Rational Reasoning with Finite Conditional Knowledge Bases: Theoretical and Implementational Aspects

383 Pages·2018·2.661 MB·English

by  Christian Eichhorn

Christian Eichhorn
Rational Reasoning
with Finite Conditional
Knowledge Bases
Theoretical and
Implementational Aspects
Rational Reasoning with Finite
Conditional Knowledge Bases
Christian Eichhorn
Rational Reasoning
with Finite Conditional
Knowledge Bases
Theoretical and Implementational
Aspects
Christian Eichhorn
Iserlohn, Germany
Zugleich Dissertation an der Fakultät Informatik, TU Dortmund, Dortmund (DE) 2018
unter dem Titel “Qualitative Rational Reasoning with Finite Conditional Knowledge
Bases – Theoretical and Implementational Aspects”
This work was supported by Grant KI1413/5-1 of Deutsche Forschungsgemeinschaft
(DFG) to Gabriele Kern-Isberner as part of the priority program “New Frameworks of
Rationality” (SPP 1516).
ISBN 978-3-476-04823-3 ISBN 978-3-476-04824-0 (eBook)
https://doi.org/10.1007/978-3-476-04824-0
Library of Congress Control Number: 2018965124
J.B. Metzler
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Acknowledgements
There a many people to whom I am very grateful for their support during
thepastyears. FirstofallIwanttothankGabrieleKern-Isbernerforbeing
my supervisor both scientifically as with respect to my career. If she did
not had believed that I could acquire the necessary formal skills to work in
knowledge representation and reasoning, this thesis would never have been.
Apart from that she supported my research, pulled me back when ideas
went astray, guided me round various obstacles of scientific research, and
supportedmewheneverIwasinneedforsupport. IthankMarcoRagnifor
beingmysecondreviewer,forbeinganexcellenthostinawonderfulcityand
a marvellous discussion partner. Furthermore I thank Gu¨nter Rudolph and
Peter Padawitz for agreeing to be part of my disseration committee. I also
thank Lars Hildebrand, who accepted being my mentor, for his insights, be
itscientificallyorwithrespecttootherissuesduringmydissertationperiod.
Many thanks go to my colleagues Daan Apeldoorn, Tanja Bock, Diana
Howey, Steffen Schieweck, Andre Thevapalan, and Marco Wilhelm of the
Information Engineering group as well as the whole Chair 1 of Computer
Science at TU Dortmund for their discussions, ideas, and joint publica-
tions. I thank Christoph Beierele, Steven Kutsch, and Kai Sauerwald of
the Lehrgebiet Wissensbasierte Systeme from the University of Hagen for
interesting discussions, joint publications, and enjoyable travels.
I thank my wife, Claudia Eichhorn, for having endured the busy and
stressful time of me having to finish this dissertation, for raising me up
when this project pushed me down, and generally being there.
I am grateful that I had the possibility to conduct my research in the
communityofthepriorityprogram“NewFrameworksofRationality”andI
thank all friends and colleagues for the great time we had. Finally, I thank
all co-authors of joint articles, cited or not cited in this thesis, for working
with me on all these fascinating and interesting things.
ThisworkwassupportedbyGrantKI1413/5-1ofDeutscheForschungsge-
meinschaft(DFG)toGabrieleKern-Isberneraspartofthepriorityprogram
“New Frameworks of Rationality” (SPP1516).
Contents
1 Introduction 1
1.1 Context and Motivation . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Own contributions to joint articles . . . . . . . . . . . . . . . 10
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Preliminaries 15
2.1 Propositional Logic and Satisfaction . . . . . . . . . . . . . . 15
2.2 Conditionals and Knowledge Bases . . . . . . . . . . . . . . . 19
2.3 Inference Relations . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Graphs and Hypergraphs . . . . . . . . . . . . . . . . . . . . 32
2.5 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Basic Techniques 37
3.1 Preferential Models and Preferential Inference . . . . . . . . . 37
3.2 Conditional Structures . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Ordinal Conditional Functions (OCF) . . . . . . . . . . . . . 44
3.3.1 Definition and Properties . . . . . . . . . . . . . . . . 45
3.3.2 Inference with OCF . . . . . . . . . . . . . . . . . . . 53
3.3.3 Inductive Approaches to Generating OCF . . . . . . . 54
3.3.3.1 System Z . . . . . . . . . . . . . . . . . . . . 54
3.3.3.2 System Z+ . . . . . . . . . . . . . . . . . . . 55
3.3.3.3 c-Representations . . . . . . . . . . . . . . . 59
3.4 Probabilistic Reasoning . . . . . . . . . . . . . . . . . . . . . 64
