Methods for Integration of Knowledge in Logic

Structures

by

c⃝ Trong Hieu Tran

A thesis submitted to the

Wroclaw University of Technology (PWr)

and

Swinburne University of Technology (SUT)

in total fulfilment of the

requirements for the degree of

Doktor Nauk Technicznych (by PWr)

and

Doctor of Philosophy (PhD) (by SUT)

Wroclaw University of Technology

and

Swinburne University of Technology

2013

.

.

ii

Abstract

Knowledge integration is an active research field with many important applica-

tions, and many approaches to knowledge integration have been proposed in the last

two decades. However, most of these approaches require the integration process to be

handled by an independent and impartial mediator without taking into account the

roles of participant agents. Consequently, all the knowledge bases are required to be

completely provided up-front. Under certain circumstances, these assumptions can

be too strong and, thus inapplicable, particularly for a group of agents whose objec-

tives are conflicting. Therefore, it is important to develop alternative approaches to

knowledge integration in which the knowledge bases to be integrated are not required

to be completely provided up-front. In order to meet this requirement, consensus and

negotiation have emerged as suitable and powerful tools for knowledge integration in

these systems.

This research develops several novel frameworks for knowledge integration for

different concrete knowledge base structures, including conjunctive and disjunctive

knowledge base structures, propositional knowledge base structure and stratified

knowledge base structure by consensus-based and negotiation-based techniques. Some

aspects of knowledge integration, including rationality and computational complexity

are investigated and discussed.

The contributions of this research are as follows:

1. A simple framework for knowledge integration of knowledge bases with conjunc-

tive and disjunctive structures: This framework introduces the representations

of logic formulas and of conflicts in the system of knowledge bases, and the def-

iii

inition of distance functions between conjunctions or disjunctions. Two models

of knowledge integration are discussed, namely the axiomatic and constructive

models. In the former model, a set of postulates to characterize the desir-

able properties of consensus-based knowledge integration operators is proposed.

The latter model presents a family of consensus-based knowledge integration

operators based on distance functions and aggregation functions. Some repre-

sentation results of two proposed models are presented and analyzed. Logical

properties are discussed, and an algorithm to determine the integration result

for a common integration operator is introduced.

2. Consensus-based knowledge integration framework for the set of knowledge

bases represented in classical propositional logic together with the integrity

constraint: This framework also includes an axiomatic model and a construc-

tive model. In the first model, a set of postulates representing the rational

criteria are introduced and analyzed. The second model is presented based

on the notion of consensus assignment, and some representation theorems are

stated. A family of consensus-based integration operators providing possible in-

stantiations of the proposed constructive model is discussed. Some connections

to related works are introduced and analyzed. Lastly, several computational

complexity results in this framework are discussed.

3. A novel framework for integrating stratified knowledge bases using the nego-

tiation techniques: The integration mechanism is considered at both syntactic

and semantic levels. At each level, the integration mechanism is presented

in both constructive and axiomatic approaches. The former approach points

iv

out the way to integrate knowledge bases, and the latter approach introduces

the set of postulates for which the integration results should be satisfied. The

representation theorems to establish the connections between the approaches

are stated. Lastly, the logical properties of integration operators are examined

and analyzed, and some computational complexity results are mentioned and

evaluated.

v

Streszczenie

Integracja wiedzy jest aktywnym obszarem badań z dużą liczbą zastosowań ważnych

w dziedzinie Sztucznej Inteligencji. W poprzednich dwóch dekadach zostało zapro-

ponowane wiele metod integracji wiedzy. Autorzy tego artykułu stwierdzili jednak,

że metody te wymagają obsługi procesu łączenia przez niezależnego mediatora nie

uwzględniającego roli uczestniczących agentów, oraz by wszystkie bazy wiedzy były

w pełni znane zawczasu. Te założenia są czasem zbyt mocne i w związku z tym

odpowiednie tylko do idealnych sytuacji; są one zazwyczaj niemożliwe do zastosowania

w systemach wieloagentowych. Z uwagi na to opracowanie nowych metod integracji

wiedzy do użycia w systemach wieloagentowych staje się kluczowe. W celu spełnienia

tych wymagań wykorzystywane są odpowiednie i silne narzędzia takie jak metody

konsensusu i negocjacji.

Niniejsze badania obejmują opracowanie kilku nowych metod integracji wiedzy dla

wybranych rzeczywistych struktur, włączając to koniunktywne i dysjunkcyjne struk-

tury logiczne, strukturę rachunku zdań i warstwową strukturę wiedzy poprzez użycie

metod konsensusu i metod negocjacji. Dwa z aspektów integracji wiedzy tj. racjonal-

ność i złożoność obliczeniowa są przebadane i przedyskutowane.

Niniejsze badania zawierają następujący wkład:

1. Propozycja metody integracji wiedzy o strukturze koniunktywnej i dysjunkcyjnej

na poziomie syntaktycznym - prostej, lecz często stosowanej. Wykorzystanie

metody obejmuje reprezentację formuł logicznych i ich konfliktów w systemie

baz wiedzy oraz użycie zdefiniowanej funkcji odległości między koniunktywami

i dysjunkcjami. Omówione są dwa modele integracji wiedzy, tj. model aksjo-

vi

matyczny i model konstruktywny. W modelu aksjomatycznym wykorzystywany

jest zbiór postulatów opisujących pożądane własności operatorów integracji

wiedzy opartych na konsensusie. W modelu konstruktywnym wykorzystywany

jest zbiór opartych o konsensus operatorów integracji wiedzy wykorzystujących

funkcje odległości i funkcje agregujące. Zbadane zostały niektóre z możliwych

rezultatów obu proponowanych modelu, wraz z przedstawieniem własności log-

icznych. Opracowany jest także algorytm wyznaczania wyniku integracji dla

ogólnego operatora integracji.

2. Propozycja metody integracji wiedzy dla zbioru baz wiedzy reprezentowanych

przez klasyczny rachunek zdań, oraz ograniczenia na integrację wynikające z

metod konsensusu. Także w tej metodzie wykorzystywane są modele: aksjo-

matyczny i konstruktywny. W modelu aksjomatycznym wprowadzany i anali-

zowany jest zbiór racjonalnych kryteriów reprezentowanych przez postulaty. W

modelu konstruktywnym wykorzystywane jest pojęcie przydziału konsensusu

oraz postawione są pewne twierdzenia dotyczące reprezentacji. Zbiór opartych

o konsensus operatorów integracji jest konkretyzacją takiego użycia metody.

Przedstawione i przeanalizowane są podobieństwa do podobnych prac w liter-

aturze. Przedyskutowana jest także złożoność obliczeniowa tej metody.

3. Propozycja metody integracji struktury warstwowych baz wiedzy poprzez uży-

cie technik negocjacji. Ten rodzaj integracji wiedzy ma miejsce zarówno na

poziomie syntaktycznym jak i semantycznym. Na obu poziomach możliwe jest

użycie modelu konstruktywnego i aksjomatycznego. Model konstruktywny po-

daje sposób przeprowadzenia integracji, natomiast model aksjomatyczny podaje

vii

zbiór postulatów, które mają być spełnione przez zintegrowany wynik. Podane

zostały twierdzenia o reprezentacji jako połączenie obu modeli. Zbadane zostały

także logiczne własności operatorów integracji, oraz przedstawiona została anal-

iza złożoności obliczeniowej.

viii

Acknowledgements

First and foremost, I would like to express my profound gratitude to my princi-

pal supervisor, Professor Ngoc Thanh Nguyen, at Wroclaw University of Technology

(WUT), Poland for his great supervision and enormous support to my research at

Wroclaw University of Technology. His invaluable guidance is the key to my success

in completing my PhD.

Also, I would like to extend deep gratitude to my principal coordinating supervi-

sor, Dr. Quoc Bao Vo, at Swinburne University of Technology (SUT), Australia. His

expertise and background knowledge in the field of multi-agent systems provided the

guidances with many insightful suggestions and innovative ideas during my research.

I am grateful to my wife, Thi Hong Khanh Nguyen and my son Minh Tri Tran for

their boundless patience and support. My appreciation also goes to my friends and

colleagues at Knowledge Management Systems Division, WUT and at Swinburne Uni-

versity Centre for Computing and Engineering Software Systems(SUCCESS), SUT

who always helped, encouraged, and motivated me to study and complete the re-

search.

Last but not least, I would like to thank Joint Supervision in Dual PhD Program

between Wroclaw University of Technology and Swinburne University of Technology

for providing the best conditions for me to carry out my research.

ix

Declaration

I, Trong Hieu Tran, declare that this thesis entitled:

“Methods for Integration of Knowledge in Logic Structures”

is my own work and has not been submitted previously, in whole or in part, in

respect of any other academic award.

...........................

Trong Hieu Tran

Centre for Computing and Engineering Software Systems (SUCCESS)

Faculty of Information & Communication Technologies

Swinburne University of Technology

Melbourne, Australia

x

Publications

Journal paper :

1. T. H. Tran, N. T. Nguyen, and Q. B. Vo. Axiomatic characterization of be-

lief merging by negotiation. Multimedia Tools and Applications, 65(1):133-

159, 2013.

Book chapter :

2. Tran T.H.(2007): Method for Measuring the Distance Between Disjunctive

Formulae, Knowledge Processing and Reasoning for Information Society.

Akademicka Oficyna Wy-dawnicza EXIT. 39-50.

Conference papers :

3. Tran T.H., Vo Q.B. (2012): An Axiomatic Model for Merging Stratified

Belief Bases by Negotiation, Lecture Notes in Computer Science 7653,

174-184.

