https://doi.org/10.31449/inf.v46i1.3591 Informatica 46 (2022) 57–67 57 A Novel Fuzzy Modifier Interpolation Rule for Computing with Words Tanvi Dadu, Swati Aggarwal * and Nisha Aggarwal E-mail: tanvid.co.16@nsit.net.in, swati1178@gmail.com, nishaaggarwal2810@gmail.com Netaji Subhas University of Technology, Dwarka, New Delhi 110078, India Keywords: type-1 fuzzy sets, interval type 2 fuzzy sets, fuzzy modifiers, interpolation rule, computing with words Received: June 15, 2021 Computing with words is a concept that is used to solve problems with input in natural language. Modifiers are transformation functions with predefined labels used extensively in decision-making to specify the desired value of a linguistic variable defined by fuzzy sets. In past years, few efforts have been made to study the application of Computing with Words (CW) in many domains ranging from fraud detection systems to diagnosis systems in medicine. However, the application of CW in these fields with modified Fuzzy sets did not give satisfactory results. When applied to modified Fuzzy sets, the existing interpolation rule does not cover the extreme left and extreme right-shifted fuzzy sets. Hence, there is a need to introduce a new interpolation rule when working with modifiers. This paper introduces a new Fuzzy Modifier Interpolation Rule to Type-1 Fuzzy sets and Interval Type-2 (IT-2) Fuzzy Sets to enhance the quality of results obtained when modifiers are applied. Povzetek: V prispevku je predstavljeno novo interpolacijsko pravilo za mehke modifikatorje v mehkih nizih tipa 1 in intervalnega tipa 2. 1 Introduction Computing with Words (CW) is a methodology that allows using words in place of numbers for computing and reason- ing. It was introduced as an extension to Fuzzy Logic Sys- tems by Zadeh [1][2][3]. It is a system of computation that offers the capability to compute with information present in a natural language. The advancement in CW has allowed a certain degree of fuzziness to the input and the propositions present in the database of the CW inference engine. The propositions in CW are specified using linguistic variables [4][5][6], whose values are words in natural language. CW engine consists of Rule Base, Fuzzy Inference Engine, and Output Generator. The IF-THEN rules in the CW engine are specified using natural language, which is modeled us- ing either Type - 1 or Type - 2 Fuzzy sets (Interval Type-2 and General Type-2) [7]. Computing with Words (CW) has a wide variety of applications ranging from household appliances like fraud detection systems to biomedical instrumentation [8]. Medicine is one of the domains where the applications of Computing with Words has been recognized [8]. The un- certainty found in these applications is appropriately cap- tured by fuzzy set theory. Therefore, in the past few years, new techniques in CW have been extensively applied in decision-making systems. Modifiers are transformation functions with predefined labels applied to a fuzzy set defined on a linguistic variable to specify the desired value of a variable. Modifiers help us define input when it is not present in the given fuzzy * Corresponding author sets. Modifiers like VERY , EXTREMELY , and connec- tive like NOT, OR can be used to enhance the capability of the meaning specified by Fuzzy sets [9][10][11]. How- ever, the direct application of the Fuzzy Interpolation rule on the modified fuzzy set gives poor results. While com- puting results, it does not incorporate the contribution from fuzzy sets shifted to the extreme left or extreme right on the application of modifiers as detailed in Section 4. This paper introduces a novel Fuzzy Modifier Interpola- tion rule for Type-1 Fuzzy sets and Interval Type-2 Fuzzy sets to get better results and reduce errors when modifiers are applied to the fuzzy sets. In Section 2, we present re- cent relevant works followed by the proposed Fuzzy Modi- fier Interpolation rule for Type-1 and Interval Type-2 Fuzzy sets in Section 3. In Section 4, we present results of exper- iments performed on the UCLA dataset for heart disease [27] to show improvement in graphs or inference obtained upon application of our new Fuzzy Modifier Interpolation Rule. In Section 5, we discuss the results obtained for our proposed Fuzzy Modifier Interpolation rule with Fuzzy In- terpolation rule. Finally, we discuss the advantages and limitations of our proposed work in Section 6 and Section 7, respectively. 