3.5 Bayesian Networks and LEG Networks . . . . . . . . . . . . . 68
3.6 Ceteris Paribus Networks . . . . . . . . . . . . . . . . . . . . 73
3.7 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Properties of Nonmonotonic Inference Relations 77
4.1 Axiom Systems for Nonmonotonic Reasoning . . . . . . . . . 77
4.1.1 Introduction and Common Properties . . . . . . . . . 77
4.1.2 System O . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.1.3 System C . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.1.4 System P . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.5 System R . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 Selected Formal Properties of Nonmonotonic Inference . . . . 106
4.3 Properties of Structural Inference . . . . . . . . . . . . . . . . 111
VIII Contents
4.4 Properties of Ranking Inference . . . . . . . . . . . . . . . . . 119
4.5 Properties of Inference with System Z . . . . . . . . . . . . . 123
4.6 Properties of Inference with c-Representations . . . . . . . . . 126
4.7 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Reasoning with Sets of c-Representations 133
5.1 c-Representations as Constraint Satisfaction Problems . . . . 133
5.2 Properties of Inference with Sets of c-Representations . . . . 137
5.3 Reasoning with Sets of Preferred c-Representations . . . . . . 149
5.4 Interim Summary and Discussion . . . . . . . . . . . . . . . . 154
6 Normal Forms of Conditional Knowledge Bases 159
6.1 Equivalences for Conditional Knowledge Bases . . . . . . . . 159
6.2 Transformation Systems for Knowledge Bases . . . . . . . . . 161
6.3 Interim Summary . . . . . . . . . . . . . . . . . . . . . . . . . 167
7 Networks for Compact Representation and Implementation169
7.1 OCF-Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.1.1 Structure and Properties . . . . . . . . . . . . . . . . 171
7.1.2 Inductive Generation of OCF-Networks . . . . . . . . 175
7.1.2.1 ...Network Component . . . . . . . . . . . . 176
7.1.2.2 ...OCF-Component w/ Semi-Quantitat. KB 179
7.1.2.3 Complexity Results . . . . . . . . . . . . . . 187
7.1.2.4 Correctness Results . . . . . . . . . . . . . . 189
7.1.2.5 ...OCF-component w/ Qualitative KB . . . 195
7.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.1.4 Interim Summary. . . . . . . . . . . . . . . . . . . . . 199
7.2 Comparing OCF- and CP-networks . . . . . . . . . . . . . . . 201
7.2.1 Plain Generation OCF-Networks ↔ CP-Networks . . 201
7.2.2 Bottom-Up Induction of OCF-Networks . . . . . . . . 212
7.2.3 Interim Summary and Discussion . . . . . . . . . . . . 222
7.3 OCF-LEG Networks . . . . . . . . . . . . . . . . . . . . . . . 225
7.3.1 Basic Definition and Properties . . . . . . . . . . . . . 225
7.3.2 Inductive Generation . . . . . . . . . . . . . . . . . . . 230
7.3.3 Further Reflections on the Consistency Condition . . . 245
7.3.4 Discussion and Comparison to OCF-Networks. . . . . 248
7.3.5 Interim Summary. . . . . . . . . . . . . . . . . . . . . 250
7.4 Interim Summary of the Chapter . . . . . . . . . . . . . . . . 252
Contents IX
8 Connections to Psychology and Cognition 253
8.1 Simulating Human Inference in the Suppression Task . . . . . 254
8.1.1 Modelling the Suppression Task. . . . . . . . . . . . . 255
8.1.2 Background Knowledge in the Suppression Task . . . 265
8.1.3 Strengthening / Weakening in the Suppression Task . 269
8.2 Inference Patterns . . . . . . . . . . . . . . . . . . . . . . . . 275
8.3 Interim Conclusion and Discussion . . . . . . . . . . . . . . . 288
9 Summary and Final Remarks 291
9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
9.2 Further and Future Work . . . . . . . . . . . . . . . . . . . . 291
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
A Proofs of Technical Findings 295
A.1 Proofs for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . 295
A.2 Proofs for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . 298
A.3 Proofs for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 305
A.4 Proofs for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . 313
A.5 Proofs for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . 324
A.6 Proofs for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . 327
A.7 Proofs for Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . 346
Bibliography 347
List of Figures
1.1 Relations between the topics of this thesis . . . . . . . . . . . 12
2.1 Illustration of the running example (Diekmann, 2012) . . . . 16
2.2 Knowledge Base Consistency Test Algorithm . . . . . . . . . 27
2.3 Illustration of directed and undirected graphs . . . . . . . . . 33