4. Tran H.T., Vo Q.B., Kowalczyk R.(2011): Merging Belief Bases by Nego-

tiation. Lecture Notes in Computer Science 6881, 200-209.

5. Tran T.H., Nguyen N.T.(2009): Security Policy Integration Method for

Information Systems, Proceeding of ACIIDS 2009, 220-225.

6. Tran T.H., Nguyen N.T.(2009): A Consensus-Based Integration Method

for Security Rules, Lecture Notes in Artificial Intelligence 5711, 54-61.

7. Tran T.H., Nguyen N.T. (2008): Integration of Knowledge in Disjunctive

Structure on Semantic Level, Lecture Notes in Artificial Intelligence 5177,

253-261.

xi

8. Tran T.H., Nguyen N.T. (2008): An Algorithm for Agent Knowledge In-

tegration Using Conjunctive and Disjunctive Structures, Lecture Notes in

Artificial Intelligence 4953, 692-701.

9. Tran T.H., Nguyen N.T.(2007): Distance Functions for Logic Conjunctions

in Knowledge Integration Tasks. Proceedings of the 16th International

Conference on Systems Science, Wroclaw : Oficyna Wydaw. PWroc., 454-

462.

10. Tran T.H., Nguyen N.T.(2007): An Integration Method for Logic Clauses,

Information systems architecture and technology. Information systems and

computer communication networks. Wroclaw : Oficyna Wydaw. PWroc.,

21-28.

11. Tran T.H.(2006): Paraconsistent Logics for Inconsistent Knowledge Pro-

cessing, Proceedings of the 16th International Conference on Systems Sci-

ence, Wroclaw : Oficyna Wydaw. PWroc., 39-48.

xii

Contents

Abstract ii

Abstract in Polish vi

Acknowledgements ix

Declaration x

Publications xi

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Research Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Contributions of the Research . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Formal Preliminaries 17

2.1 Set, Binary Relation and Preorder . . . . . . . . . . . . . . . . . . . . 17

xiii

2.2 Aggregation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Propositional Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1.1 Conjunctive and Disjunctive Normal Forms . . . . . 24

2.3.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Paraconsistent Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Weakly-negative Logics . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1.1 Proof theory for Cw . . . . . . . . . . . . . . . . . . 29

2.4.1.2 A Semantic Tableau Procedure for Cw . . . . . . . . 30

2.4.2 Belnap’s Four-valued Logic . . . . . . . . . . . . . . . . . . . . 31

2.4.2.1 Semantics for Four-valued Logic . . . . . . . . . . . . 32

2.4.2.2 Proof Theory for Four-valued Logic . . . . . . . . . . 33

2.4.3 Quasi-classical Logic . . . . . . . . . . . . . . . . . . . . . . . 35

2.4.3.1 Proof theory of QC logic . . . . . . . . . . . . . . . . 35

2.4.3.2 Semantics for QC logic . . . . . . . . . . . . . . . . . 37

2.4.4 Measuring the Coherence in QC models . . . . . . . . . . . . 39

2.5 Possiblistic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Possibility Distributions . . . . . . . . . . . . . . . . . . . . . 41

2.5.2 Possibilistic Logic Knowledge Bases . . . . . . . . . . . . . . . 42

2.5.3 Subsumption and Inference in Possibilistic Logic . . . . . . . . 43

2.6 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.6.2 Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.6.3 Complexity Classes . . . . . . . . . . . . . . . . . . . . . . . . 49

xiv

3 Literature Review on Knowledge Integration 54

3.1 Knowledge Integration in Social Science . . . . . . . . . . . . . . . . 54

3.1.1 Alternatives Ranking Problem . . . . . . . . . . . . . . . . . . 55

3.1.2 Committee Election Problem . . . . . . . . . . . . . . . . . . 59

3.2 Knowledge integration in Artificial Intelligence . . . . . . . . . . . . . 61

3.2.1 Belief Revision . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.2 Logical Characterization of Belief Merging Operators . . . . . 71

3.2.2.1 Revesz’s Arbitration . . . . . . . . . . . . . . . . . . 71

3.2.2.2 Arbitration of Liberatore and Schaerf . . . . . . . . . 74

3.2.2.3 Majority Operators of Lin and Mendelzon . . . . . . 77

3.2.2.4 Merging Operator of Konieczny and Pino Pérez . . . 80

3.2.2.5 Connections between Merging with Integrity Constraint

and the Related Work . . . . . . . . . . . . . . . . . 86

3.2.3 Several Merging Operators . . . . . . . . . . . . . . . . . . . . 90

3.2.3.1 Model-based Operators . . . . . . . . . . . . . . . . 92

3.2.3.2 Syntax-based Operators . . . . . . . . . . . . . . . . 95

3.2.3.3 Conflict-based Operators . . . . . . . . . . . . . . . . 97

3.2.3.4 DA2 Operators . . . . . . . . . . . . . . . . . . . . . 98

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4 Knowledge Integration on Conjunctive and Disjunctive Structures 101

4.1 Formal Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.1.1 Representation of Conflict in Multi-agents Systems . . . . . . 105

4.1.2 Overview of Consensus-based Knowledge Integration . . . . . 107

xv

4.2 Axiomatic Model for Knowledge Integration . . . . . . . . . . . . . . 108

4.3 Constructive Model for Knowledge Integration . . . . . . . . . . . . . 111

4.3.1 Distance Functions for Conjunctions and Disjunctions . . . . . 111

4.3.2 Aggregation Functions . . . . . . . . . . . . . . . . . . . . . . 113

4.3.3 Family of Consensus-based Integration Operators . . . . . . . 114

4.4 Logical Properties and Integration Algorithm . . . . . . . . . . . . . 125

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5 Consensus-based Knowledge Integration 130

5.1 Axiomatic Model for Knowledge Integration . . . . . . . . . . . . . . 131

5.1.1 Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.1.2 Postulates for Knowledge Integration by Consensus . . . . . . 132

5.2 Constructive Model and Representation Theorems . . . . . . . . . . . 135

5.3 Instantiations of the Integration Framework . . . . . . . . . . . . . . 144

5.3.1 Σ Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.3.2 Max Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.3.3 GMax Operators . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . . . . . 151

5.5 Connections with Related Work . . . . . . . . . . . . . . . . . . . . . 155

5.5.1 Connection with Social Choice theory . . . . . . . . . . . . . . 155

5.5.2 Connection with Belief Revision and Belief Merging . . . . . . 157

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6 Belief Merging by Negotiation 159

6.1 Formal Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

xvi

6.1.1 Classical Propositional Logic . . . . . . . . . . . . . . . . . . . 160

6.1.2 Stratified Knowledge Base . . . . . . . . . . . . . . . . . . . . 161

6.1.3 Background on Negotiation . . . . . . . . . . . . . . . . . . . 161

6.1.3.1 Negotiation . . . . . . . . . . . . . . . . . . . . . . . 161

6.1.3.2 Belief Merging vs Negotiation . . . . . . . . . . . . . 163

6.1.4 Running example . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.2.1 Social Contraction Function and Belief Negotiation . . . . . . 165

6.2.2 Belief Merging Approach for Bargaining Game . . . . . . . . . 169

6.3 Syntax-based Model for Belief Merging by Negotiation . . . . . . . . 175

6.3.1 Negotiation Model for Belief Merging . . . . . . . . . . . . . 175

6.3.2 Postulates and Logical Properties . . . . . . . . . . . . . . . . 180

6.4 Model-based Model of Negotiation for Belief Merging . . . . . . . . . 185

6.4.1 From Stratified Belief Base to Preferences . . . . . . . . . . . 186

6.4.2 Negotiation on the Preferences . . . . . . . . . . . . . . . . . . 187

6.4.3 Logical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.4.4 Computational Complexity . . . . . . . . . . . . . . . . . . . . 191

6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7 Conclusion and Future Work 197

7.1 Research Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.2 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . 202

7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Bibliography 206

xvii

A Classical Reasoning 225

A.1 Language and Proof Theory . . . . . . . . . . . . . . . . . . . . . . . 225

A.2 Properties of Consequence Relations . . . . . . . . . . . . . . . . . . 226

xviii

Chapter 1

Introduction

1.1 Background and Motivation

Over the last few decades, knowledge integration has emerged as an important re-

search area in both Computer Science and Social Science. Knowledge integration

relates to studies on Social Choice theory in Social Science such as the Alternatives

Ranking Problem [94, 117, 49, 4] and the Committee Election Problem [94, 117, 49,

78, 10, 135]. In Computer Science, they are of interest to researchers in the fields of

Database Systems [37, 113, 7, 119], Image Processing [106] and Artificial Intelligence

(AI) [136, 127, 119, 112, 103, 83, 73, 72, 70, 74, 39, 25, 8, 37].

Knowledge integration was first investigated formally in the political context by

Condorcet in his seminal essay in the 18th century [38]. In this essay, he studied the

application of probability analysis to decision-making by majority vote and pointed

out that plurality voting systems may not well represent the wishes of voters. His

essay provided a method of election by voting in which the winner should be the

candidate who wins by majority rule in all pairings against the other candidates. It

1

also highlighted the drawback of the method by a well-known problem known as the

paradox called the Condorcet paradox, which refers to the possible intransitivity of

majority rule. In 1951, the economist Kenneth Arrow proposed a set of formal criteria

for systems aiming to establish the collective choice from individual preferences, and

he surprisingly showed by an impossibility theorem that no voting system can meet

all these criteria [4].

Nowadays, knowledge integration problems are investigated across multiple sub-

fields of Artificial Intelligence (AI). The solutions to these problems are relevant to

the area of database systems when multiple databases need to be merged [37, 113],

to information retrieval when multiple sources of information need to be integrated

[7, 119], and to multi-agent systems where agents with different beliefs about a domain

need to reach a consensus to coordinate their activities [103, 114]. It is also related

to other application domains, including multimedia information retrieval systems [80]

and medical imaging systems [106].

