Y. Xu, D. Ruan, K. Qin, J. Liu

Lattice-Valued Logic

Springer-Verlag Berlin Heidelberg GmbH

Y. Xu, D. Ruan, K. Qin, J. Liu

Lattice-Valued Logic

Springer-Verlag Berlin Heidelberg GmbH

Studies in Fuzziness and Soft Computing, Volume 132 http://www.springer.de/cgi-bin/search_book.pl?series=2941

Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl

Further volumes of this series can be found on our homepage

Vol. 113. A. Abraham, 1.C. Jain and J. Kacprzyk (Eds.) Recent Advances in Intelligent Paradigms and Applications", 2003 ISBN 3-7908-1538-1

Vol. 114. M. Fitting and E. Orowska (Eds.) Beyond Two: Theory and Applications of Multiple Valued Logic, 2003 ISBN 3-7908-1541-1

Vol. 115. J.J. Buckley Fuzzy Probabilities, 2003 ISBN 3-7908-1542-X

Vol. 116. C. Zhou, D. MaravaH and D. Ruan (Eds.) Autonomous Robotic Systems, 2003 ISBN 3-7908-1546-2

Vol 117. O. Castillo, P. Melin Soft Computing and Fractal Theory for Intelligent Manufacturing, 2003 ISBN 3-7908-1547-0

Vol. 118. M. Wygralak Cardinalities of Fuzzy Sets, 2003 ISBN 3-540-00337-1

Vol. 119. Karmeshu (Ed.) Entropy Measures, Maximum Entropy Principle and Emerging Applications, 2003 ISBN 3-540-00242-1

Vol. 120. H.M. Cartwright, L.M. Sztandera (Eds.) Soft Computing Approaches in Chemistry, 2003 ISBN 3-540-00245-6

Vol. 121. J. Lee (Ed.) Software Engineering with Computational Intelligence, 2003 ISBN 3-540-00472-6

Vol. 122. M. Nachtegael, D. Van der Weken, D. Van de Ville and E.E. Kerre (Eds.) Fuzzy Filters for Image Processing, 2003 ISBN 3-540-00465-3

Vol. 123. V. Torra (Ed.) Information Fusion in Data Mining, 2003 ISBN 3-540-00676-1

Vol. 124. X. Yu, J. Kacprzyk (Eds.) Applied Decision Support with Soft Computing, 2003 ISBN 3-540-02491-3

Vol. 125. M. Inuiguchi, S. Hirano and S. Tsumoto (Eds.) Rough Set Theory and Granular Computing, 2003 ISBN 3-540-00574-9

Vol. 126. J.-1. Verdegay (Ed.) Fuzzy Sets Based Heuristics for Optimization, 2003 ISBN 3-540-00551-X

Vol 127. 1. Reznik, V. Kreinovich (Eds.) Soft Computing in Measurement and Information Acquisition, 2003 ISBN 3-540-00246-4

Vol 128. J. Casillas, O. Cord6n, F. Herrera, L. Magdalena (Eds.) Interpretability Issues in Fuzzy Modeling, 2003 ISBN 3-540-02932-X

Vol 129. J. Casillas, O. Cord6n, F. Herrera, 1. Magdalena (Eds.) Accuracy Improvements in Linguistic Fuzzy Modeling, 2003 ISBN 3-540-02933-8

Vol 130. P.S. Nair Uncertainty in Multi-Source Databases, 2003 ISBN 3-540-03242-8

Vol 131. J.N. Mordeson, D.S. Malik, N. Kuroki Fuzzy Semigroups, 2003 ISBN 3-540-03243-6

Studies in Fuzziness and Soft Computing, Volume 132 http://www.springer.de/cgi-bin/search_book.pl?series=2941

Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: kacprzyk@ibspan.waw.pl

Further volumes of this series can be found on our homepage

Vol. 113. A. Abraham, 1.C. Jain and J. Kacprzyk (Eds.) Recent Advances in Intelligent Paradigms and Applications", 2003 ISBN 3-7908-1538-1

Vol. 114. M. Fitting and E. Orowska (Eds.) Beyond Two: Theory and Applications of Multiple Valued Logic, 2003 ISBN 3-7908-1541-1

Vol. 115. J.J. Buckley Fuzzy Probabilities, 2003 ISBN 3-7908-1542-X

Vol. 116. C. Zhou, D. MaravaH and D. Ruan (Eds.) Autonomous Robotic Systems, 2003 ISBN 3-7908-1546-2

Vol 117. O. Castillo, P. Melin Soft Computing and Fractal Theory for Intelligent Manufacturing, 2003 ISBN 3-7908-1547-0

Vol. 118. M. Wygralak Cardinalities of Fuzzy Sets, 2003 ISBN 3-540-00337-1

Vol. 119. Karmeshu (Ed.) Entropy Measures, Maximum Entropy Principle and Emerging Applications, 2003 ISBN 3-540-00242-1

Vol. 120. H.M. Cartwright, L.M. Sztandera (Eds.) Soft Computing Approaches in Chemistry, 2003 ISBN 3-540-00245-6

Vol. 121. J. Lee (Ed.) Software Engineering with Computational Intelligence, 2003 ISBN 3-540-00472-6