2 Related works The concept of modifier to represent linguistic hedges employing a mathematical transformation of membership functions was first introduced by Zadeh [12]. The au- thor proposed using linguistic hedges as a power functions based operator, which applied to fuzzy sets, represents 58 Informatica 46 (2022) 57–67 T. Dadu et al. the meaning of its operands. While applying transforma- tions, the author used pure post modification of member- ship functions to represent linguistic hedges. The modifiers proposed by the author did not cover all forms of linguis- tic hedges. To mitigate this, another type of modifier called shifting modifiers was put forward in [13]. These modifiers are translatory modifiers that involve pure pre-modification of the membership functions. However, traditional modifiers like powering and shift- ing modifiers could not handle similar categories distin- guished by subtle differences. To mitigate this, De Cock and Kerre [14] introduced a new form of fuzzy modifiers, where weakening adverbs (more or less, roughly) and in- tensifying adverbs (very, extremely) are modeled in the in- clusive and the non-inclusive interpretation. Another en- hancement on powering and shifting modifiers was L-fuzzy modifiers introduced in [15] to model linguistic hedges in the L-fuzzy sets. They proposed using context through L- fuzzy relations to ensure L-fuzzy modifiers are endowed with clear inherent semantic. The proposed modifiers out- performed the traditional ones from the semantic point of view. Since their advent, fuzzy modifiers have been used ex- tensively in fuzzy rule-based and control systems [18, 19], analogy-Based Reasoning systems [16] and image process- ing or image-based retrieval systems [21]. They are used to obtain more interpretable results, limit the number of premises, dynamically modify the shape of membership functions and reduce the rule base [17, 20]. They have been used in designing databases as well, like SQLf - a well- known query language database [23, 22]. Fuzzy Modifiers have a wide range of practical applications, and therefore, it is imperative to account for the edge cases when modifiers are applied in real-world applications. 3 Approach 3.1 Preliminaries Modifiers are a necessary part of Computing with Words. They allow us to define values in between givenN Fuzzy sets. Very, extremely, slightly, relatively, somewhat, quiet, and rather are some of the frequently used modifiers. They are generally denoted bym. XismA!Xisf(A) (1) Modifier Type Functionf(x) Extremely/Highly x 3 Very/Rather x 2 Relatively/Quite x 3=2 Slightly x 1=2 Somewhat x 1=3 Table 1: Functions of The Modifiers where f(A) is used to modify the behaviour of Fuzzy sets. The f(x) for frequently used modifiers is given in Table 1. Figure 1: Modifier Applied to Type-1 Fuzzy Set Figure 1 shows the Modifiers mentioned in Table 1 i.e. Extremely andSomewhat applied to Fuzzy set Low. The f(x) isx 3 in case of Extremely andx 1=3 in case of Some- what. The membership curve has shifted towards left for extremely modifier, and it has shifted towards the right for somewhat modifier. This is in accordance with the context of the application of modifiers. Modifiers are extensively used in decision systems, espe- cially in the domain of medicine. The interpolation rule for the Fuzzy set is unable to effectively deal when modifiers are applied. It cannot take contributions from the extreme left and extreme right-shifted fuzzy sets on the application of modifiers. Therefore a new Fuzzy Modifier Interpolation Rule was introduced for Type-1 Fuzzy sets. 3.2 Proposed fuzzy modifier interpolation rule on type-1 fuzzy set In this subsection, we introduce our proposed Fuzzy Inter- polation rule for Type-1 Fuzzy sets. The structure of the proposed rule consists of: 1. A set of IF-THEN rules, which is the database for in- ference mechanism. 2. An Antecedent with fuzzy modifier applied on it. 3. The proposed Fuzzy Modifier Interpolation rule, which consists of two cases. (a) Edge Case: It refers to the fuzzy sets having val- ues near the boundaries of the domain. (b) Non-Edge Case: It refers to fuzzy set not having values near the boundaries of the domain. A Novel Fuzzy Modifier Interpolation Rule. . . Informatica 46 (2022) 57–67 59 Our proposed Fuzzy Modifier Interpolation technique predominantly deals with edge cases-Left edge case and the right edge case. Block diagram for proposed Fuzzy Mod- ifier Interpolation rule is given in Fig 2 where m in mA i antecedent is a modifier applied to the Fuzzy set inputA i . There are two separate cases - edge case and non-edge case, which givesY isB as consequent. Figure 2: Block diagram for the application of Fuzzy Mod- ifier Interpolation rule 3.2.1 Edge case The edge case is defined as the case where the fuzzy sets have values near the boundaries of the domain. Formally, it is defined to be present when the following condition is satisfied: IfmA i > A i wheni =normA i