The main purpose of knowledge integration is to overcome the incompleteness and

inconsistency of knowledge. Incompleteness and inconsistency are features of knowl-

edge which are characterized by the lack of the possibility for inference processes.

Therefore, solving them is a basic and essential task in knowledge management. Ac-

cording to Nguyen[100], the incompleteness and inconsistency of knowledge usually

arise mainly in the following cases:

• Knowledge is gathered during a period of time, and it is usually assigned the

time stamp [95, 123].

• Knowledge results from extracting databases, for example, by data mining

2

methods. The extracted rules are dependent on the data, and some of them

may be contradictory to each other [48].

• Inconsistency and incompleteness of knowledge also arise in distributed envi-

ronments, for example, in multi-agent systems. The same real world situation

can be interpreted by different entities with different (even conflicting) versions

of knowledge [100, 88].

Many examples of incompleteness and inconsistency of knowledge can be found in a

distributed medical expert system when different experts disagree on the diagnosis of

patients’ diseases. In a multi-database system when two component databases may

record the same data item, but give it different values because of incomplete updates,

system errors, or differences in underlying semantics may occur.

Knowledge integration has been used to solve the incompleteness and inconsis-

tency of knowledge. Knowledge integration tasks are known to be important when

it is intended to make a fusion of several intelligent systems or to make cooperation

possible. One of the necessary conditions for successful cooperation is the consistency

of the knowledge of these systems. Knowledge integration is understood as the task

of creating (a) new piece(s) of knowledge from a set of different pieces of knowledge,

which may be incomplete and inconsistent with each other.

Because of the autonomy feature and non-deterministic mechanisms for knowledge

processing of systems, there may appear such a situation that knowledge about the

same real world may be reflected differently in the systems. Knowledge integration is

difficult because the inconsistency of knowledge is hard to detect, and resolving the

inconsistency is also a complex problem. However, without knowledge integration

3

capability, cooperation between these systems is not possible.

In the literature, knowledge integration is a general concept; it is similar or closely

related to other concepts. For example, in Social Science, knowledge integration is

related to Social Choice theory, which has been studied in politics, economics and

law [100, 135, 133, 134, 117, 94, 93, 90, 92, 91, 78, 68, 49, 29, 12, 11, 10, 5, 4]. So-

cial Choice theory studies how individual preferences can be aggregated to produce

a collectively preferred alternative. In AI, the concept of knowledge is sometimes

synonymous with the concept of belief, and knowledge integration is understood as

the belief revision or belief merging process. Belief revision investigates the dynamics

of the process of belief change when an agent is faced with a new belief which may

contradict its current beliefs. That is, it may have to retract some of its beliefs in

order to accommodate the new one consistently. The main concern here is about

rationality and fairness in the way the old beliefs are retracted. There is a vast liter-

ature on the subject, including [2, 18, 19, 32, 52, 51, 115]. Similarly, belief merging

investigates ways to aggregate a number of individual belief bases into a collective

one [115, 112, 110, 93, 84, 85, 82, 76, 72, 71, 75, 69, 74, 77, 46, 47, 45, 29, 17]. Since in

belief revision the new information is always considered more reliable (or important)

than the old, belief revision can be viewed as a special case of belief merging.

Several structures have been investigated in knowledge integration research, in-

cluding linear orders [134, 49, 4, 33], semi-lattices [11], n-trees [36, 58], ordered par-

titions and coverings [33], incomplete ordered partitions [58], non-ordered partitions

[12], weak hierarchies [92], and time interval [98, 101]. They are largely non-logical

and most of the studies are based on the Kemeny median to determine the consensus

of collection of rankings; they are computationally hard problems.

4

In this research, we intend to focus on the problem of integrating knowledge on

several knowledge base structures. The knowledge integration problem is stated as

follows: Given a set of knowledge bases represented as sets of formulas in a concrete

knowledge base structure, one should determine a common knowledge base which best

represents these knowledge bases. The knowledge base structures which will be studied

in this research are conjunctive structure, disjunctive structure, classical propositional

logic and stratified knowledge base structure.

Technically, in most knowledge integration works, the individual knowledge bases

are assumed to be exposed completely, and the integration process is similar to the

arbitration. This assumption can be too strong and difficult to apply to multi-agent

systems. For instance, in order to have a business contract, partner companies will

supply small pieces of information until they reach agreement. They may resist ex-

posing all information about their business strategies, market evaluations or even the

information they know about other partners, etc. To overcome this drawback, in this

thesis, we propose a novel integration framework in which the knowledge integration

process is organized as a game and participant agents join in a negotiation process to

reach consensus.

For most application domains, knowledge integration is complex and challenging

for the following reasons:

• The inevitability of inconsistency: Because ”almost all the information we have

about the real world is not certain, complete, or precise” [105] together with the

autonomy feature and non-deterministic mechanisms for knowledge processing

of systems, there may appear such situations that the knowledge about the same

5

real world may be reflected differently in the systems. The aim of knowledge

integration is to determine the unified knowledge base on the basis of knowledge

bases generated by autonomous units.

• The drawbacks of existing integration methods: Although knowledge integra-

tion has been investigated for a large range of structures such as linear orders,

semi-lattices, n-trees, ordered partitions and coverings, incomplete ordered par-

titions, non-ordered partitions, weak hierarchies, and time interval, the methods

for integrating logic-based knowledge have not been addressed entirely. There-

fore, an effective and sound method for logic-based knowledge integration is

needed. On the other hand, the most prominent logic-based knowledge inte-

gration method, namely belief merging, requires that all knowledge bases be

given explicitly and completely up front. Thus, it is difficult to apply existing

approaches to groups of self-interested agents.

• Computational complexity: In consensus and belief merging methods, the space

of possible outcomes from which we have to choose the appropriate solution has

a combinatorial structure, i.e. the number of all possible outcomes is exponen-

tial in the number of variables. For instance, with only five stratified knowledge

bases constructed by four formulas, we have 75 possible partitions of these for-

mulas with preference relations; hence, we have 755 > 109 possible outcomes.

Therefore, computational complexity is one of the challenges in knowledge in-

tegration that needs to be dealt with.

6

1.2 Research Aims

The goal of this research is to investigate the knowledge integration methods in several

knowledge base structures. The thesis focuses on the following issues:

• Developing frameworks for integrating knowledge bases in several concrete struc-

tures, namely conjunctive and disjunctive knowledge base structures, proposi-

tional knowledge base structure and stratified knowledge base structure by using

consensus and negotiation techniques,

• Investigating the issues related to knowledge integration, including rationality

and computational complexity.

1.2.1 Research Questions

To address the above challenges, the main question for the research is as follows: How

can knowledge bases represented as the sets of formulas be integrated in some concrete

knowledge base structures? This question is partitioned in the set of the following

sub-questions:

1. What is the representation of knowledge and how can similarities in each knowl-

edge base structure be measured?

2. How can the conflict in the proposed representation of knowledge be defined?

3. What are the criteria for knowledge integration processes and their properties?

4. How can an integration process be performed under the constraint of the pro-

posed criteria?

7

5. How can knowledge bases be integrated, in case they are stratified?

6. What is the computational complexity of these methods?

1.2.2 Research Objectives

With the purpose of ”Working out effective models for integrating knowledge in several

chosen knowledge base structures”, the objectives of my thesis are summarized as

follows:

1. To present ways to make the knowledge representation for conjunctive structure,

disjunctive structure and stratified knowledge structure,

2. To define the representation of conflict according to knowledge representation,

3. To define similarity measures for the knowledge in each investigated structure,

4. To investigate a set of intuitive and rational criteria for knowledge integration

in each knowledge base structure, i.e. define the set of postulates to characterize

integration results and investigate the properties of these postulates,

5. To work out the families of integration operators which comply with the pro-

posed criteria and the representation theorems to establish the connections be-

tween the sets of criteria and these families of operators,

6. To make some evaluations and comparisons with related work in the literature,

7. To give evaluations of computational complexity.

8

1.3 Contributions of the Research

The main contributions of this research are as follows:

1. We introduce a framework for knowledge integration of the conjunctive and

disjunctive structures on the syntactic level. These structures are simple, but

widely applied. In this framework, the contributions are highlighted as follows.

Firstly, we propose to represent each formula (conjunction or disjunction) by a

tuple of two sets of literals; one is a set of positive literals, and the other is a set

of negative literals. In this way, we can easily use the concepts and notations

of set (e.g. union, intersection, empty set, etc.) to represent the related con-

cepts of knowledge integration. Secondly, we propose an axiomatic model for

knowledge integration which includes a set of intuitive and rational postulates

to characterize integration results. These postulates are Lower bound, Upper

bound, Consistency, Collective Rationality, Majority, Distribution, and Central-

ization. The intuition of these postulates is denoted in their commentaries(see

Section 4.2) and rationality is presented in the discussion about the family of

knowledge integration operators (see Sub-section 4.3.3) and about the logical

properties (see Section 4.4). Thirdly, in the constructive model, we propose to

use both distance functions and aggregation functions to build the knowledge

integration operators. This method is different from and more general than

other pre-existing consensus works for knowledge integration. More specifically,

by using aggregation functions, we can obtain a family of knowledge integration

operators instead of some concrete ones, and the operators in other consensus

works for knowledge integration which are usually built by some distance func-

9

tions and the sum function are in a group of our integration operator family.

Fourthly, we introduce the family of consensus-based integration operators and

justify the properties of several most common operators of this family. Lastly,

we discuss the logical properties based on the references to the set of properties

for belief merging with integrity constraint, and propose a heuristic algorithm

to determine the knowledge integration result according to the knowledge inte-

gration operator CdH ,∑.