Vol. 122. M. Nachtegael, D. Van der Weken, D. Van de Ville and E.E. Kerre (Eds.) Fuzzy Filters for Image Processing, 2003 ISBN 3-540-00465-3

Vol. 123. V. Torra (Ed.) Information Fusion in Data Mining, 2003 ISBN 3-540-00676-1

Vol. 124. X. Yu, J. Kacprzyk (Eds.) Applied Decision Support with Soft Computing, 2003 ISBN 3-540-02491-3

Vol. 125. M. Inuiguchi, S. Hirano and S. Tsumoto (Eds.) Rough Set Theory and Granular Computing, 2003 ISBN 3-540-00574-9

Vol. 126. J.-1. Verdegay (Ed.) Fuzzy Sets Based Heuristics for Optimization, 2003 ISBN 3-540-00551-X

Vol 127. 1. Reznik, V. Kreinovich (Eds.) Soft Computing in Measurement and Information Acquisition, 2003 ISBN 3-540-00246-4

Vol 128. J. Casillas, O. Cord6n, F. Herrera, L. Magdalena (Eds.) Interpretability Issues in Fuzzy Modeling, 2003 ISBN 3-540-02932-X

Vol 129. J. Casillas, O. Cord6n, F. Herrera, 1. Magdalena (Eds.) Accuracy Improvements in Linguistic Fuzzy Modeling, 2003 ISBN 3-540-02933-8

Vol 130. P.S. Nair Uncertainty in Multi-Source Databases, 2003 ISBN 3-540-03242-8

Vol 131. J.N. Mordeson, D.S. Malik, N. Kuroki Fuzzy Semigroups, 2003 ISBN 3-540-03243-6

Yang Xu Da Ruan Keyun Qin Jun Liu

Lattice-Valued Logic An Alternative Approach to Treat Fuzziness and Incomparability

Springer

Yang Xu Da Ruan Keyun Qin Jun Liu

Lattice-Valued Logic An Alternative Approach to Treat Fuzziness and Incomparability

Springer

Professor Dr. Yang Xu E-mail: yxu@home.swjtu.edu.cn Professor Dr. Keyun Qin E-mail: keyunqin@263.net

Southwest Jiaotong University

Department of Applied Mathematics

Chengdu

610031 Sichuan

PRChina

Professor Dr. Da Ruan E-mail: druan@sckcen.be

Belgian Nuclear Research Center (SCK*CEN)

Boeretang 200

2400 Mal

Belgium

Dr.Jun Liu

The University of Manchester Institute

of Science and Technology

Department of Mathematics

PO Box 88

Manchester M60 1 QD, UK

ISBN 978-3-642-07279-6 ISBN 978-3-540-44847-1 (eBook) DOI 10.1007/978-3-540-44847-1

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© Springer-Verlag Berlin Heidelberg 2003 Originally published by Springer-Verlag Berlin Heidelberg New York in 2003

Softcover reprint of the hardcover lst edition 2003

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Preface

One of the fundamental goals of artificial intelligence (AI) is to build arti­ficially computer-based systems which make computer simulate, extend and expand human's intelligence and empower computers to perform tasks which are routinely performed by human beings. Due to the fact that human intel­ligence actions are always involved with uncertainty information processing, one important task of AI is to study how to make the computer simulate hu­man being to deal with uncertainty information. Among major ways in which human being deal with uncertainty information, the uncertainty inference be­comes an essential mechanism in AI. From the viewpoint of symbolism, it is highly necessary to study and establish the logical foundation for uncertainty inference. Note that classical logic has already been the foundation of cer­tainty inference, logical foundation for uncertainty inference should be the extension and development of classical logic, which are often generally called non-classical logics, where many-valued logic has been one kind of important non-classical logics.

Fuzziness and incomparability are two kinds of uncertainty often associ­ated with human's intelligent activities in the real world, and they exist not only in the processed object itself, but also in the course of the object being dealt with. Therefore, it is necessary to investigate the logical foundation and the corresponding uncertainty inference theory and methods for characteriz­ing and dealing with not only the fuzziness and incomparability associated with the object itself, but also the uncertainty involved within the course of the object being processed as well. These have become the motivation of our research work. Accordingly, we started the corresponding research work from 1980's. We proposed and studied a many-valued logic-lattice-valued logic based on lattice implication algebra and further studied the theories and methods of uncertainty inference and automated reasoning. Up to now, there has been a lot of work being done. In this present book, we systemati­cally summarize our research work over last decades mainly by our research group, but also include some other researchers' work in this subject.

We expect the book to be useful for AI research and as a reference book for logicians, mathematicians and computer scientists in uncertainty information processing. This book may be used as a text book for graduate students in the relevant areas. The book is divided into three parts.

Preface

One of the fundamental goals of artificial intelligence (AI) is to build arti­ficially computer-based systems which make computer simulate, extend and expand human's intelligence and empower computers to perform tasks which are routinely performed by human beings. Due to the fact that human intel­ligence actions are always involved with uncertainty information processing, one important task of AI is to study how to make the computer simulate hu­man being to deal with uncertainty information. Among major ways in which human being deal with uncertainty information, the uncertainty inference be­comes an essential mechanism in AI. From the viewpoint of symbolism, it is highly necessary to study and establish the logical foundation for uncertainty inference. Note that classical logic has already been the foundation of cer­tainty inference, logical foundation for uncertainty inference should be the extension and development of classical logic, which are often generally called non-classical logics, where many-valued logic has been one kind of important non-classical logics.

