-
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
.
-
.
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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-
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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
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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.
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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-
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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
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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.
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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
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A Classical Reasoning 225
A.1 Language and Proof Theory . . . . . . . . . . . . . . . . . . . . . . . 225
A.2 Properties of Consequence Relations . . . . . . . . . . . . . . . . . . 226
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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
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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
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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
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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.
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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
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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.
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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?
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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.
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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-
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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
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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
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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.
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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).
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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,
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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.
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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.
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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
.
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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.
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• 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,
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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.
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• 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
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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.
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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),
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• ¬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)).
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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
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• 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)}.
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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
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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
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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.
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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}}
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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.
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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.
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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
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∧ 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
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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.
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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.
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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.
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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}}
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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.
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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.
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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
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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:
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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.
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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