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2001 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

A Logical Theory of Nonmonotonic Inference and Belief Change

verfasst von: Dr. Alexander Bochman

Verlag: Springer Berlin Heidelberg

Buchreihe : Artificial Intelligence

Enthalten in: Professional Book Archive

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Über dieses Buch

The main subject and objective of this book are logical foundations of non­ monotonic reasoning. This bears a presumption that there is such a thing as a general theory of non monotonic reasoning, as opposed to a bunch of systems for such a reasoning existing in the literature. It also presumes that this kind of reasoning can be analyzed by logical tools (broadly understood), just as any other kind of reasoning. In order to achieve our goal, we will provide a common logical basis and semantic representation in which different kinds of non monotonic reasoning can be interpreted and studied. The suggested framework will subsume ba­ sic forms of nonmonotonic inference, including not only the usual skeptical one, but also various forms of credulous (brave) and defeasible reasoning, as well as some new kinds such as contraction inference relations that express relative independence of pieces of data. In addition, the same framework will serve as a basis for a general theory of belief change which, among other things, will allow us to unify the main approaches to belief change existing in the literature, as well as to provide a constructive view of the semantic representation used. This book is a monograph rather than a textbook, with all its advantages (mainly for the author) and shortcomings (for the reader).

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction
Abstract
Taken in a broader perspective, a theory of nonmonotonic reasoning gives us (or, more exactly, should give) a more direct and adequate description of the actual ways we think about the world. More often than not we reason and act in situations where we do not or even cannot have complete information. So, deductive reasoning, taken by itself, simply cannot help us in such situations. Still, we usually need to act in such situations in a reasonable way, and it is here that nonmonotonic reasoning finds its place. This means, in particular, that the necessity of nonmonotonic reasoning does not stem from computational considerations, as is sometimes supposed in Artificial Intelligence; it is not the question of speeding or simplifying our reasoning. Rather, it is a matter of vital necessity: we have to act reasonably in situations of partial knowledge in order to achieve our practical and theoretical goals.
Alexander Bochman

The Framework

Frontmatter
2. Consequence Relations
Abstract
In this chapter we will describe the formalism of consequence relations that will serve as a logical basis for our approach to nonmonotonic reasoning and belief change. We are not intending, however, to give a comprehensive theory of consequence relations. Rather, the following exposition can be seen as a compendium of notions and results that will be used in what follows. In fact, due to this biased description, a lot of the notions described below, such as semi-classicality, groundedness and base-generation, will be quite new and unusual in a general theory of consequence relations. We will show, in particular, that Scott consequence relations is a powerful formalism that allows a concise logical description of many interesting concepts outside their traditional range of applications. Actually, the results of this chapter will also show that common descriptions of consequence relations still only scratch the surface of this notion. And subsequent applications will demonstrate that it is worth being studied in depth.
Alexander Bochman
3. Epistemic States
Abstract
Semantic interpretation constitutes one of the main components of a viable reasoning system, monotonic or not. A formal inference engine, though important in its own right, can be effectively used for representing and solving commonsense reasoning tasks only if its basic inference steps possess a clear meaning allowing us to discern them from a description of a situation at hand. This also means that not all formally adequate semantic representations will do on the final score, but only those which provide a human-friendly framework of concepts that can be easily and systematically discerned from (or imposed upon) the informal description.
Alexander Bochman
4. Similarity, Equivalence and Decomposition of Epistemic States
Abstract
Since epistemic states will be used in this book for representing nonmonotonic inference and belief change, we will be primarily interested in the properties of epistemic states that could influence the behavior of generated inference relations and belief change operations. In order to single out the properties of epistemic states that are relevant, we will introduce in this chapter a number of notions of equivalence for epistemic states that will preserve the generated inference relations and belief sets. We will start with the strongest notion of similarity for epistemic states that will be invariant under the basic belief change operations. In other words, similar epistemic states will produce the same response under any future change made to these states; borrowing the terminology of [FUKV86], they will be equivalent forever. We will also show that any epistemic state is decomposable in this sense into a set of linear epistemic states. In addition, we will introduce the notion of a selection function which will determine the ‘inference profile’ of an epistemic state. Epistemic states generating the same selection function will be called equivalent. The notions of equivalence and similarity will supply us with powerful tools for investigating properties of epistemic states that are essential in determining their behavior in nonmonotonic inference and belief change.
Alexander Bochman
5. Epistemic Entrenchment and Its Relatives
Abstract
In this chapter we will lay down general foundations for an approach to nonmonotonic reasoning and belief change based on establishing certain expectation relations on the set of propositions. As we mentioned in Chapter 3, such an approach arises naturally when we adopt a coherentist position according to which dependencies and justification relations are disregarded. However, we have also seen that the dependence structure is also describable in terms of certain dependence relations on propositions. This creates a rather involved picture in which these relations are actually interdefinable. A first sketch of the emerging picture will be drawn in this chapter. Further details will be added in subsequent chapters when we will consider the role being played by these relations in describing nonmonotonic reasoning and belief change.
Alexander Bochman