Fuzziness and incomparability are two kinds of uncertainty often associ­ated with human's intelligent activities in the real world, and they exist not only in the processed object itself, but also in the course of the object being dealt with. Therefore, it is necessary to investigate the logical foundation and the corresponding uncertainty inference theory and methods for characteriz­ing and dealing with not only the fuzziness and incomparability associated with the object itself, but also the uncertainty involved within the course of the object being processed as well. These have become the motivation of our research work. Accordingly, we started the corresponding research work from 1980's. We proposed and studied a many-valued logic-lattice-valued logic based on lattice implication algebra and further studied the theories and methods of uncertainty inference and automated reasoning. Up to now, there has been a lot of work being done. In this present book, we systemati­cally summarize our research work over last decades mainly by our research group, but also include some other researchers' work in this subject.

We expect the book to be useful for AI research and as a reference book for logicians, mathematicians and computer scientists in uncertainty information processing. This book may be used as a text book for graduate students in the relevant areas. The book is divided into three parts.

II Preface

Part I (Chapter 1) provides the academic background of this book and the major methodologies of AI, where the related research work outline of many­valued logic, especially lattice-valued logic as well as the related uncertainty inference and automated reasoning based on many-valued logic are reviewed.

Part II (Chapters 2-8) introduces the logic algebra-lattice implication al­gebra and studies its properties. Concretely, Chapter 2 proposes the concept of lattice implication algebra by combining lattice and implication algebra, and gives some typical examples of lattice implication algebra, e.g., Boolean algebra, Lukasiewicz implication algebra. Furthermore, it discusses some ba­sic properties of lattice implication algebras and lattice H implication algebras as well as homomorphisms between lattice implication algebras. The fact that all lattice implication algebras form a proper class is proved. Chapter 3 dis­cusses a special kind of substructures in lattice implication algebra-filters, and gives various kinds of filters and their properties. It is shown that filter is a kind of abstract of MP rule. Chapter 4 discusses the dual structure of filter, i.e., LI-ideal, and introduces some special LI-ideal structures in lattice implication algebra and their properties. Chapter 5 devotes to the homomor­phism between lattice implication algebras, discusses the lattice implication quotient algebras based on the congruence relations induced by filters, fuzzy filters, LI-ideals and fuzzy LI-ideals, and gives the corresponding homomor­phism theorem and the isomorphism theorem of lattice implication algebras. Moreover it discusses the characteristics of proper lattice implication algebra. Chapter 6 studies lattice implication algebra from the viewpoint of topology, and discusses the filter spaces for lattice implication algebras and their topo­logical properties, such as count ability, separability, compactness and con­nectedness. It further investigates prime topological space based on prime filters. Chapter 7 describes the relations between lattice implication algebra and BCK-algebra, MV-algebra, FI-algebra, Ro-algebra, (2, 2, 2, 1,0,0)­type algebra, (2, 1, 0, O)-type algebra, (2, 0, O)-type algebra, respectively, and gives several equivalent definitions of lattice implication algebras. Chap­ter 8 treats the category theories of lattice implication algebras, fuzzy lattice implication algebras, and fuzzy power set theory based on lattice implication algebra. It discusses the adjoint semi groups induced by lattice implication al­gebras, and formalizes the lattice implication algebra theory in the first-order language with identity.

Part III (Chapters 9-11) introduces the algebraic logic -lattice-valued log­ics based on lattice implication algebra and their applications in uncertainty reasoning and automated reasoning. Concretely, Chapter 9 establishes the lattice-valued propositional logic LP(X) and the gradational lattice-valued propositional logic L vpl based on lattice implication algebra and discusses their properties. The main results include the Soundness Theorem, Deduc­tion Theorem and Consistent Theorem of LP(X), the (ex, (3)-i type Complete­ness and Soundness Theorem, (ex, (3)-i type Consistent Theorem, (ex, (3, 8)-i type Deduction Theorem of L vpl ' It also discusses the compactness in L vpl ,

II Preface

Part I (Chapter 1) provides the academic background of this book and the major methodologies of AI, where the related research work outline of many­valued logic, especially lattice-valued logic as well as the related uncertainty inference and automated reasoning based on many-valued logic are reviewed.

