Ebook: Knowledge Representation and Inductive Reasoning using Conditional Logic and Sets of Ranking Functions
A core problem in Artificial Intelligence is the modeling of human reasoning. Classic-logical approaches are too rigid for this task, as deductive inference yielding logically correct results is not appropriate in situations where conclusions must be drawn based on the incomplete or uncertain knowledge present in virtually all real world scenarios.
Since there are no mathematically precise and generally accepted definitions for the notions of plausible or rational, the question of what a knowledge base consisting of uncertain rules entails has long been an issue in the area of knowledge representation and reasoning. Different nonmonotonic logics and various semantic frameworks and axiom systems have been developed to address this question.
The main theme of this book, Knowledge Representation and Inductive Reasoning using Conditional Logic and Sets of Ranking Functions, is inductive reasoning from conditional knowledge bases. Using ordinal conditional functions as ranking models for conditional knowledge bases, the author studies inferences induced by individual ranking models as well as by sets of ranking models. He elaborates in detail the interrelationships 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 realize and implement the entailment relations obtained.
In this work, “Steven Kutsch convincingly presents 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 advanced Java library (…). He significantly advances the state of the art in this field.” – Prof. Dr. Christoph Beierle of the FernUniversität in Hagen
The area of knowledge representation and reasoning is concerned with simulating intelligent behaviour in automated agents using explicit, formal representations of domain or evidential knowledge. We consider rules with exceptions, called conditionals, within the broader context of inductive reasoning for representing and reasoning with uncertain knowledge. Inductive reasoning techniques transform a finite 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 implausibility 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 better, we introduce a classification of qualitative conditionals with respect to semantic properties of conditionals. The idea of redundant conditionals proposed in the literature is elaborated upon, while investigating 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 employing 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 considering inference postulates proposed in the literature and by presenting 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.