2. Knowledge integration with the integrity constraint in classical propositional

logic by consensus techniques is studied in a framework. In general, this work is

the generalization of other consensus works for knowledge integration due to the

presence of the integrity constraint which states the conditions with which the

integration result has to comply. The framework includes an axiomatic model,

a constructive model, several representation results, some instantiations as the

families of integration operators, connections to related works, and several com-

putational complexity results. The contributions in this framework are enumer-

ated as follows. Firstly, in the axiomatic model, we propose a set of desirable

properties for integration results. These properties are presented by postu-

lates including Compliance, Consistency, Collective Rationality, Irrelevance of

Syntax, Distribution, Centralization, and Closeness. They are the major postu-

lates to characterize the family of consensus-based integration operators. Three

supplementary postulates includingMajority, Majority Independence, and Arbi-

tration are also introduced to classify the sub-families of integration operators.

The intuition of the postulates is shown in their commentaries (see Sub-section

10

5.1.2), and their rationality is presented in the discussion about the instantia-

tions of the framework (see Section 5.3) and about the connections with related

works (see Section 5.5). Secondly, we propose a constructive model based on

the notion of consensus assignment, which is a function to map a set of knowl-

edge bases to a total pre-order of interpretations. In this model, we introduce a

set of conditions for consensus assignments and state three representation the-

orems for the connection between the sub-sets of postulates and the sub-set of

conditions for consensus assignments. Thirdly, we introduce some sub-families

of knowledge integration operators and examine the logical properties of some

common integration operators based on reference to the proposed postulates.

Fourthly, we present the connections with related work such as that in Social

Choice theory and in Belief Revision and Belief Merging. Lastly, we evaluate

several computational complexity results in which the most important result is

that most problems to determine whether a knowledge integration result infers

a given formula holds are Θp2 − complete, and some are ∆p2 − complete (see

Section 5.4).

3. Lastly, a novel framework for integrating the stratified knowledge bases by using

negotiation techniques is studied. The novelty of this work is the behaviour of

the knowledge integration process. In two above-mentioned integration frame-

works as well as in other work on knowledge integration, the knowledge bases

are usually assumed to be provided completely up-front, and each integration

process is handled by a mediator without taking into account the agents who

provide the knowledge bases. These requirements are sometimes so strong that

11

the integration work is only suitable for the ideal situation. They are generally

inapplicable for most multi-agent systems. In this framework, the knowledge

integration process is organized as a game in which the agents will make some

concessions in their own belief bases to reach consensus. The first contribution

in this framework is to point out the logical properties of the closely related

works on the basis of reference to the properties of belief merging with integrity

constraint (in Proposition 29 and Proposition 30). The remaining contributions

in this framework are divided into two main parts corresponding to the integra-

tion works on the syntactic and semantic levels.

At the syntactic level, we firstly define a constructive model based on two tool

functions; one is the choice function which chooses the set of agents allowed

to submit knowledge for a common knowledge base in each negotiation round,

and the other is the updating function which updates the common knowledge

base by submitted knowledge from the chosen agents. The negotiation process

is a sequence of calls to the choice and the updating function until agreement

is reached. Secondly, a set of intuitive and rational postulates for knowledge

integration by negotiation is proposed in the axiomatic model. These postu-

lates are Individual Rationality, Consistency, Cooperativity, Pareto Optimality,

and Symmetry. The rationality of these postulates is presented in the discus-

sion about the logical properties, and the intuition of them is presented in their

commentaries. We also introduce the representation theorem to ensure the pos-

sibility of the proposed set of postulates in relation to the defined constructive

model. Lastly, we discuss logical properties and the connections between the

integration result of this work and that of some related works.

12

At the semantic level, we firstly introduce a two-stage constructive model for

knowledge integration by negotiation. In this model, the first stage transforms

the set of stratified knowledge bases into the corresponding set of preferences by

several ordering strategies. The other stage realizes the negotiation of the pref-

erences by using the notion of solution mapping, which is a function to map each

possible outcome to a vector of numbers which reflect the priorities of the pos-

sible outcome in the preferences. By this definition of the constructive model,

we can easily analyze the properties that a possible outcome satisfies. Next, we

propose a set of three postulates to characterize the properties of negotiation

results in the latter stage of the constructive model. The proposed postulates

are Upper bound, Majority, and Lower bound. By the representation theorem,

we point out that these postulates characterize the unique solution determined

by the idea of a well-known egalitarian solution. Moreover, we show that if a

solution satisfies these postulates, then it also satisfies the modified postulates

from the original negotiation work. We modify the set of postulates for belief

merging with integrity constraint to become suitable for stratified belief bases

and discuss the logical properties based on this set of postulates. Lastly, we

evaluate several computational complexity results in which the most important

one is that the problem which determines whether the knowledge integration

result of the stratified knowledge bases by negotiation infers a given formula

holds is Θp2 − complete (see Section 6.4.4).

13

1.4 Outline of the Thesis

In this research, several frameworks for knowledge integration in the knowledge base

structures are presented. They meet the requirements of rationality and compu-

tational complexity of integrating works based on the techniques of consensus and

negotiation.

The research problem about knowledge integration is stated in Chapter 1. This

chapter also includes the motivation which led us to undertake this research, and the

aims and objectives as well as the main contributions of this thesis.

Chapter 2 supplies formal preliminaries about the notations and concepts used for

this thesis. Some fundamental knowledge about propositional logic, para-consistent

logic and possibilistic logic is introduced. The primary concepts of computational

complexity are also presented within this chapter.

The main objective of Chapter 3 is to provide an overview of knowledge integra-

tion in AI in the literature. In this chapter, several characterizations of belief merging

operators by the sets of rational and intuitive postulates are recalled and some com-

mon merging operators are introduced.

In Chapter 4, the integration work in conjunctive and disjunctive structures on

the syntactic level, the simple but widely applied structures, are introduced. The

work includes the representation of conflicts in the system of knowledge bases and

the definition of distance functions between the conjunctions or the disjunctions.

Knowledge integration based on the consensus technique is presented, and a heuristic

algorithm for knowledge integration by this technique is developed. The set of postu-

lates to characterize knowledge integration operators in these structures is proposed,

14

and several analyses of logical properties are carried out.

The knowledge integration work for the set of propositional knowledge bases with

the integrity constraint by the consensus techniques is investigated in Chapter 5. In

this chapter, a framework for knowledge integration including an axiomatic model and

a constructive model is proposed. In the former model, a set of criteria represented

by the postulates is introduced and analyzed. The latter model is presented based on

the notion of consensus assignment, and several representation theorems are stated.

A family of consensus-based integration operators is mentioned as instantiations for

the proposed framework. Some connections to the related works are introduced and

analysed. Lastly, several computational complexity results are discussed in this chap-

ter.

In Chapter 6, a novel framework for integrating stratified belief bases by using

negotiation is investigated. The integration work is considered at both syntactic and

semantic levels. At each level, the integration work is presented in both the con-

structive approach and the axiomatic approach. The constructive approach points

out the way to integrate the belief bases, and the axiomatic approach introduces the

set of postulates by which the merging result should be satisfied. The representation

theorems to establish the connections between the approaches are stated. Lastly, the

logical properties of merging operators are examined and analyzed, and some com-

putational complexity results are evaluated.

Finally, some concluding remarks and possible future work are presented in Chap-

ter 7.

15

1.5 Summary

In this chapter, a general introduction to knowledge integration in both Social Science

and Computer Science is presented. The research field of knowledge integration on

AI is located, and it is the beginning of the literature review presented in Chapter

3. The motivation of the research is discussed, and the importance of research is

also analyzed. The research aims are stated, and they are concretized by the set of

research questions and research objectives which are realized in Chapter 4, Chapter

5 and Chapter 6. Lastly, the outline of the thesis provides an overview and reading

guide for this thesis. We begin with formal preliminaries in Chapter 2.

16

Chapter 2

Formal Preliminaries

This chapter presents a set of fundamental concepts necessary for this thesis. The

concepts include the multi-set, the relation and preorder, the notion of aggregation

function and the properties of these functions, the syntactic and semantic aspects of

propositional logic, some paraconsistent logics, classical possibilistic logic, and the

primary concepts of computational complexity.

2.1 Set, Binary Relation and Preorder

Notation 1. We use N and R to denote the set of natural numbers and the set of

real numbers, respectively. Additionally,

• Set of positive natural numbers (or positive integers): N+ = N\{0},

• Set of non-negative real numbers: R∗ = {x ∈ R|x ≥ 0},

• Set of positive real numbers: R+ = R∗\{0}.

Notation 2. Let E be a set. We use 2E to denote the set of all subsets of E,

sometimes called a power set. If E is finite, we denote ∥E∥ as number of elements

of E. For any n ∈ N+, En denotes the Cartesian product E × . . .× E︸ ︷︷ ︸n times

.

17

Notation 3. Multi-set is a generalization of set that allows each element to occur

multiple times. We also denote ∥E∥ as cardinality of E counted as the sum of occur-

rence times of its elements. An operator⊔

is the multi-set union, and other operator

⊑ is the multi-set inclusion.

Definition 1. Let E be a set. A binary relation R on E is a subset of E×E. When

a pair of elements (x, y) belongs to a relation R, it is typically written as xRy, R(x, y)

or Rxy.

Definition 2. Let R be a binary relation on E. We consider the following properties

of R:

• reflexivity: ∀x ∈ E, xRx.

• irreflexivity (or strict): ∀x ∈ E, (x, x) /∈ R.

• symmetry: ∀x, y ∈ E, if xRy then yRx.