Part II (Chapters 2-8) introduces the logic algebra-lattice implication al­gebra and studies its properties. Concretely, Chapter 2 proposes the concept of lattice implication algebra by combining lattice and implication algebra, and gives some typical examples of lattice implication algebra, e.g., Boolean algebra, Lukasiewicz implication algebra. Furthermore, it discusses some ba­sic properties of lattice implication algebras and lattice H implication algebras as well as homomorphisms between lattice implication algebras. The fact that all lattice implication algebras form a proper class is proved. Chapter 3 dis­cusses a special kind of substructures in lattice implication algebra-filters, and gives various kinds of filters and their properties. It is shown that filter is a kind of abstract of MP rule. Chapter 4 discusses the dual structure of filter, i.e., LI-ideal, and introduces some special LI-ideal structures in lattice implication algebra and their properties. Chapter 5 devotes to the homomor­phism between lattice implication algebras, discusses the lattice implication quotient algebras based on the congruence relations induced by filters, fuzzy filters, LI-ideals and fuzzy LI-ideals, and gives the corresponding homomor­phism theorem and the isomorphism theorem of lattice implication algebras. Moreover it discusses the characteristics of proper lattice implication algebra. Chapter 6 studies lattice implication algebra from the viewpoint of topology, and discusses the filter spaces for lattice implication algebras and their topo­logical properties, such as count ability, separability, compactness and con­nectedness. It further investigates prime topological space based on prime filters. Chapter 7 describes the relations between lattice implication algebra and BCK-algebra, MV-algebra, FI-algebra, Ro-algebra, (2, 2, 2, 1,0,0)­type algebra, (2, 1, 0, O)-type algebra, (2, 0, O)-type algebra, respectively, and gives several equivalent definitions of lattice implication algebras. Chap­ter 8 treats the category theories of lattice implication algebras, fuzzy lattice implication algebras, and fuzzy power set theory based on lattice implication algebra. It discusses the adjoint semi groups induced by lattice implication al­gebras, and formalizes the lattice implication algebra theory in the first-order language with identity.

Part III (Chapters 9-11) introduces the algebraic logic -lattice-valued log­ics based on lattice implication algebra and their applications in uncertainty reasoning and automated reasoning. Concretely, Chapter 9 establishes the lattice-valued propositional logic LP(X) and the gradational lattice-valued propositional logic L vpl based on lattice implication algebra and discusses their properties. The main results include the Soundness Theorem, Deduc­tion Theorem and Consistent Theorem of LP(X), the (ex, (3)-i type Complete­ness and Soundness Theorem, (ex, (3)-i type Consistent Theorem, (ex, (3, 8)-i type Deduction Theorem of L vpl ' It also discusses the compactness in L vpl ,

Preface iii

and gives the gradational Lukasiewicz propositional logic Lu and F-valued Lukasiewicz propositional logic FLu as the special cases. Chapter 10 fur­ther introduces the corresponding lattice-valued first-order logics based on lattice implication algebra, i.e., lattice-valued first-order logic LF(X) and the gradational lattice-valued first-order logic Lvjl, and discusses the analo­gous properties in Chapter 9 accordingly. Also the gradational Lukasiewicz first-order logic Luj and F-valued Lukasiewicz first-order logic F Luj are introduced as the special cases. Chapter 11 starts with the uncertainty rea­soning approaches based on lattice-valued logic LP(X) corresponding to four kinds of uncertainty reasoning models, then introduces the uncertainty rea­soning theory and approach based on Lvpl, where the reasoning scheme may be implemented according to semantic interpretation as well as syntactic de­duction in Lvpl. Finally, it establishes the a-resolution principles based on lattice-valued logic LP(X) and LF(X).

The outcome of this book involves many people's efforts. Our thanks are due to Dr. Jun Ma for his excellent typesetting and editing of the manuscript with some help from Wei Wang and Xuefang Wang; to Xiaodong Guan, Dr. Zheng Pei, Dr. Tianrui Li, Dr. Wenjiang Li, Shuwei Chen, Dan Meng, Yongchuan Tang, Xiaoping Qiu, and Xiaowei Yang for their assistance; to Prof. Young Bae Jun at Gyeongsang National University, Korea and Prof. Yiquan Zhu at Zhao Qing Normal College, China for providing us with some research materials in line with the subject of this book. Furthermore, we ac­knowledge gratefully for great support on our research by National Natural Science Foundation Committee of China (No. 60074014), Science & Technol­ogy Department of Sichuan Province, China, and the Flanders-China Coop­eration Project, Belgium. Special thanks are also due to all the support in the preparation of the book by Southwest Jiaotong University of China, the Belgian Nuclear Research Centre (SCKeCEN), and UMIST in the UK. In addition, we acknowledge all the authors whose research papers and books are cited in this book. We do realize that many scholars' research results on the subject are not yet fully cited in the book, but will be part of our future research tasks. Last but not least, our thanks go to Prof. Janusz Kacprzyk (Editor-in-chief), to Katharina Wetzel-Vandai and Dr. Thomas Ditzinger (Editors, Springer-Verlag) for their kind consideration and advice to include this book into their "Studies in Fuzziness and Soft Computing" series.

Chengdu e Mol e Manchester Yang Xu, Da Ruan, Keyun Qin, Jun Liu March 2003

Preface iii

and gives the gradational Lukasiewicz propositional logic Lu and F-valued Lukasiewicz propositional logic FLu as the special cases. Chapter 10 fur­ther introduces the corresponding lattice-valued first-order logics based on lattice implication algebra, i.e., lattice-valued first-order logic LF(X) and the gradational lattice-valued first-order logic Lvjl, and discusses the analo­gous properties in Chapter 9 accordingly. Also the gradational Lukasiewicz first-order logic Luj and F-valued Lukasiewicz first-order logic F Luj are introduced as the special cases. Chapter 11 starts with the uncertainty rea­soning approaches based on lattice-valued logic LP(X) corresponding to four kinds of uncertainty reasoning models, then introduces the uncertainty rea­soning theory and approach based on Lvpl, where the reasoning scheme may be implemented according to semantic interpretation as well as syntactic de­duction in Lvpl. Finally, it establishes the a-resolution principles based on lattice-valued logic LP(X) and LF(X).

