Modeling and implementation of Z-number

Abstract

Computing with words provides symbolic and semantic methodology to deal with imprecise information associated with natural languages. It encapsulates various fuzzy logic techniques developed in past decades and formalizes them. Z-number is an emerging paradigm that has been utilized in computing with words among others. The concept of a Z-number is intended to provide a basis for computation with numbers, specifically with reliability of information. Z-numbers are in confluence between the two most prominent approaches to uncertainty, probability and possibility, that allow computations on complex statements. Certain computations related to Z-numbers are ambiguous and complicated leading to its slow adaptation into areas such as computing with words. The biggest contributing factor to the complexity is the usage of probability distributions in the computations. This paper seeks to provide an applied model of Z-number based on certain realistic assumptions regarding probability distributions. Algorithms are presented to implement this model and integrate it into an expert system shell for computing with words called CWShell. CWShell is a software tool that abstracts the underlying computation required for computing with words, and provides a convenient way to represent and compute with unstructured natural language using specialized language called Generalized constraint language (GCL). This paper introduces new constructs for Z-numbers to GCL and provides detailed inference mechanism and computation strategy on those constructs. We present two case studies to demonstrate the working and feasibility of the approach.

Appendix: Algorithms

1.1 Probability measure of fuzzy event

In the procedure, zDist is an instance of ZDistribution and values is a fuzzy set. The array values[0] is the universe of a fuzzy set, while values[1] is the receptive membership value. The probability measure of a fuzzy event is calculated by iterating over all the elements of the fuzzy set (code 8).

1.2 Probability qualification

Code 9 presents the approach to calculate probability qualification, wherein distributions is the pool of distributions and $a$ is the fuzzy value specified for the Z-number. Probability qualification is a list created by calculating probability measure on all the distributions in $distributions$ for the fuzzy value $a$.

1.3 Induced probability

The probability qualification, described in Code 9, is used in Code 10 for calculating probQuali array. The method, resolveMembershipValue, is used to calculate the membership of probQuali[i] in the fuzzy set, $b$, which in turn is stored in the final array called results.

1.4 Inference based on granular probability

The Code 11 outlined the basic inference mechanism with single argument. The array, $\mathrm{myQuali}$, stores the probability qualification of the Z-number in the antecedent of the inference rule, while $\mathrm{otherQuali}$ holds the probability qualification of the Z-numbers in the precedent, $\mathrm{other}$. Next step constructs the fuzzy set in the array, $\mathrm{results}$. Values of $\mathrm{myQuali}$ form the element of the set, while the values in $\mathrm{otherQuali}$ are used to calculate corresponding membership values for the set.

It is likely that the fuzzy set, $\mathrm{results}$, at this point might has repeating elements, in results[0], with possible different membership values, in results[1]. This is the case because the calculations for probability qualifications are carried out on large pools of probability distribution. The method, $\mathrm{maxSortFuzzySet}$ outlined in Code 12, sorts the fuzzy set and applies max on the membership value for duplicate elements. The first loop in the procedure corresponds to the supreme operation in (28). The procedure iteratively creates a map, $\mathrm{sortMap}$, for the element in fuzzy set, $\mathrm{input}$. If the element is already in the map, then it puts in the max of the two values. Note that the map, $\mathrm{sortMap}$, is of type TreeMap. In Java, TreeMap is an Red-Black tree-based hash map that stores the keys in their natural order. The method call, $\mathrm{sortMap.entrySet()}$, returns the already sorted key value, and hence the second loop converts the map back to the array of fuzzy set, which is sorted by the value of its elements. The use of TreeMap in this procedure allowed us to design this procedure in O$(n)$, which otherwise would require an additional sorting. As will be seen later, this algorithm is critical piece and also used in other inference procedures.

The inference mechanism with multiple Z-numbers is presented in Code 13. The procedure iterates over all the Z-numbers, $\mathrm{other}$, which are in the antecedent of the inference rules, and stores the min of their induced probability values in the array, $otherInduced$. The array, $\mathrm{otherInduced}$ along  with the probability qualification of the Z-number in the precedent of the rule, forms the fuzzy set, $\mathrm{result}$. As described in Code 11, $\mathrm{maxSortFuzzySet}$ is applied to the $\mathrm{result}$ to construct the fuzzy set.

1.5 Inference based on extension principle

In the Code 14, the lines 3–4 calculate induced probability measure for the Z-numbers in the antecedent of the rules. It returns a map wherein the ZDistribtuion is the key and induced probability distribution is value. Line 6 calculates membership function of the probability distribution that is listed in detail in Code 15. Code 15 loops through the induced probability distributions, incudedProbA and incudedProbB, to create (1) ZConvolutionConstrained distribution, say $p_Z$ and (2) membership value for $p_Z$. Finally, the lines 8–20 create membership function for the fuzzy value $b$. For all the ZDistributions, $p_Z$, the probability measure of the fuzzy value $a$ constructs the domain, $\mathrm{xVals}$, and the membership values of probability membership function, $probabilityMemFunc$, construct membership function, $\mathrm{yVals}$, for the fuzzy value $B$. As with all the $\mathrm{inferB}$ algorithms, $\mathrm{maxSortFuzzySet}$ method is called to perform supreme operation.