The Quintuple Implication Principle of fuzzy reasoning

Zhou, B; Xu, G; Li, S

The Quintuple Implication Principle of fuzzy reasoning

Zhou, B Xu, G Li, S

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Publication Type:: Journal Article
Citation:: Information Sciences, 2015, 297 pp. 202 - 215
Issue Date:: 2015-03-10

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Field	Value	Language
dc.contributor.author	Zhou, B	en_US
dc.contributor.author	Xu, G	en_US
dc.contributor.author	Li, S https://orcid.org/0000-0002-3332-2546	en_US
dc.date.issued	2015-03-10	en_US
dc.identifier.citation	Information Sciences, 2015, 297 pp. 202 - 215	en_US
dc.identifier.issn	0020-0255	en_US
dc.identifier.uri	http://hdl.handle.net/10453/32846
dc.description.abstract	© 2014 Elsevier Inc. Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT) are two fundamental patterns of approximate reasoning. Suppose A and B are fuzzy predicates and "IF A THEN B" is a fuzzy rule. Approximate reasoning often requires to derive an approximation A∗ of B from a given approximation A∗ of A, or vice versa. To solve these problems, Zadeh introduces the well-known Compositional Rule of Inference (CRI), which models fuzzy rule by implication and computes A∗ (A∗, resp.) by composing A∗ (A∗, resp.) with A→B. Wang argues that the use of the compositional operation is logically not sufficiently justified and proposes the Triple Implication Principle (TIP) instead. Both CRI and TIP do not explicitly use the closeness of A and A∗ (or that of B and A∗) in the process of calculating the consequence, which makes the thus computed approximation sometimes useless or misleading. In this paper, we propose the Quintuple Implication Principle (QIP) for fuzzy reasoning, which characterizes the approximation A∗ of B (A∗ of A, resp.) as the formula which is best supported by A→B,A∗→A and A∗ (A→B,B→A∗ and A∗, resp.). Based upon Monoidal t-norm Logic (MTL), this paper applies QIP to solve FMP and FMT for four important implications. Most importantly, we show that QIP, when using Gödel implication, computes exactly the same approximation as Mamdani-type fuzzy inference does. This is surprising as Mamdani interprets fuzzy rules in terms of the minimum operation, while CRI, TIP and QIP all interpret fuzzy rules in terms of implication.	en_US
dc.relation.ispartof	Information Sciences	en_US
dc.relation.isbasedon	10.1016/j.ins.2014.11.024	en_US
dc.subject.classification	Artificial Intelligence & Image Processing	en_US
dc.title	The Quintuple Implication Principle of fuzzy reasoning	en_US
dc.type	Journal Article
utslib.citation.volume	297	en_US
utslib.for	080203 Computational Logic and Formal Languages	en_US
utslib.for	0102 Applied Mathematics	en_US
utslib.for	01 Mathematical Sciences	en_US
utslib.for	08 Information and Computing Sciences	en_US
utslib.for	09 Engineering	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Strength - QSI - Centre for Quantum Software and Information
utslib.copyright.status	closed_access
pubs.publication-status	Published	en_US
pubs.volume	297	en_US

Abstract:

© 2014 Elsevier Inc. Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT) are two fundamental patterns of approximate reasoning. Suppose A and B are fuzzy predicates and "IF A THEN B" is a fuzzy rule. Approximate reasoning often requires to derive an approximation A∗ of B from a given approximation A∗ of A, or vice versa. To solve these problems, Zadeh introduces the well-known Compositional Rule of Inference (CRI), which models fuzzy rule by implication and computes A∗ (A∗, resp.) by composing A∗ (A∗, resp.) with A→B. Wang argues that the use of the compositional operation is logically not sufficiently justified and proposes the Triple Implication Principle (TIP) instead. Both CRI and TIP do not explicitly use the closeness of A and A∗ (or that of B and A∗) in the process of calculating the consequence, which makes the thus computed approximation sometimes useless or misleading. In this paper, we propose the Quintuple Implication Principle (QIP) for fuzzy reasoning, which characterizes the approximation A∗ of B (A∗ of A, resp.) as the formula which is best supported by A→B,A∗→A and A∗ (A→B,B→A∗ and A∗, resp.). Based upon Monoidal t-norm Logic (MTL), this paper applies QIP to solve FMP and FMT for four important implications. Most importantly, we show that QIP, when using Gödel implication, computes exactly the same approximation as Mamdani-type fuzzy inference does. This is surprising as Mamdani interprets fuzzy rules in terms of the minimum operation, while CRI, TIP and QIP all interpret fuzzy rules in terms of implication.

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http://hdl.handle.net/10453/32846