The outcome of this book involves many people's efforts. Our thanks are due to Dr. Jun Ma for his excellent typesetting and editing of the manuscript with some help from Wei Wang and Xuefang Wang; to Xiaodong Guan, Dr. Zheng Pei, Dr. Tianrui Li, Dr. Wenjiang Li, Shuwei Chen, Dan Meng, Yongchuan Tang, Xiaoping Qiu, and Xiaowei Yang for their assistance; to Prof. Young Bae Jun at Gyeongsang National University, Korea and Prof. Yiquan Zhu at Zhao Qing Normal College, China for providing us with some research materials in line with the subject of this book. Furthermore, we ac­knowledge gratefully for great support on our research by National Natural Science Foundation Committee of China (No. 60074014), Science & Technol­ogy Department of Sichuan Province, China, and the Flanders-China Coop­eration Project, Belgium. Special thanks are also due to all the support in the preparation of the book by Southwest Jiaotong University of China, the Belgian Nuclear Research Centre (SCKeCEN), and UMIST in the UK. In addition, we acknowledge all the authors whose research papers and books are cited in this book. We do realize that many scholars' research results on the subject are not yet fully cited in the book, but will be part of our future research tasks. Last but not least, our thanks go to Prof. Janusz Kacprzyk (Editor-in-chief), to Katharina Wetzel-Vandai and Dr. Thomas Ditzinger (Editors, Springer-Verlag) for their kind consideration and advice to include this book into their "Studies in Fuzziness and Soft Computing" series.

Chengdu e Mol e Manchester Yang Xu, Da Ruan, Keyun Qin, Jun Liu March 2003

List of symbols

II incomparable

rh disjoint

o empty set

~ partial order relation

complementary, or negation

-+ implication

o residuated operation

V supremum, or disjunction

1\ infimum, or conjunction

[a) generated filter by the element a (a) generated ideal by a ord( a) order of the element a aD inner point of a

Ii the closure of a

[A) generated filter by the set A (A) generated ideal by the set A c(A) the closure of A

d(A) the derived sets of A i(A) inner of A

ob(A) the set of all objects of A

P(A) power set of the set A

L Lattice; short for lattice implication algebra

C(L) the set of completely normal fuzzy LI-ideal

Dat(L) the set of all dual atoms of L

I F( L) the set of all implicative filters of lattice implication algebra L N(L) the set of normal fuzzy LI-ideal of L PF(L) the set of all prime filters of lattice implication algebra L

LC the category of lattice implication algebra

List of symbols

II incomparable

rh disjoint

o empty set

~ partial order relation

complementary, or negation

-+ implication

o residuated operation

V supremum, or disjunction

1\ infimum, or conjunction

[a) generated filter by the element a (a) generated ideal by a ord( a) order of the element a aD inner point of a

Ii the closure of a

[A) generated filter by the set A (A) generated ideal by the set A c(A) the closure of A

d(A) the derived sets of A i(A) inner of A

ob(A) the set of all objects of A

P(A) power set of the set A

L Lattice; short for lattice implication algebra

C(L) the set of completely normal fuzzy LI-ideal

Dat(L) the set of all dual atoms of L

I F( L) the set of all implicative filters of lattice implication algebra L N(L) the set of normal fuzzy LI-ideal of L PF(L) the set of all prime filters of lattice implication algebra L

LC the category of lattice implication algebra

vi List of symbols

FLC the category of fuzzy lattice implication algebra

F(LC) the forgetful category of LC

I the greatest element of lattice implication algebra; the greatest element of lattice

o the smallest element of lattice implication algebra; the smallest element of lattice

[a, b 1 the interval in lattice

~A congruence relation induced by the fuzzy LI-ideal A == I congruence induced by the mapping of f D- ker(f) dual kernel of mapping f ker(f) kernel of mapping f XA the characteristic function of the set A §dU) the set of all L-fuzzy subsets on U

§(X) the set of all fuzzy subsets on X

L/ F lattice implication quotient algebra induced by filter F hom( L1, L2) the set of all lattice implication homomorphisms from L1 to L2 max maximum

min minimum

TF(L) generated topology by F(L) Ux the neighborhood base of x

(L, A) topology lattice

v valuation

F, IF semantical implies

f-, If- syntactical implies

Fa, I Fa a semantically implies

f-a, If-a a syntactical implies

Fq strong a syntactical implies

LP(X) the lattice-valued propositional logic

LF(X) the lattice-valued first-ordered logic

Lvpl the gradational lattice-valued propositional logic

Lvii the gradational lattice-valued first-ordered logic

F the set of formulae for propositional logic LP(X) § the set of formulae for first-ordered logic LF(X) §p the set of formulae for gradational lattice-valued propositional logic Lvpl

FI the set of formulae for gradational lattice-valued first-ordered logic Lvii

Con the set of consequence

Ded the set of deduction

vi List of symbols

FLC the category of fuzzy lattice implication algebra

F(LC) the forgetful category of LC

I the greatest element of lattice implication algebra; the greatest element of lattice

o the smallest element of lattice implication algebra; the smallest element of lattice