2.4 Illustration of a (graph theoretic) tree . . . . . . . . . . . . . 34
2.5 Illustration of hypergraph and hypertree. . . . . . . . . . . . 35
3.1 Preferential relation for Example 3.7. . . . . . . . . . . . . . . 40
3.2 Introductory example to conditional structures . . . . . . . . 44
3.3 Illustration of conditional independence axioms . . . . . . . . 49
3.4 Flowchart for System Z+ . . . . . . . . . . . . . . . . . . . . 58
3.5 Network for the car start example . . . . . . . . . . . . . . . 69
3.6 Bayesian network of the car start example . . . . . . . . . . . 70
3.7 Hypergraph to the car start example . . . . . . . . . . . . . . 72
3.8 CP-network for the dinner example . . . . . . . . . . . . . . . 74
4.1 Axioms of System O . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Properties of System O . . . . . . . . . . . . . . . . . . . . . 82
4.3 Axioms of System C . . . . . . . . . . . . . . . . . . . . . . . 87
4.4 Structural preference for the penguin example . . . . . . . . . 89
4.5 Properties of System C. . . . . . . . . . . . . . . . . . . . . . 92
4.6 Axioms of System P . . . . . . . . . . . . . . . . . . . . . . . 96
4.7 Properties of System P . . . . . . . . . . . . . . . . . . . . . . 98
4.8 Mutual inclusion of axiom systems O, O+, C, P, and R. . . . 103
4.9 Properties of System R. . . . . . . . . . . . . . . . . . . . . . 104
4.10 Properties implied by (M), (T), and (CPS) . . . . . . . . . . 107
4.11 Example for preferential inference violating (WD) . . . . . . 108
4.12 Example for preferential inference violating (WCPS) . . . . . 109
4.13 Example for preferential inference violating (CI) . . . . . . . 110
4.14 Example for preferential inference violating (RM) . . . . . . . 110
4.15 Example for structural inference violating (DR) . . . . . . . . 114
4.16 Example for structural inference violating (NR) . . . . . . . . 116
4.17 Structural inference from inconsistent knowledge base . . . . 118
4.18 Example for structural inference violating (SM) . . . . . . . . 119
4.19 Properties of structural inference . . . . . . . . . . . . . . . . 131
4.20 Properties of κ-inference . . . . . . . . . . . . . . . . . . . . . 132
XII List of Figures
5.1 Set inclusions betw. inferences with sets of c-representations . 155
6.1 Rules of transformation system T . . . . . . . . . . . . . . . . 163
7.1 OCF-network of the car start example . . . . . . . . . . . . . 174
7.2 Flowchart of Algorithm 7.1 . . . . . . . . . . . . . . . . . . . 177
7.3 Network component generated from the car start example . . 179
7.4 Illustration of both graphs from Example 7.9 . . . . . . . . . 179
7.5 Flowchart of Algorithm 7.2 . . . . . . . . . . . . . . . . . . . 182
7.6 Flowchart of Algorithm 7.3 . . . . . . . . . . . . . . . . . . . 184
7.7 Illustration of Algorithm 7.4 . . . . . . . . . . . . . . . . . . . 186
7.8 Car start solutions from System Z+ and c-representations . . 188
7.9 Generated graph from Example 7.23 . . . . . . . . . . . . . . 196
7.10 OCF-network of Example 7.23 . . . . . . . . . . . . . . . . . 197
7.11 Flowchart of Algorithm 7.5 . . . . . . . . . . . . . . . . . . . 204
7.12 OCF-network with induced CP-network of Example 7.30 . . . 205
7.13 Flowchart of Algorithm 7.6 . . . . . . . . . . . . . . . . . . . 208
7.14 CP-network with induced OCF-network . . . . . . . . . . . . 209
7.15 Flowchart of Algorithm 7.7 . . . . . . . . . . . . . . . . . . . 216
7.16 Markov Property vs. CP-independence . . . . . . . . . . . . . 218
7.17 Flowchart of Algorithm 7.8 . . . . . . . . . . . . . . . . . . . 221
7.18 Relations between Algorithm 7.7 and Algorithm 7.8 . . . . . 222
7.19 Hypergraph to the car start example in Example 7.49 . . . . 226
7.20 Hypergraph for Example 7.51 . . . . . . . . . . . . . . . . . . 228
7.21 Flowchart of Algorithm 7.9 . . . . . . . . . . . . . . . . . . . 233
7.22 Cutgraph to Example 7.56 . . . . . . . . . . . . . . . . . . . . 235
7.23 Flowchart of Algorithm 7.10 . . . . . . . . . . . . . . . . . . . 238
7.24 Flowchart of Algorithm 7.11 . . . . . . . . . . . . . . . . . . . 241
7.25 Hypergraph of Example 7.64 . . . . . . . . . . . . . . . . . . 246
7.26 Hypergraph for Example 7.66 . . . . . . . . . . . . . . . . . . 247
8.1 Process trees for formalising the Suppression Task . . . . . . 261
8.2 Constraints imposed by pattern % . . . . . . . . . . . . . . 278
B89
8.3 Preference relations of all inference patterns. . . . . . . . . . 280
8.4 Algorithm Explanation Generator illustration . . . . . . . . . 284
A.1 Structure the proof for Theorem 7.43 . . . . . . . . . . . . . . 341