Knowledge Representation and Inductive Reasoning using Conditional Logic and Sets of Ranking Functions

186 Pages·2021·5.526 MB·English

by  Steven Kutsch

KNOWLEDGE REPRESENTATION AND INDUCTIVE
REASONING USING CONDITIONAL LOGIC AND SETS OF
RANKING FUNCTIONS
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ISSN 0941-5769 (print)
ISSN 2666-2175 (online)
Knowledge Representation and Inductive
Reasoning using Conditional Logic and
Sets of Ranking Functions
Steven Kutsch
Lehrgebiet Wissensbasierte Systeme, FernUniversität in Hagen, Deutschland
© 2021 Akademische Verlagsgesellschaft AKA GmbH, Berlin
All rights reserved. No part of this book may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, without prior written permission from the publisher.
ISBN 978-3-89838-760-6 (AKA, print)
ISBN 978-1-64368-162-7 (IOS Press, print)
ISBN 978-1-64368-163-4 (IOS Press, online)
Bibliographic information available from the Katalog der Deutschen Nationalbibliothek (German
National Library Catalogue) at https://www.dnb.de
Dissertation, approved by FernUniversität in Hagen
Date of the defense: 22 October 2020
Supervisor: Prof. Dr. Christoph Beierle
ORCID iD of the author: https://orcid.org/0000-0003-1670-737X
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Abstract
Theareaofknowledgerepresentationandreasoningisconcernedwith
simulating intelligent behaviour in automated agents using explicit,
formal representations of domain or evidential knowledge. We con-
sider rules with exceptions, called conditionals, within the broader
context of inductive reasoning for representing and reasoning with
uncertain knowledge. Inductive reasoning techniques transform a fi-
nite and incomplete knowledge base into a complete representation of
the knowledge and beliefs of the agent operating with this knowledge
base.
We employ Spohn’s ranking functions that assign degrees of im-
plausibility to the interpretations of the propositional language used
to formulate the defeasible rules representing uncertain knowledge.
A special kind of ranking functions, the c-representations introduced
by Kern-Isberner, will take a central role in our investigations.
In particular, we investigate how reasoning using sets of ranking
models can be understood, researched, implemented, and optimised.
We define modes of inference that, together with the choice of the
actual set of ranking models, give us two dimensions along which to
position inference relations induced by conditional knowledge bases.
To understand the relationships among these inference relations bet-
ter, we introduce a classification of qualitative conditionals with re-
spect to semantic properties of conditionals. The idea of redundant
conditionals proposed in the literature is elaborated upon, while in-
vestigating properties of c-inference relations. We show that adding
or removing redundant conditionals from knowledge bases can change
the induced c-inference relations, therefore highlighting that explicit
and inferred information is considered different within the framework
of c-representations. We exploit our classification of conditionals to
optimise the task of calculating complete inference relations, also em-
v
vi ABSTRACT
ploying a novel data structure for representing inference relations.
To optimise the task of answering queries with respect to sets of c-
representations, we introduce a compact representation of conditional
knowledge bases, designed to capture the computational complexity
of answering queries. To make implementing c-inference practical,
we introduce a finite domain variant of the constraint satisfaction
problem used to calculate c-representations and characterise kinds of
upper bounds used in solving it. We investigate formal properties
of inference relations defined over sets of ranking models by consid-
ering inference postulates proposed in the literature and by present-
ing an approach for experimentally investigating inference systems.
We show for instance that credulous and weakly skeptical inference
over arbitrary sets of ranking models satisfies the properties (REF),
(LLE), (RW), (VCM) and (WAND). We also show that c-inference
relations satisfy the property weak rational monotony (WRM), which
is not generally satisfied by inference relations defined over arbitrary
sets of ranking models. Finally, we present an implementation, called
InfOCF-Lib, featuring all proposed techniques, algorithms, and data
structures in form of an easy to use Java library.
Foreword
A core problem in Artificial Intelligence is the modelling of human
reasoning. It is rather obvious that classic-logical approaches are too
rigid for this task. While deductive inference always yields logically
correct results, this kind of reasoning is not appropriate in situations
where only incomplete or uncertain knowledge is available and where
conclusions must be drawn based on this knowledge. Such situations
are typical for human reasoning, because in virtually all realistic sce-
narios, there are all kinds of uncertainty and vagueness, or there is
only partial information available.
For representing uncertain or incomplete knowledge and for en-
abling reasoning with it, often expressions of the form If A, then
usually B are used. Such conditionals can be seen as default rules,
establishing a plausible, but defeasible relationship between the an-
tecedent A and the consequent B. Given a knowledge base consist-
ing of a set of such conditionals, one would like to draw conclusions
that are plausible and rational on the basis of the available knowl-
edge. Such conclusions are nonmonotonic because they may have to
be withdrawn when additional new information becomes available.
However, for the notions of plausible or rational, there are no mathe-
matically precise, generally accepted definitions. What a knowledge
base entails has therefore been a longstanding question in the area
of knowledge representation and reasoning. To address this question,
different nonmonotonic logics and various semantic frameworks as
well as axiom systems specifying desirable inference properties have
been developed. Drawing inferences licensed by a specific inference
method on the basis of the conditionals in a knowledge base induc-
tivelycompletestheknowledgeexplicitlygivenintheknowledgebase.
Inductive reasoning from conditional knowledge bases is the main
theme of this work. Using ordinal conditional functions, also called
vii
viii FOREWORD
ranking functions, as semantics for conditionals, Steven Kutsch stud-
ies inferences induced by single or by sets of ranking models, or by
specific subclasses of ranking models, in particular c-representations,
governedbyvariousinferencemodes. Heelaboratesindetailtheinter-
relationships among the resulting inference relations and shows their
formal properties with respect to established inference axioms. Based
on the introduction of a novel classification scheme for conditionals,
he also addresses the question of how to realise and to implement the
obtained entailment relations.
Throughoutthewholethesis,StevenKutschconvincinglypresents
his ideas, provides illustrating examples for them, rigorously defines
the introduced concepts, formally proves all technical results, and
fully implements every newly introduced inference method in an ad-
vanced Java library. Given the numerous remarkable results, con-
cerning both theoretical and practical aspects of inductive reasoning
from conditional knowledge bases, with his thesis, Steven Kutsch sig-
nificantly advances the state of the art in this field.
Hagen, November 2020 Christoph Beierle