[a, b 1 the interval in lattice

~A congruence relation induced by the fuzzy LI-ideal A == I congruence induced by the mapping of f D- ker(f) dual kernel of mapping f ker(f) kernel of mapping f XA the characteristic function of the set A §dU) the set of all L-fuzzy subsets on U

§(X) the set of all fuzzy subsets on X

L/ F lattice implication quotient algebra induced by filter F hom( L1, L2) the set of all lattice implication homomorphisms from L1 to L2 max maximum

min minimum

TF(L) generated topology by F(L) Ux the neighborhood base of x

(L, A) topology lattice

v valuation

F, IF semantical implies

f-, If- syntactical implies

Fa, I Fa a semantically implies

f-a, If-a a syntactical implies

Fq strong a syntactical implies

LP(X) the lattice-valued propositional logic

LF(X) the lattice-valued first-ordered logic

Lvpl the gradational lattice-valued propositional logic

Lvii the gradational lattice-valued first-ordered logic

F the set of formulae for propositional logic LP(X) § the set of formulae for first-ordered logic LF(X) §p the set of formulae for gradational lattice-valued propositional logic Lvpl

FI the set of formulae for gradational lattice-valued first-ordered logic Lvii

Con the set of consequence

Ded the set of deduction

V interpretation

.f interpretation

(Tn, t n) n-ary inference rule

fJl the set of inference rules

val(w) the value the proof w

l ( w) the length of proof w Ra(Dj, D k ) a-resolvent of Dj and Dk

a-@ a-empty clause

~o denumerable infinity ~ be defined as

N the set of {a, 1, ... }

N+ the set of {I, 2, ... }

List of symbols vii

V interpretation

.f interpretation

(Tn, t n) n-ary inference rule

fJl the set of inference rules

val(w) the value the proof w

l ( w) the length of proof w Ra(Dj, D k ) a-resolvent of Dj and Dk

a-@ a-empty clause

~o denumerable infinity ~ be defined as

N the set of {a, 1, ... }

N+ the set of {I, 2, ... }

List of symbols vii

List of Figures

1.1 Research Structure Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Hasse Diagram of L = {O,a,b,c,d,I} ...................... 30 2.2 Hasse Diagram of L = {O, a, b, I} ........................ " 31

11.1 Commutative Graph 1 for UH . ...............•............ 352 11.2 Commutative Graph 2 for UH . ...............•............ 353 11.3 Commutative Graph for v H . .............................. 353 11.4 Graph for L N+ .......................................... 355 11.5 Lift Lemma of a-Resolution Principle ...................... 358

List of Figures

1.1 Research Structure Graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Hasse Diagram of L = {O,a,b,c,d,I} ...................... 30 2.2 Hasse Diagram of L = {O, a, b, I} ........................ " 31

11.1 Commutative Graph 1 for UH . ...............•............ 352 11.2 Commutative Graph 2 for UH . ...............•............ 353 11.3 Commutative Graph for v H . .............................. 353 11.4 Graph for L N+ .......................................... 355 11.5 Lift Lemma of a-Resolution Principle ...................... 358

List of Tables

1.1 Comparison of the Characteristics of Symbolism, Connectionism and Behaviorism. .. . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Implication Operator of L = {O, a, b, c, I}. . . . . . . . . . . . . . . . . .. 29 2.2 Implication Operator of L = {O,a,b,c,d,I} ................ 30 2.3 Implication Operator of L = {O, a, b,I} .. , . . . . . . . . . . . . . . . .. 30 2.4 Operators of L = {O, a, b, I} . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31

11.1 Nine Cases for x, y, z ..................................... 330

List of Tables

1.1 Comparison of the Characteristics of Symbolism, Connectionism and Behaviorism. .. . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Implication Operator of L = {O, a, b, c, I}. . . . . . . . . . . . . . . . . .. 29 2.2 Implication Operator of L = {O,a,b,c,d,I} ................ 30 2.3 Implication Operator of L = {O, a, b,I} .. , . . . . . . . . . . . . . . . .. 30 2.4 Operators of L = {O, a, b, I} . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31

11.1 Nine Cases for x, y, z ..................................... 330

Table of Contents

Preface....................................................... iv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi

Part I Introduction

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Major Methodologies in Artificial Intelligence. . . . . . . . . . . . . . 3 1.2 Basic Academic Ideas. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Some Related Concepts ................................. 7 1.4 Many-Valued Logic and Lattice-Valued Logic .............. 12 1.5 Uncertainty Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16

1.5.1 Probability-Based Uncertainty Reasoning. . . . . . . . . . .. 16 1.5.2 Fuzzy Set Based Uncertainty Reasoning. . . . . . . . .. . .. 19 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning. .. 20

1.6 Automated Reasoning in Many-Valued Logic. . . . . . . . . . . . . .. 22

Part II Lattice Implication Algebras

2 Concepts and Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27 2.1 Lattice Implication Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27

2.1.1 Concepts and Examples. . . . . . . . . . . . . . . . . . . . . . . . . .. 27 2.1.2 Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

2.2 Lattice H Implication Algebras. . . . . . . . . . . . . . . . . . . . . . . . .. 47 2.3 Lattice Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 2.4 Homomorphisms....................................... 53

Table of Contents

Preface....................................................... iv

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. viii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xi

