new word generated from it by applying a hedge. In other words, we would claim that while hedges are a simple syntactic device for generating new terms there is no equally simple semantic device for generating the associated new meanings. Hence, in the following we consider only fixed finite term sets where all the labels and their associated meanings are predefined. This does not mean that we do not permit labels such as very small in the term set but rather that we would take its meaning as being predefined instead of being determined from that of small in a functional way. This gives µ M(small) (25)=1 and µ M(Medium) (25)=0.5
Now this voting parameter can be represented by a mass assignment on the power set of {small, medium, large},{small, medium}:0.5,{small}:0.5(i.e. 50% of voters select both small and medium as possible labels for 25 and 50% select only small). This in turn represents a fuzzy set on the set of words, namely Small/1+medium/0.5 Hence, in practice we need only define the fuzzy sets M (small) , M( medium) and M (large) from which we can determine any linguistic description. Fig. 1 illustrates how a linguistic description can be ``read’’ from the fuzzy set meanings of the words. Here the value 25 has membership 1 in M (small) , 0.5 in M( medium) and zero in M(large)(and all other labels) giving a linguistic description of small/1 + medium/0.5. (1) In cases where the linguistic variable is fixed we drop the subscript L and write des(x). This notion can be extended to the case where the value given is a crisp set or a fuzzy subset of Ω in which case the appropriate linguistic description is defined as follows. Here we are using the notation for piecewise linear function where [x 1 :y 1 ,….x n : y n ] denotes a function F(x)
In natural language linguistic quantifiers are used to indicate the speaker’s level of belief in a statement or to express the degree to which it is applicable.
For instance, the following are typical English sentences: Most good musicians like dance, It is highly likely that India will win the world cup.
In both these cases we can interpret the quantifier as a linguistic description of the probability of the statement. That is the quantifiers are words or labels describing a probability value and their meanings are given by fuzzy subsets of the interval [0, 1]. In the case of the first of the two sentences above we might argue that it is more natural to think of most as describing the proportion of the set of all mathematicians who like music. However, clearly we can equally think of such a proportion as the probability of picking a mathematician at random who likes music. It should also be noted that we restrict linguistic quantifiers to descriptions of probabilistic belief values and do not allow quantification over fuzzy truth values as illustrated by terms such as quite true and very true [8]. It is our view, as is consistent with the voting model, that truth values are emergent properties and can only be defined across a population of voters. Hence, it is not meaningful to discuss truth values with individual voters and in particular it is not meaningful to ask individual voters to provide labels for truth values. We would also suggest that this position is in keeping with natural language usage since, at least in English, quantifiers suc
This content is AI-processed based on open access ArXiv data.