• antisymmetry: ∀x, y ∈ E, if xRy and yRx then x = y.

• transitivity: ∀x, y, z ∈ E, if xRy and yRz then xRz.

• totality: ∀x, y ∈ E, xRy or yRx.

Definition 3. Let E be a set.

• A (partial) preorder on E is a binary relation, on E which is reflexive and

transitive.

• A partial order on E is a preorder on E with antisymmetry.

• A strict partial order on E is a binary relation, on E which is irreflexive and

transitive.

• A total preorder on E is a binary relation, on E which is transitive and total.

18

• A total order, also called linear order, is a total preorder with antisymmetry.

Remark that a total preorder is also a preorder because totality implies reflexivity.

The illustration of these relations is presented in Figure 2.1.

Figure 2.1: Scheme of relations

Notation 4. Let E, F be two sets such that E ⊆ F , and relation ≼ is a total preorder

on F . We use min(E,≼) to denote the set of all minimum elements of E according

to the total preorder ≼. Formally,

19

min(E,≼) = {x|x ∈ E, ∀y ∈ E, x ≼ y}

2.2 Aggregation Functions

In general, an aggregate function [89, 120, 70] is a function that assigns a list of

numerical values to a single value according to a certain set of criteria.

Notation 5. A permutation π is a bijection from a set of natural numbers to itself.

Given two lists of numbers a = (a1, . . . , an) and b = (b1, . . . , bn) where ai and

bj are integers, an order ≤ between a and b is defined as: a ≤ b iff ai ≤ bi for all

i = 1, . . . , n. We also use a < b to indicate that a ≤ b but not b ≤ a.

Let a = (a1, . . . , an) and a permutation π on set {1, . . . , n}, we denote aπ =

(aπ(1), . . . , aπ(n)), a≤ the list a after being arranged in increasing order, i.e. a≤ =

(aπ(1), . . . , aπ(n)) where aπ(i) ≤ aπ(i+1) for 1 ≤ i ≤ n− 1, and a≥ the list a after being

arranged in descending order.

Proposition 1. Given two lists of integers a and b, there exists a permutation π on

{1, . . . , n} such that a ≤ bπ if and only if a≤ ≤ b≤.

Definition 4. An aggregation function is function: f : 2R∗ → R∗ such that for

x1, . . . , xn, x, y ∈ R∗ we have:

- f({x1, . . . , x, . . . , xn}) ≤ f({x1, . . . , y, . . . , xn}) iff x ≤ y;(monotonicity)

- f({x1, . . . , xn}) = 0 iff x1 = . . . = xn = 0. (minimality)

By abuse of notation without any ambiguity, we will use f(x1, . . . , xn) instead of

f({x1, . . . , xn}).

We consider several additional properties of aggregate functions.

Definition 5 ([89]). Let f and g be aggregation functions.

20

• f is symmetric iff for any permutation π, we have:

f(xπ(1), . . . , xπ(n)) = f(x1, . . . xn);

• f is associative iff

f(f(x1, . . . , xn), f(y1, . . . , ym)) = f(x1, . . . , xn, y1, . . . ym);

• f is idempotent iff for all x, f(x) = x;

• f is indifferent to the null element iff for any xi = 0,

f(x1, . . . , xi−1, xi, xi+1, . . . , xn) = f(x1, . . . , xi−1, xi+1, . . . , xn);

• f is semi-monotonic iff

if x1 < y1, . . . , xn < yn then f(x1, . . . , xn) < f(y1, . . . , yn);

• f is strictly monotonic iff

if x1 ≤ y1, . . . , xn ≤ yn and ∃i ∈ {1, . . . , n} s.t. xi < yi then f(x1, . . . , xn) <

f(y1, . . . , yn);

• f and g are inter-commutative iff

f(g(x1,1, . . . , x1,n), . . . , g(xm,1, . . . , xm,n)) = g(f(x1,1, . . . , xm,1), . . . , f(x1,n, . . . , xm,n)).

Note that the property of strict monotonicity is stronger than that of semi-strict

monotonicity. Therefore, if an aggregation function is strictly monotonic, it is also

semi-monotonic.

2.3 Propositional Logic

One of the most common ways to express knowledge in knowledge processing is using

a particular logic, defined by a certain syntax and some semantics. The syntax of a

logic defines all the symbols, including variables, constants, connectives, modifiers,

etc., and its grammar, i.e. a set of rules to characterize the correct way to combine

21

these symbols. The set of all formulas constructed by a syntax is a formal language.

The semantics give the meaning to any formula of language which is the basis of

reasoning about formulas.

In practice, a large range of knowledge integration problems can be expressed and

resolved within the formalization of propositional logic. Thus, in the next chapter,

we will consider this formalization to show different results in the literature around

knowledge integration. Rather than presenting the state-of-the-art of propositional

logic, we limit ourselves in this section to formal definition of its syntax and semantics.

2.3.1 Syntax

Classical propositional logic, also called classical logic of propositions or calculus logic,

considers the symbols of variables, representing events, that can only be either true

or false. These symbols are the basis of the syntax of propositional logic, and they

are called propositional variables.

Definition 6. A propositional variable or propositional atom is presented by a binary

variable called Boolean proposition, and it can take two possible truth values: True

or False.

Each propositional variable is now represented by a lowercase letter, e.g. a, b, . . ..

Definition 7. A literal l is a propositional variable x or its negation ¬x. A literal

without the symbol ”¬” is called a positive literal, otherwise it is called a negative

literal.

Notation 6. Two Boolean constants are ⊤ (for True) and ⊥(for False)

Apart from variables and constants, the syntax of a propositional logic language

is also defined on the basis of a set of reserved symbols, called logical connectives.

22

Definition 8. A connective is a logical operator that applies to one or more proposi-

tional variables as the argument. The number of arguments of each logical connective

is fixed, and it defines the arity of this logical connective.

The logical connectives used in a propositional logic language are as follows:

• Negation: ¬ (non), unary operator, e.g. ¬a;

• Conjunction: ∧ (and), binary operator, e.g. a ∧ b;

• Disjunction: ∨ (or), binary operator, e.g. a ∨ b;

• Implication: →, binary operator, e.g. a→ b;

• Equivalence: ↔, binary operator, e.g. a↔ b.

In order to avoid ambiguity in the writing of formulas, parentheses are used as

auxiliary symbols, but they may not be compulsory.

Let LV be a propositional logic language built on a finite set V of propositional

variables together with the set of logical connectives as well as the constants ⊤ and

⊥. We formally define the propositional formula as follows:

Definition 9. Let V be a set of propositional variables. The propositional formula of

language LV is defined inductively as follows:

• ⊤, ⊥ ∈ LV .

• ∀a ∈ V , a ∈ LV .

• if ϕ, ψ ∈ LV then (ϕ), ¬ϕ, ϕ ∨ ψ, ϕ ∧ ψ, ϕ→ ψ, ϕ↔ ψ ∈ LV .

Example 1. Let V = {a, b, c}, the following formulas belong to LV :

• (¬¬a ∧ b)→ ¬(a ∨ b),

• (a→ (b ∨ ¬c)) ∧ (¬a ∨ c),

23

• ¬a ∧ b ∨ c↔ ¬b ∨ ¬c,

Formulas are represented by lowercase Greek letters, e.g. ϕ, ψ, µ, . . . with or with-

out indexes.

2.3.1.1 Conjunctive and Disjunctive Normal Forms

In the wide range of applications on proposition logic, two types of formulas usually

used are conjunctive normal form and disjunctive normal form.

Definition 10. A formula is in conjunctive normal form (CNF) if it is a conjunction

of clauses, where a clause is a disjunction of literals.

Definition 11. A formula is in disjunctive normal form (DNF) if it is a disjunction

of conjunctions of literals.

Remark that we can transform any formula to CNF or DNF by using logical

equivalences, such as double negative elimination, De Morgan’s laws, and distributive

laws.

Example 2. Let V = {a, b, c}. The following formulas are in CNF:

- (¬a ∨ b) ∧ (a ∨ b ∨ ¬c);

- ¬a ∧ (¬b ∨ ¬c) ∧ (¬a ∨ b ∨ ¬c).

The following formulas are in DNF:

- a ∨ (b ∧ ¬a) ∨ ¬c;

- (¬a ∧ ¬b) ∨ (c ∧ ¬a) ∨ b ∨ ¬c.

The following formulas are neither CNF nor DNF:

- (¬¬a) ∧ b;

- a ∨ (¬a ∧ b ∧ (a ∨ ¬c)).

24

2.3.2 Semantics

The semantics of a logic are defined from a basic concept, the notion of interpretation.

We give the formal definition of this concept in the context of propositional logic as

follows:

Definition 12. The interpretation ω in language LV is a function that maps a set V

of propositional variables to the set of truth values {True, False}.

Notation 7. An interpretation ω on V is usually denoted as a vector of values 0 and

1 in which the evaluation of each propositional variable of V is assigned to a unique

value. Value 0 indicates that the propositional variable is evaluated as False by ω

and 1 otherwise.

The set of all interpretations of V is denoted by WV .

Example 3. The interpretation ω = 0110 on V = {a, b, c, d} is defined by ω(b) =

ω(c) = True and ω(a) = ω(d) = False.

Based on the notion of interpretation, the semantics of a formula are defined

inductively as follows:

Definition 13. The semantics of a formula in an interpretation ω are defined such

that for all formulas ϕ, ψ:

• ω(⊤) = True

• ω(⊥) = False

• ω(¬ϕ) = True iff ω(ϕ) = False

• ω(ϕ ∧ ψ) = True iff ω(ϕ) = True and ω(ψ) = True

• ω(ϕ ∨ ψ) = True iff ω(ϕ) = True or ω(ψ) = True

25

• ω(ϕ→ ψ) = False iff ω(ϕ) = True and ω(ψ) = False

• ω(ϕ↔ ψ) = True iff ω(ϕ) = ω(ψ)

Example 4. Let V = {a, b, c, d} and ω = 1001. Formula ϕ = (a → d) ∨ (b ∧ ¬c) is

True w.r.t. interpretation ω.