Part I Introduction

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Major Methodologies in Artificial Intelligence. . . . . . . . . . . . . . 3 1.2 Basic Academic Ideas. .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Some Related Concepts ................................. 7 1.4 Many-Valued Logic and Lattice-Valued Logic .............. 12 1.5 Uncertainty Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 16

1.5.1 Probability-Based Uncertainty Reasoning. . . . . . . . . . .. 16 1.5.2 Fuzzy Set Based Uncertainty Reasoning. . . . . . . . .. . .. 19 1.5.3 Non-Monotonic Logic Based Uncertainty Reasoning. .. 20

1.6 Automated Reasoning in Many-Valued Logic. . . . . . . . . . . . . .. 22

Part II Lattice Implication Algebras

2 Concepts and Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27 2.1 Lattice Implication Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27

2.1.1 Concepts and Examples. . . . . . . . . . . . . . . . . . . . . . . . . .. 27 2.1.2 Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

2.2 Lattice H Implication Algebras. . . . . . . . . . . . . . . . . . . . . . . . .. 47 2.3 Lattice Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50 2.4 Homomorphisms....................................... 53

xiv Table of Contents

3 Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.1 Filters and Implicative Filters. . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.2 Generated Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 3.3 Positive Implicative Filters and Associative Filters. . . . . . . . .. 69 3.4 Prime Filters and Ultra-Filters. . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.5 I-Filters, Involution Filters and Obstinate Filters. . . . . . . . . .. 77 3.6 Fuzzy Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80

4 LI-Ideals. . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85 4.1 LI-Ideals.............................................. 85 4.2 Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 4.3 Normal Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90 4.4 Intuitionistic Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94

5 Homomorphisms and Representations .................... 101 5.1 Congruence Relations ................................... 101

5.1.1 Congruence Relations Induced by Filters ............ 101 5.1.2 Congruences Relations Induced by LI-ideals ......... 111 5.1.3 Congruence Relations Induced by Fuzzy Filters ...... 113 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals .... 119

5.2 Proper Lattice Implication Algebras ...................... 122 5.3 Representations ........................................ 127

6 Topological Structure of Filter Spaces .................... 135 6.1 Filter Spaces ........................................... 135

6.1.1 Basic Concepts ................................... 135 6.1.2 Topological Properties ............................ 138

6.2 Product Topology and Quotient Topology ................. 141 6.3 Lattice Topology ....................................... 144 6.4 Prime Spaces .......................................... 145

7 Connections with Related Algebras ....................... 153 7.1 Lattice Implication Algebras and BCK-Algebras ........... 153 7.2 Lattice Implication Algebras and MV-Algebras ............ 157 7.3 Lattice Implication Algebras and Related Algebras ......... 162

8 Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.1 Category of Lattice Implication Algebras .................. 171 8.2 Category of Fuzzy Lattice Implication Algebras ............ 178 8.3 Fuzzy Power Sets ....................................... 185 8.4 Adjoint Semigroups ..................................... 192 8.5 Logical Properties ...................................... 200

xiv Table of Contents

3 Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.1 Filters and Implicative Filters. . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 3.2 Generated Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65 3.3 Positive Implicative Filters and Associative Filters. . . . . . . . .. 69 3.4 Prime Filters and Ultra-Filters. . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.5 I-Filters, Involution Filters and Obstinate Filters. . . . . . . . . .. 77 3.6 Fuzzy Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80

4 LI-Ideals. . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85 4.1 LI-Ideals.............................................. 85 4.2 Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88 4.3 Normal Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90 4.4 Intuitionistic Fuzzy LI-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . .. 94

5 Homomorphisms and Representations .................... 101 5.1 Congruence Relations ................................... 101

5.1.1 Congruence Relations Induced by Filters ............ 101 5.1.2 Congruences Relations Induced by LI-ideals ......... 111 5.1.3 Congruence Relations Induced by Fuzzy Filters ...... 113 5.1.4 Congruence Relations Induced by Fuzzy LI-ideals .... 119

5.2 Proper Lattice Implication Algebras ...................... 122 5.3 Representations ........................................ 127

6 Topological Structure of Filter Spaces .................... 135 6.1 Filter Spaces ........................................... 135

6.1.1 Basic Concepts ................................... 135 6.1.2 Topological Properties ............................ 138

6.2 Product Topology and Quotient Topology ................. 141 6.3 Lattice Topology ....................................... 144 6.4 Prime Spaces .......................................... 145

7 Connections with Related Algebras ....................... 153 7.1 Lattice Implication Algebras and BCK-Algebras ........... 153 7.2 Lattice Implication Algebras and MV-Algebras ............ 157 7.3 Lattice Implication Algebras and Related Algebras ......... 162

8 Related Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8.1 Category of Lattice Implication Algebras .................. 171 8.2 Category of Fuzzy Lattice Implication Algebras ............ 178 8.3 Fuzzy Power Sets ....................................... 185 8.4 Adjoint Semigroups ..................................... 192 8.5 Logical Properties ...................................... 200

Table of Contents xv

Part III Lattice-Valued Logic Systems

9 Lattice-Valued Propositional Logics ....................... 207 9.1 Lattice-Valued Propositional Logic LP(X) . ................ 207

