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The above features are given in a rather descriptive than formal format and should then be treated as a collection of useful guidelines rather than completely strict definitions. Especially, some threshold levels (like ε, need to be specified numerically).

3.16.2. Main properties

Considering the family of the linguistic labels encapsulated in the same frame of cognition, several properties are worth underlining.

Specificity: the frame of cognition is more specific than ’ if the elements of are more specific than the elements of . The specificity of the fuzzy set A can be conveniently evaluated using the specificity measure as discussed by Yager (1981;1982). An example of and is shown in Fig. 3.15. In a more descriptive way: the granularity of the elements of ’ is higher than that associated with .


Figure 3.15  Frame of cognition of different levels of granularity

The concept of granularity and its importance is best exemplified via images, Fig. 3.16. Any perspective one assumes depends on the problem at hand. Too great a distance does not allow perception of an image as many, even quite important, aspects are not seen and recognized. Too close a distance forces us to view an image as nothing but a collection of pixels - in this way its content is missing.


Figure 3.16  Cognitive perspective and its role in knowledge representation

Focus of attention: a focus of attention (scope of perception) set up by A1 in is defined as an α-cut of this fuzzy set. By moving A along X while not changing its membership function we can focus attention on a certain region of X. This phenomenon is illustrated in Fig. 3.17.

Information hiding: this idea is directly linked with the previous notion. By modifying the membership function of A being an element of , we can reach an important effect of achieving an equivalence of the elements lying within some regions of X. Consider a trapezoidal fuzzy set A in R with its 1-cut distributed between a1 and a2. Observe that all the elements falling within this interval are now made nondistinguishable (equivalent) once expressed via A; beyond doubt A(x) = 1 for x ∈ [a1, a2]. Thus at the assumed level of granularity, the processing module does not distinguish between any two elements in the 1-cut of A, hence the more detailed information ( namely that about the position of x within this interval) becomes hidden. By modulating (namely increasing or decreasing) the level of the α-cut we can accomplish an α-information hiding. Again, Fig. 3.18 provides more clarification to this effect.


Figure 3.17  Focus of attention (scope of perception) realized through A


Figure 3.18  Effect of information hiding realized with fuzzy sets

The selection of the linguistic labels as well as their distribution in the universe of discourse (space) is very much problem dependent. This feature emphasizes the flexibility of fuzzy sets and their abilities to reflect and encapsulate domain knowledge. Let us look into the problem of navigating a robot through a series of verbal commands. The robot and a target are situated in the same room, Fig. 3.19.


Figure 3.19  A robot - target scenario (i), frame of cognition (ii), and fuzzy relations of different granularity (iii)

By monitoring a location of the robot and estimating its actual distance from the target, we navigate by issuing a series of voice commands such as: move about x cm forward, turn left a bit, etc. Evidently, the closer the robot approaches the target, the more detailed and precise the linguistic commands should be. The frame of cognition, Fig. 3.19(ii) composed of several highly specific fuzzy sets that are centered around zero is suitable for this type of navigation. The linguistic terms PB and NB need not be very specific as capturing situations being quite remote from the target. To the contrary, too detailed and precise labels do not do any good and may even have a detrimental effect by slowing down the navigation mechanism. Similarly, one can assume fuzzy sets of angle (β) with the higher granularity (specificity) of the terms centered around zero values. The use of fuzzy relations (as opposed to fuzzy sets defined in the individual spaces of distance and angles) is helpful in reducing a number of the linguistic landmarks necessary to describe the navigation process, Fig. 3.19(iii). Notice again that the granularity of the linguistic terms (fuzzy relations) is higher around zero angles and distances.

3.16.3. Approximation aspects of the frame of cognition

The frame of cognition can be approximated by a binary (set-based) partition. The sets of this partition are included by the original sets of . We derive as a solution to an optimization problem. Let us assume, which is realistic, that a nonzero overlap occurs between successive fuzzy sets of . The overlap between other fuzzy sets is negligible.


Figure 3.20  Fuzzy sets and their set approximation

In other words we get

The above allows us to decompose the problem into a series of subproblems involving only two fuzzy sets. The set approximation of Ai and Ai+1 is formulated as the optimization problem

The two sets, Bi and Bi+1 approximate the original fuzzy sets; the approximation of each of them yields an error with two components:

  the error of elevating the membership values of Ai to 1,

  the error of neglecting low values of Ai by their replacement by 0,

The same two components are also present in the approximation of Ai+1. The necessary condition for the minimum of Q is obtained from the relationship

Note that for any function “g” for which the integral makes sense we get the relationship

where z is in [x, x +Δx]. In limit, when Δx attains zero, this expression returns g(xi). The computations of the derivative lead to the expression

The final simplification produces a surprisingly simple observation


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