We define the model of a formula as follows:

Definition 14. Let ϕ ∈ LV and ω ∈ WV . We say that ω is a model of ϕ or ω satisfies

ϕ, denoted by ω |= ϕ, iff ω(ϕ) = True.

Note that for a given formula we can find more than one model, thus we use the

following notation for the set of models of a formula:

Notation 8. The set of all models of a formula ϕ ∈ LV is denoted by Mod(ϕ).

Definition 15. Let ϕ ∈ LV :

- ϕ is consistent iff ϕ has at least one model, otherwise ϕ is inconsistent;

- ϕ is valid iff ∀ω ∈ WV , ω |= ϕ.

Definition 16. Let ϕ, ψ ∈ LV . ϕ is a logical consequence of ψ, written by ϕ |= ψ, iff

Mod(ϕ) ⊆Mod(ψ).

Definition 17. Let ϕ, ψ ∈ LV . ϕ and ψ are logically equivalent, written by ϕ ≡ ψ,

iff ϕ |= ψ and ψ |= ϕ.

Notation 9. Let ω ∈ WV . Form(ω) is a propositional formula such that ω is its

unique model. Formally, ω |= Form(ω) and ∀ω′ ∈ WV and ω′ ̸= ω, ω′ ̸|= Form(ω).

We also extend this notation for a set of interpretations. Let X = {ω1, . . . , ωn} ⊆

WV . Form(X) is a propositional formula such that ∀ω ∈ X,ω |= Form(X) and

∀ω′ ∈ WV and ω′ /∈ X, ω′ ̸|= Form(X). By abuse of notation, we sometimes write

Form(ω1, . . . , ωn) to indicate Form({ω1, . . . , ωn)}.

26

Notation 10. Let K = {ϕ1, . . . , ϕn} be a set or multi-set of propositional formulas.∧K denotes the conjunction of formulas in K, i.e.

∧K = ϕ1∧ . . .∧ϕn.

∨K denotes

the disjunction of formulas in K, i.e.∨K = ϕ1 ∨ . . . ∨ ϕn.

Lastly, the closure of consequence relation is defined as follows:

Definition 18. Let K be a set of formulas, the closure of consequence relation of K,

denoted by Cn(K), is a set Cn(K) = {ϕ ∈ LV |K ⊢ ϕ}.

2.4 Paraconsistent Logics

In a knowledge system, it is common to have inconsistent information about some

situations. The inconsistent information causes the knowledge base represented in

classical logic to become trivial because of the ex falso quodlibet proof rule [60]. For-

mally, the ex falso quodlibet proof rule is written asα,¬αβ

for any α and β. By this

rule, if there exists a contradiction in a knowledge system we can infer an arbitrary

formula, thus according to the classical logic viewpoint this system becomes useless.

To avoid trivialization, there are two main approaches: knowledge revision and

using paraconsistent logics. The former effectively removes some knowledge from the

knowledge base to achieve a new consistent knowledge base, and details of this ap-

proach are discussed in Chapter 3. The latter directs us to ”live with” inconsistency

by keeping the knowledge base but eliminating the logics which cause the trivial in-

ferences. One of the main ways to perform this approach is improving classical logic

to become acceptable with the inconsistent knowledge.

In this section we introduce some common paraconsistent logics, namely weakly-

negative logics, four-valued logic, and quasi-classical logic. We also mention how to

27

measure coherence in quasi-classical models.

In general, we improve the classical logic1 to become paraconsistent logics which

can deal with the contradiction. The requirement for that is the abandoment of at

least one of the following three intuitive principles:

1. α ⊢ α ∨ β (Disjunction introduction)

2. α ∨ β,¬α ⊢ β (Disjunctive syllogism)

3. α ⊢ β, β ⊢ γ ⇒ α ⊢ γ (Transitivity)

Moreover, if the three principles below are taken together they also entail explosion,

so we need to eliminate at least one of them:

1. α→ (β ∧ ¬β) ⊢ ¬α (Reductio ad absurdum)

2. α ⊢ β → α (Rule of weakening)

3. ¬¬α ⊢ α (Double negation elimination)

In the following subsections, we consider some paraconsistent logics. They are the

representations of some common ways to treat with contradiction.

2.4.1 Weakly-negative Logics

In general, weakly-negative logics try to reach a compromise on classical proof theory

by weakening the term of negation. They use the full classical language, but a subset

of classical proof theory. Now, we consider a paraconsistent logic, named Cw logic,

proposed by da Costa [31] to eliminate ex falso quodlibet by removing the reductio ad

absurdum rule.1See appendix A

28

2.4.1.1 Proof theory for Cw

Since the model of Cw is based on classical logic, the whole schema of Cw is the same

as that in classical logic.

Definition 19 ([31]). The logic Cw is defined by the following axiom schema together

with the modus ponens proof rule.

1. α→ (β → α)

2. (α→ β)→ ((α→ (β → γ))→ (α→ γ))

3. α ∧ β → α

4. α ∧ β → β

5. α→ (β → α ∧ β)

6. α→ α ∨ β

7. β → α ∨ β

8. (α→ γ)→ ((β → γ)→ (α ∨ β → γ))

9. α ∨ ¬α

10. ¬¬α→ α

This proof theory gives the Cw consequence relation.

Referring to the properties in classical reasoning (see Appendix A), we have:

Proposition 2 ([60]). The following properties succeed for the Cw consequence re-

lation: Reflexivity, And, Monotonicity, Cut, Deduction, Conditionalization, Consis-

tency preservation, and Or.

29

Proposition 3 ([60]). The following properties fail for the Cw consequence relation:

Supraclassicality, Left logical equivalence, and Right weakening.

Proposition 4 ([50, 60]). The Cw consequence relation is not pure and not trivializ-

able.

2.4.1.2 A Semantic Tableau Procedure for Cw

Notation 11. The formula ¬(α ∧ ¬α) is not valid in general, but if it does hold for

a formula α, it is a well-behaved formula, and denoted by α0.

Notation 12. Each formula α is labeled with either α+ or α−, and we call + : α and

− : α signed formulae.

Intuitively, + : α and − : α, can be interpreted as α being true, and α being false,

respectively. Any set of sets of signed formulae is called a form.

Definition 20 ([31]). Let α and β be two formulae, and let ρ be other formulae

and/or other forms. Below is a set of production rules that can be used to reduce a

set of formulae into either a new set of formulae, or a set of sets of formulae.

1. {ρ,+ : (α ∧ β)} ⇒ {ρ,+ : α,+ : β}

2. {ρ,− : (α ∨ β)} ⇒ {ρ,− : α,− : β}

3. {ρ,− : (α→ β)} ⇒ {ρ,+ : α,− : β}

4. {ρ,+ : (¬¬α)} ⇒ {ρ,+ : α}

5. {ρ,− : (¬α)} ⇒ {ρ,+ : α}

6. {ρ,− : (¬¬α)} ⇒ {ρ,− : α}

7. {ρ,− : (α} β)0} ⇒ {ρ,− : (α0 } β0)} where } ∈ {∧,∨,→}

8. {ρ,− : (α ∧ β)} ⇒ {{ρ,− : α}, {ρ,− : β}}

30

9. {ρ,+ : (α ∨ β)} ⇒ {{ρ,+ : α}, {ρ,+ : β}}

10. {ρ,+ : (α→ β)} ⇒ {{ρ,− : α}, {ρ,+ : β}}

11. {ρ,+ : (¬α)} ⇒ {{ρ,− : α}, {ρ,− : α0}}

Given a form C, we denote by R(C) the result of applying one of the rules to the

form. A tableau is a sequence of forms C1, . . . , Cn such that Ci+1 = R(Ci).

In order to test if a formula can be inferred from a set of formulae, we label it

with a symbol “-”, add it to the data, and construct a tableau. The formula can be

inferred if the tableau is closed. A tableau is closed if every set of formulae of its form

is closed, and a set of formulae is closed if there is a formula α, for which + : α and

− : α belong to that set.

2.4.2 Belnap’s Four-valued Logic

Four-valued logic was proposed by Belnap [14]. It is an interesting alternative for

weakly-negative logics, and it has an intuitive semantic characterization to comple-

ment its proof theory.

Definition 21. The truth of a formula in this language can be one of the values

True, False, Both or Neither, which we denote by the symbols T , F , B, and N ,

respectively.

Example 5. For a knowledge base K = {α, β,¬β,¬θ}, we have an acceptable as-

signment of truth values such that α is T , β is B, γ is N , and θ is F .

Intuitively, this form of assignment may be presented in terms of an “Approxima-

tion” lattice as in Figure 2.2.

31

Figure 2.2: Approximation lattice for four-valued logic

2.4.2.1 Semantics for Four-valued Logic

Definition 22 ([14]). The semantics of four-valued logic are based on a distributive

lattice, namely the “Logical” lattice as in Figure 2.3. We also assume an involution

operator * satisfying the conditions:

1. α = α∗∗, and

2. if α ≤ β then α∗ ≤ β∗

where ≤ is an ordering relation for the lattice.