9.1.1 Language ........................................ 207 9.1.2 Semantics ....................................... 207 9.1.3 Syntax .......................................... 214 9.1.4 Examples ....................................... 226

9.2 Gradational Lattice-Valued Propositional Logic Lvpl ........ 227 9.2.1 Language ........................................ 227 9.2.2 Rules of Inference ................................ 228 9.2.3 Semantics ....................................... 233 9.2.4 Syntax .......................................... 240 9.2.5 Satisfiability and Consistency ...................... 246 9.2.6 Deduction Theorem .............................. 249 9.2.7 Compactness .................................... 251 9.2.8 Examples ....................................... 256

10 Lattice-Valued First-Order Logics ........................ 259 10.1 Lattice-Valued First-Order Logic LF(X) .................. 259

10.1.1 Language ........................................ 259 10.1.2 Interpretation .................................... 260 10.1.3 Semantics ....................................... 261 10.1.4 Syntax .......................................... 265 10.1.5 Properties of Model Theory ........................ 272

10.2 Gradational Lattice-Valued First-Order Logic LvII • ..•...... 278 10.2.1 Language ........................................ 278 10.2.2 Interpretation .................................... 278 10.2.3 Semantics ....................................... 280 10.2.4 Standardization of Formulae ....................... 290 10.2.5 Syntax .......................................... 294 10.2.6 Soundness and Completeness ...................... 300 10.2.7 Satisfiability and Consistency ...................... 301 10.2.8 Deduction Theorem .............................. 302 10.2.9 Compactness .................................... 302 1O.2.10Examples ....................................... 303

11 Uncertainty and Automated Reasoning ................... 305 11.1 Uncertainty Reasoning Based on LP(X) .................. 305 11.2 Uncertainty Reasoning Based on L vpl .•................... 310

Table of Contents xv

Part III Lattice-Valued Logic Systems

9 Lattice-Valued Propositional Logics ....................... 207 9.1 Lattice-Valued Propositional Logic LP(X) . ................ 207

9.1.1 Language ........................................ 207 9.1.2 Semantics ....................................... 207 9.1.3 Syntax .......................................... 214 9.1.4 Examples ....................................... 226

9.2 Gradational Lattice-Valued Propositional Logic Lvpl ........ 227 9.2.1 Language ........................................ 227 9.2.2 Rules of Inference ................................ 228 9.2.3 Semantics ....................................... 233 9.2.4 Syntax .......................................... 240 9.2.5 Satisfiability and Consistency ...................... 246 9.2.6 Deduction Theorem .............................. 249 9.2.7 Compactness .................................... 251 9.2.8 Examples ....................................... 256

10 Lattice-Valued First-Order Logics ........................ 259 10.1 Lattice-Valued First-Order Logic LF(X) .................. 259

10.1.1 Language ........................................ 259 10.1.2 Interpretation .................................... 260 10.1.3 Semantics ....................................... 261 10.1.4 Syntax .......................................... 265 10.1.5 Properties of Model Theory ........................ 272

10.2 Gradational Lattice-Valued First-Order Logic LvII • ..•...... 278 10.2.1 Language ........................................ 278 10.2.2 Interpretation .................................... 278 10.2.3 Semantics ....................................... 280 10.2.4 Standardization of Formulae ....................... 290 10.2.5 Syntax .......................................... 294 10.2.6 Soundness and Completeness ...................... 300 10.2.7 Satisfiability and Consistency ...................... 301 10.2.8 Deduction Theorem .............................. 302 10.2.9 Compactness .................................... 302 1O.2.10Examples ....................................... 303

11 Uncertainty and Automated Reasoning ................... 305 11.1 Uncertainty Reasoning Based on LP(X) .................. 305 11.2 Uncertainty Reasoning Based on L vpl .•................... 310

xvi Table of Contents

11.2.1 Another Kind of Interpretation of X IF Y ........... 310 11.2.2 Basic Theory .................................... 311 11.2.3 Examples ....................................... 319 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning322

Models and Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Semantical Interpretation and Syntactical Proof. . . . . . 324

11.3 a-Resolution Principle Based on LP(X) ................... 328 11.3.1 a-Resolution Principle ............................ 328 11.3.2 Soundness and Completeness ...................... 333

11.4 a-Resolution Principle Based on LF(X) . .................. 349 11.4.1 Interpretation of Formulae. . . . . . . . . . . . . . . . . . . . . . . . . 350 11.4.2 a-Resolution Principle ............................ 353

References . ................................................... 361

Index ......................................................... 389

xvi Table of Contents

11.2.1 Another Kind of Interpretation of X IF Y ........... 310 11.2.2 Basic Theory .................................... 311 11.2.3 Examples ....................................... 319 11.2.4 Multi-Dimensional and Multiple Uncertainty Reasoning322

Models and Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 Semantical Interpretation and Syntactical Proof. . . . . . 324

11.3 a-Resolution Principle Based on LP(X) ................... 328 11.3.1 a-Resolution Principle ............................ 328 11.3.2 Soundness and Completeness ...................... 333

11.4 a-Resolution Principle Based on LF(X) . .................. 349 11.4.1 Interpretation of Formulae. . . . . . . . . . . . . . . . . . . . . . . . . 350 11.4.2 a-Resolution Principle ............................ 353

References . ................................................... 361

Index ......................................................... 389