[Paper Review] On Quantified Linguistic Approximation

Ryszard Kowalczyk|arXiv (Cornell University)|Jan 23, 2013

Fuzzy Logic and Control Systems11 references3 citations

TL;DR

This paper proposes quantified linguistic approximation as an advanced method for interpreting fuzzy set outputs in fuzzy systems by integrating linguistic quantifiers based on fuzzy and non-fuzzy cardinalities. It extends traditional linguistic approximation beyond simple, modifier, and composite rules to include quantification rules, enabling more informative and natural linguistic interpretations of complex fuzzy conclusions, with two distinct methods demonstrated through illustrative examples.

ABSTRACT

Most fuzzy systems including fuzzy decision support and fuzzy control systems provide out-puts in the form of fuzzy sets that represent the inferred conclusions. Linguistic interpretation of such outputs often involves the use of linguistic approximation that assigns a linguistic label to a fuzzy set based on the predefined primary terms, linguistic modifiers and linguistic connectives. More generally, linguistic approximation can be formalized in the terms of the re-translation rules that correspond to the translation rules in ex-plicitation (e.g. simple, modifier, composite, quantification and qualification rules) in com-puting with words [Zadeh 1996]. However most existing methods of linguistic approximation use the simple, modifier and composite re-translation rules only. Although these methods can provide a sufficient approximation of simple fuzzy sets the approximation of more complex ones that are typical in many practical applications of fuzzy systems may be less satisfactory. Therefore the question arises why not use in linguistic ap-proximation also other re-translation rules corre-sponding to the translation rules in explicitation to advantage. In particular linguistic quantifica-tion may be desirable in situations where the conclusions interpreted as quantified linguistic propositions can be more informative and natu-ral. This paper presents some aspects of linguis-tic approximation in the context of the re-translation rules and proposes an approach to linguistic approximation with the use of quantifi-cation rules, i.e. quantified linguistic approxima-tion. Two methods of the quantified linguistic approximation are considered with the use of lin-guistic quantifiers based on the concepts of the non-fuzzy and fuzzy cardinalities of fuzzy sets. A number of examples are provided to illustrate the proposed approach.

Motivation & Objective

• To address the limitation of existing linguistic approximation methods that rely only on simple, modifier, and composite re-translation rules.
• To investigate the potential of incorporating quantification rules into linguistic approximation for improved interpretability of complex fuzzy outputs.
• To formalize linguistic approximation using re-translation rules corresponding to translation rules in computing with words, particularly focusing on quantification.
• To propose and evaluate two methods of quantified linguistic approximation using linguistic quantifiers grounded in fuzzy and non-fuzzy cardinalities.
• To demonstrate the practical utility of quantified linguistic approximation through concrete examples in fuzzy decision-making and control systems.

Proposed method

• Introduces a framework for linguistic approximation using re-translation rules, including quantification rules derived from the translation rules in computing with words.
• Proposes two methods for quantified linguistic approximation: one based on non-fuzzy cardinality and another on fuzzy cardinality of fuzzy sets.
• Applies linguistic quantifiers (e.g., 'most', 'many', 'few') to represent the degree of membership or coverage of a fuzzy set in linguistic terms.
• Uses fuzzy cardinality to assess the size of a fuzzy set in a linguistic context, enabling quantified statements like 'most elements satisfy the condition'.
• Employs re-translation rules to map fuzzy set outputs into quantified linguistic propositions, enhancing interpretability and naturalness.
• Validates the approach through multiple examples showing how complex fuzzy outputs are transformed into meaningful linguistic statements using quantification.

Experimental results

Research questions

• RQ1 Why are existing linguistic approximation methods insufficient for interpreting complex fuzzy sets in practical fuzzy systems?
• RQ2 Can linguistic quantification improve the informativeness and naturalness of linguistic interpretations of fuzzy outputs?
• RQ3 How can linguistic quantifiers be formally grounded in fuzzy and non-fuzzy cardinalities to support reliable approximation?
• RQ4 What are the practical differences between using non-fuzzy versus fuzzy cardinality in quantified linguistic approximation?
• RQ5 In what ways do quantification rules enhance the re-translation process compared to traditional simple, modifier, and composite rules?

Key findings

• The proposed quantified linguistic approximation approach enables more informative and natural linguistic interpretations of fuzzy set outputs than traditional methods.
• Using linguistic quantifiers based on fuzzy cardinality allows for a more nuanced and context-sensitive interpretation of fuzzy sets in linguistic terms.
• The method based on non-fuzzy cardinality provides a simpler, yet effective, alternative for cases where crisp cardinality is sufficient for interpretation.
• Examples demonstrate that quantified linguistic approximations can express complex fuzzy conclusions in a way that aligns better with human reasoning and natural language.
• The integration of quantification rules into linguistic approximation extends the scope of computing with words, supporting richer linguistic expressions in fuzzy systems.
• The approach shows promise in enhancing decision support and control systems by improving the interpretability of fuzzy inference results.