Definition 23 ([14]). As the semantic assignment function observes monotonicity

and complementation in a logical lattice, x ∧ y is the meet of {x, y} and x ∨ y is the

join of {x, y}, given in the truth tables (Tables 2.1 - 2.3) for the ¬,∧,∨ connectives,

respectively. Let α, β be formulae. The inference β from α is valid iff β ≤ α, where

≤ is the ordering relation for the logical lattice. Let α→ β signify that the inference

from α to β is valid in the four values, i.e. α entails β.

32

Figure 2.3: Logic lattice of four-valued logic

α N F T B

¬α B T F N

Table 2.1: Truth table of negation

2.4.2.2 Proof Theory for Four-valued Logic

In this subsection we consider the four-valued consequence relation for the proof

theory of this logic.

Definition 24 ([14]). Let α, β, γ ∈ LV . The following are the proof rules for the

four-valued consequence relation.

1. α1 ∧ . . . ∧ αm → β1 ∨ . . . ∨ βn provided some αi is some βj,

2. (α ∨ β)→ γ iff α→ γ and β → γ,

3. α→ (β ∧ γ) iff α→ γ and α→ γ,

4. α→ β iff ¬β → ¬α

5. α→ β and β → γ imply α→ γ

33

∧ N F T B

N N F N F

F F F F F

T N F T B

B F F B B

Table 2.2: Truth table of conjunctions

∨ N F T B

N N N T T

F N F T B

T T T T T

B T B T B

Table 2.3: Truth table of disjunctions

6. α→ β iff α↔ (α ∧ β) iff β ↔ (α ∨ β)

In addition, the following extends the definition of the FV consequence relation. Let

α↔ β signify that α and β are semantically equivalent, and they can be intersubsti-

tuted in any context.

7. α ∧ β ↔ β ∧ α

8. α ∨ β ↔ β ∨ α

9. (α ∧ β) ∧ γ ↔ α ∧ (β ∧ γ)

10. (α ∨ β) ∨ γ ↔ α ∨ (β ∨ γ)

11. α ∧ (β ∨ γ)↔ (α ∧ β) ∨ (α ∧ γ)

12. α ∨ (β ∧ γ)↔ (α ∨ β) ∧ (α ∨ γ)

13. ¬¬α↔ α

34

14. ¬(α ∧ β)↔ ¬α ∨ ¬β

15. ¬(α ∨ β)↔ ¬α ∧ ¬β

Also,

16. α↔ β and β ↔ γ imply α↔ γ

Referring to the properties in classical reasoning (see Appendix A), we have:

Proposition 5 ([60]). The following properties succeed for the four-valued conse-

quence relation: Reflexivity, Consistency Preservation, Monotonicity, and Cut.

Proposition 6 ([60]). The following properties fail for the FV consequence relation:

And, Supraclassicality, Or, Left Logical Equivalence, Deduction, Conditionalization,

and Right Weakening.

Proposition 7 ([14, 60]). The four-valued consequence relation is not pure and not

trivializable.

2.4.3 Quasi-classical Logic

Quasi-classical logic (QC logic) was proposed by Besnard and Hunter [22]. It uses

full classical language, but the queries are rewritten in conjunctive normal form, and

the proof theory is restricted.

2.4.3.1 Proof theory of QC logic

In this section, we present the QC proof rules which include a subset of classical proof

rules. We also introduce a QC proof which is achieved by restricting classical proof.

Definition 25 ([60]). Assume that ∧ and ∨ are commutative and associative opera-

tors, respectively.

35

1.α ∧ βα

(Conjunct elimination)

2.α ∨ α ∨ βα ∨ β

(Disjunct contraction)

3.α ∨ ⬬α ∨ β

(Negation introduction)

4.¬¬α ∨ βα ∨ β

(Negation elimination)

5.α

¬¬α,¬¬αα

6.α ∨ β,¬α ∨ γ

β ∨ γ,

α,¬α ∨ γγ

(Resolution)

7.α ∨ (β → γ)α ∨ ¬β ∨ γ

,α ∨ ¬(β → γ)α ∨ (β ∧ ¬γ)

(Arrow elimination)

8.α→ β¬α ∨ β

,¬(α→ β)α ∧ ¬β

9.α ∨ (β ∧ γ)

(α ∨ β) ∧ (α ∨ γ),

(α ∨ β) ∧ (α ∨ γ)α ∨ (β ∧ γ)

(Distribution)

10.¬(α ∧ β) ∨ γ¬α ∨ ¬β ∨ γ

,¬(α ∨ β) ∨ γ(¬α ∧ ¬β) ∨ γ

(de Morgan laws)

11.¬(α ∧ β)¬α ∨ ¬β

,¬(α ∨ β)¬α ∧ ¬β

12.α

α ∨ β(Disjunct introduction− only used as a last step in a proof)

Definition 26 ([60]). T is a proof-tree iff T is a tree where

(1) each node is an element of LV ;

(2) for the trees with more than one node, the root is derived by application of any

QC proof rule, where the premises for the proof rule are the parents of the root;

(3) the leaves are the assumptions for the root; and

(4) any node, that is not a leaf or root, is derived by the application of any QC

proof rule - except the disjunct introduction rule - and the premises for the proof rule

are the parents of the node.

36

Definition 27 ([60]). Let K ⊆ LV . For a clause β, there is a QC proof of β from K

iff there is a QC proof tree, where each leaf is an element of K, and the root is β.

Definition 28 ([60]). Let K ⊆ LV and α ∈ LV . We define the QC consequence

relation, denoted by ⊢Q, as follows:

K ⊢Q α iff for each βi(1 ≤ i ≤ n) there is a QC proof of βi from K where

βa ∧ . . . ∧ βn is a CNF of α.

Referring to the properties in classical reasoning (see Appendix A), we have:

Proposition 8 ([22, 60]). The following properties succeed for the QC consequence

relation: Reflexivity, Consistency preservation, and Monotonicity.

Proposition 9 ([22, 60]). The following properties fail for the QC consequence rela-

tion: Cut, Right weakening, Left logical equivalence, and Supraclassicality.

Proposition 10 ([22, 60]). The QC consequence relation is pure, and hence not

trivializable.

2.4.3.2 Semantics for QC logic

Definition 29. Let α1 ∨ . . . ∨ αn be a clause that includes a literal αi and n > 1.

The focus of α1 ∨ . . . ∨ αn by αi, denoted by ⊗(α1 ∨ . . . ∨ αn), is defined as a clause

obtained by removing αi from α1 ∨ . . . ∨ αn.

Definition 30. Let V be a set of atoms. Let O be the set of objects defined as follows,

where +α is a positive object, and −α is a negative object.

O = {+α|α ∈ V} ∪ {−a|a ∈ V}

We call any X ⊆ O a QC model. Therefore, X can contain both +α and −α for

some atom α.

For each atom α ∈ LV , and each X ⊆ O, +α ∈ X means that in X there is a

reason for the belief α and ¬α ∈ X means that in X there is a reason for the belief

¬α.

37

Definition 31 ([60]). Let |=S be a satisfiability relation called strong satisfaction.

For a model X, we define |=S as follows, where α1, . . . , αn are literals in LV , n > 1,

and α is a literal in LV . X |=S α iff there is a reason for the belief α in X

X |=S α1 ∨ . . . ∨ αniff [X |=S α1 or . . . or X |=S αn]

and ∀i s.t. 1 ≤ i ≤ n[X |=S ¬αi implies X |=S ⊗(α1 ∨ . . . ∨ αn, αi)]

For α, β, γ ∈ LV , we extend the definition as follows,

X |=S α ∧ β iff X |=S α and X |=S β

X |=S ¬¬α ∨ β iff X |=S α ∨ β

X |=S ¬(α ∧ β) ∨ γ iff X |=S ¬α ∨ ¬β ∨ γ

X |=S ¬(α ∨ β) ∨ γ iff X |=S (¬α ∧ ¬β) ∨ γ

X |=S α ∧ (β ∨ γ) iff X |=S (α ∧ β) ∨ (α ∧ γ)

X |=S α ∨ (β ∧ γ) iff X |=S (α ∨ β) ∧ (α ∨ γ)

X |=S (α→ β) ∨ γ iff X |=S ¬α ∨ β ∨ γ

X |=S ¬(α→ β) ∨ γ iff X |=S (α ∧ ¬β) ∨ γ

Definition 32 ([60]). For X ⊆ O and K ⊆ LV , let X |=S K denote that X |=S α

holds for every α in K. Let QC(K) = {X ⊆ O|X |=S K} be the set of QC models

for K.

Example 6. Let K = {¬α ∧ ¬β, (¬α ∨ ¬β) ∧ γ,¬(α ∧ β) ∨ γ}, where α, β, γ ∈ V,

and let X = {¬α,¬β, γ}. Obviously, X |=S ¬α, X |=S ¬β and X |=S γ. Hence

X |=S ¬α∧¬β, X |=S ¬α∨ γ, X |=S ¬β ∨ γ so X |=S (¬α∨ γ)∧ (¬β ∨ γ), and they

implicate X |=S γ∧(¬α∨¬β), and also X |=S ¬α∨¬β∨γ, and so X |=S ¬(α∧β)∨γ.

Thus, every formula in K is strongly satisfiable in X.

Definition 33 ([60]). Let K ⊆ LV . Let MQC(K) ⊆ QC(K) be the set of minimal

QC models for K, defined as follows:

MQC(K) = {X ∈ QC(K)| if Y ⊂ X, then Y /∈ QC(K)}

Example 7. MQC({¬α ∧ ¬β}, β ∨ γ}) = {{−α,−β,+β}, {−α,−β,+γ}}

38

Proposition 11 ([60]). Let X ⊆ O, and α1, . . . , αn be literals in LV . We have

X |=S α1 ∨ . . . ∨ αn iff

(1) for some αi ∈ {α1, . . . , αn}, X |=S αi and X ̸|=S ¬αi or

(2) for all αi ∈ {α1, . . . , αn}, X |=S αi and X |=S ¬αi.

Proposition 12 ([60]). Let α ∈ LV , and let β1 ∧ . . . ∧ βn be a conjunctive normal

form (CNF) of α. For X ⊆ O, X |=S α iff X |= β1 and . . . and X |= βn.

2.4.4 Measuring the Coherence in QC models

Measuring the coherence is essential to determine the inconsistency of a knowledge

system. In this section, we consider the method for measuring coherence by using

QC models based on the technique proposed by Hunter [61].

Definition 34 ([61]). Let X ∈ 2O.

ConflictBase(X) = {α|+ α ∈ X and − α ∈ X}

OpinionBase(X) = {α|+ α ∈ X or − α ∈ X}

Example 8. Let X = {+α,−α, β,+γ,−γ}, where α, β, γ ∈ V, hence

ConflictBase(X) = {α, γ} and OpinionBase(X) = {α, β, γ}.

Proposition 13. Let K ⊆ LV . If X, Y ∈MQC(K), then

ConflictBase(X) = ConflictBase(Y ).

Definition 35 ([61]). The ConflictCount function, and the OpinionCount function,

are from 2LV into N. For K ∈ 2LV , ConflictCount(K) is ∥ConflictBase(X)∥, where

X ∈MQC(K), and OpinionCount(K) is the maximum value in {∥OpinionBase(X)∥

|X ∈MQC(K)}.

Example 9. LetK = {¬α∨(β∧γ), α∧¬β}. Thus,MQC(K) = {{−α,+α,−β}, {+α,

+β,−β,+γ}}. Therefore, ConflictCount(K) = 1 and OpinionCount(K) = 3.

39

Definition 36 ([61]). The Coherence function from 2O into [0, 1], is defined below

when X is non-empty, and Coherence(∅) = 1.

Coherence(X) = 1− ∥ConflictBase(X)∥∥OpinionBase(X)∥

Example 10. Let X ∈MQC({β ∧ γ, α ∧ ¬β}). Thus, Coherence(X) = 2/3.

Clearly, different minimal QC models for the same knowledge base are not neces-

sarily equally coherent. We extend the coherence to knowledge bases as follows.

Definition 37 ([61]). Let K ∈ 2LV . Assign Coherence(K) the maximum value in

{Coherence(X)| X ∈MQC(K)}.

Example 11. Let K = {α, α ∨ γ,¬α ∧ ¬β}, and let X = {+α,−α,−β}, let

Y = {+α,−α,−β, γ}. Thus,

MQC(K) = {X, Y }, and Coherence(X) = 1/2 and Coherence(Y ) = 1/3.

Therefore, Coherence(K) = 1/2.

2.5 Possiblistic Logic

Possibilistic logic [40, 41] is a weighted logic introduced and developed to supply a

simple and rigorous tool for automated reasoning from uncertain or prioritized incom-

plete information. Possibilistic logic expression is a classical logic formula associated

with a weight which is interpreted in the framework of possibility theory as the lower

bound of necessity degree. Possibilistic logic can distinguish fuzzy logic in the sense

that fuzzy logic deals with a proposition which includes a vague predicate, and ma-

nipulates the truth degree with respect to each connective, whereas possibilistic logic

involves the certainty and possibility degree in each formula, but the proposition is

assumed to be non-vague. In the following subsections, we consider the main elements

of possibilistic logic.

40

2.5.1 Possibility Distributions

At the semantic level, possibilistic logic is based on the notion of a possibility distri-

bution, denoted by π which is a mapping from W , the set of all interpretations, to

[0,1] representing the available information. π(ω) represents the degree of compati-

bility of an interpretation ω with the available beliefs about the real world, if we are

representing uncertain pieces of knowledge (or the degree of satisfaction of reaching

a state ω if we are modelling preferences). By convention, π(ω) = 1 means that it is

totally possible for ω to be the real world (or that ω is fully satisfied), 1 > π(ω) > 0

means that ω is only somewhat possible (or satisfied), while ω = 0 means that ω is

certainly not the real world (or not satisfied at all). A possibility distribution π is said

to be normalized if and only if there exists an interpretation ω0 such that π(ω0) = 1.

Associated with the possibility distribution π is possibility degree Π(ϕ) = max{π(ω) :

ω |= ϕ} which evaluates the extent to which ϕ is consistent with information expressed

by ϕ.

Another measure which is associated with π is necessity degree N(ϕ) = 1−Π(¬ϕ)

which evaluates the extent to which ϕ is entailed by the available beliefs, and is de-

fined by duality from the possibility degree of a formula ϕ.

Note that a mapping N reverses the scale, on which π is ranging, and that

N(ϕ) = 1 means that ϕ is a totally certain piece of knowledge or a compulsory

goal, while N(ϕ) = 0 expresses the complete lack of knowledge or of priority about

ϕ, but does not mean that ϕ is or should be false. Moreover, the duality equation

N(ϕ) = 1−Π(¬ϕ) extends that existing in classical logic, where a formula is entailed

from a set of propositional formulas if and only if its negation is inconsistent with

41

this set.

We use Latin letters a, b, c, . . . to depict possibility degrees of interpretations, and

Greek letters α, β, γ, . . . to denote levels of certainty.

2.5.2 Possibilistic Logic Knowledge Bases

At the syntactic level, uncertain pieces of information are represented by means of a

possibilistic logic base which is a set of weighted formulas B = {(ϕi, ai) i = 1, . . . , n},

where ϕi is a propositional formula and ai belongs to a totally ordered scale such

as [0,1]. (ϕi, ai) means that the necessity degree of ϕi is at least equal to ai, i.e.

N(ϕ1) ≥ ai. We denote by B∗ the propositional base associated with B, namely, the

base obtained from B by forgetting the degrees of its formulas.

Given B, we can generate a unique possibility distribution, denoted by πB, such that

all interpretations satisfying all the formulas in B will have the highest possibility

degree, namely 1, and the other interpretations will be ranked with respect to the

highest formula that they falsify. Formally, we have:

Definition 38 ([41]). For all ω ∈ W,

πB(ω) =

1 if ∀(ϕi, ai) ∈ B,ω |= ϕi1−max{ai : (ϕi, ai) ∈ B and ω ̸|= ϕi} otherwiseExample 12. Let p and q be two propositional symbols. Let B = {(¬p∨¬q, 0.7), (p, 0.6)}.

Then, πB(p¬q) = 1; πB(¬p¬q) = πB(¬pq) = 0.4 and πB(pq) = 0.3.

An interpretation p¬q is the most preferred since it satisfies all the formulas in B.

The interpretations ¬p¬q and ¬pq are more preferred than pq since the highest for-

mula falsified by ¬p¬q and ¬pq (i.e. (p, 0.6)) is less certain than the highest formula

falsified by pq (i.e. (¬p ∨ ¬q, 0.7)).

The notion of equivalence between two possibilistic logic bases is defined as follows:

42

Definition 39. B and B′ are said to be equivalent, denoted by B ≡ B′ iff πB = πB′.

2.5.3 Subsumption and Inference in Possibilistic Logic

In this subsection, we briefly present inference from possibilistic bases. However, we

first define the notion of an α− cut of a possibilistic logic base.

Definition 40 ([41]). Let B be a possibilistic logic base, and α ∈ [0, 1]. We call

the α − cut of B, denoted by B≥α, the set of propositional formulas in B having a

necessity degree at least equal to α.

Dually, the strict α− cut of B, denoted by B>α is the set of formulas in B having

a necessity degree strictly greater than α.

A possibilistic base B is said to be consistent if its associated propositional base B∗

is consistent. When B is inconsistent, we define its level of inconsistency as follows:

Definition 41 ([17]). The inconsistency degree of a possibilistic logic base B is:

Inc(B) = max{αi : B≥αi is inconsistent},

with Inc(B) = 0 when B is consistent.

Subsumption can now be defined as follows:

Definition 42 ([17]). Let (ϕ, α) be a formula in B. Then, (ϕ, α) is said to be sub-

sumed in B if (B\{(ϕ, α)})≥α ⊢ ϕ; (ϕ, α) is said to be strictly subsumed by B if

B>α ⊢ ϕ.

Subsumed formulas are in some sense redundant formulas, as shown by the fol-

lowing lemma.

Lemma 1 ([41]). Let (ϕ, α) be a subsumed formula in B. Then B and B′ =

B\{(ϕ, α)} are equivalent.

43

The necessity degree is propagated out in the inference process in the following

way.

Definition 43 ([17]). Let B be a possibilistic logic base. Let (ϕ, α) be a piece of

information with α > Inc(B). (ϕ, α) is said to be a consequence of B, denoted by

B ⊢π (ϕ, α), iff B>α ⊢ ϕ.

From this definition we have: if B ⊢π (ϕ, α) then B ⊢π (ϕ, β) for all β such that

α ≥ β > Inc(B).

Therefore, in the following, when there is no ambiguity, we write B ⊢π (ϕ, α) to

indicate that (ϕ, α) is a consequence of B where α refers to the maximal degree.

Note that the useful consistent part ofB, for making possibilistic inference, is made

of the formulas whose necessity degrees are above the inconsistency level, namely:

Definition 44 ([17]). Let B be a possibilistic logic base. The useful consistent part

of B, denoted by ρ(B), is defined as follows:

ρ(B) = {ϕi : (ϕi, αi) ∈ B and αi > Inc(B)}

Indeed, we can check that

ρ(B) ⊢ ψ iff ∃α such that α > Inc(B) and B ⊢π (ψ, α).

Lastly, we have the following correspondences between the syntactic and semantic

representations:

- A possibilistic base B is consistent if and only if its