Nonmonotonic Inference

Frontmatter
6. The Basic Inference Relation
Abstract
The approach to nonmonotonic reasoning based on describing associated inference relations forms one of the most natural and powerful tools in its study. Gabbay’s paper [Gab85], a starting point of the approach, has been followed by a number of fundamental works (notably [Mak89, Sho88]) that have reached saturation in the so-called KLM theory [KLM90, LM92] (see also [Mak94] for an overview). In these works a semantic representation of nonmonotonic inference relations was developed based on sets of world-states ordered by a preference relation: a nonmonotonic inference rule A |~ B was characterized as saying that B should be true in all preferred states satisfying A. Shoham [Sho88] has shown that such a preference-based representation allows us to unify many approaches to nonmonotonic reasoning developed in Artificial Intelligence.
Alexander Bochman
7. Skeptical Inference Relations
Abstract
In this chapter we are going to consider the notion of a skeptical inference, which constitutes a most important and studied kind of nonmonotonic inference. As we said in Chapter 3, our semantic interpretation of such an inference is derived from the traditional understanding of conditionals suggested by Ramsey and Mill. From a purely technical point of view, the representation of such an inference in terms of epistemic states could also be seen as a ‘fusion’ of the two main approaches to describing such inference suggested in the literature, namely possible worlds-based preferential models of [KLM90], and expectation-based models of [GM94]. We have already seen that possible worlds models constitute a very special case of epistemic states. On the other hand, our representation provides a generalization of the approach suggested in [GM94] that extends the latter to a broader class of nonmonotonic logics. Moreover, it shares with the latter the change of gestalt consisting in viewing preferences between worlds as merely a by-product of more basic relations holding between our beliefs. As a special case of this representation, we will establish a direct correspondence between preferential inference relations and supraclassical Tarski consequence relations.
Alexander Bochman
8. Prolegomena to a Theory of Defeasible Entailment
Abstract
A most natural kind of default assumptions we use in commonsense reasoning are conditional in nature, such as “Birds fly” or “Adults are employed”. Consequently, a theory of reasoning about such default conditionals should occupy a central place in the general theory of nonmonotonic reasoning.
Alexander Bochman
9. Credulous Nonmonotonic Inference
Abstract
Skeptical inference relations described in Chapter 7 were designed to capture a skeptical approach to nonmonotonic reasoning, according to which if there is a number of equally preferred alternatives, we infer only what is common to all of them. However, works in nonmonotonic reasoning have suggested also an alternative approach, usually called credulous or brave reasoning, according to which each of the preferred alternatives is considered as an admissible solution to the nonmonotonic reasoning task. Actually, there are many important reasoning problems in AI and beyond, such as diagnosis, abduction and explanation, that are best seen as involving search for particular preferred solutions. This idea is implicit also in the notion of an extension in default logic [Rei80] and its generalizations, as well as in similar constructs in modal nonmonotonic logics.
Alexander Bochman
10. Contraction Inference
Abstract
In this chapter we will give a detailed description of contraction inference introduced in Chapter 3. As has been said, in the general correspondence between nonmonotonic inference and belief change, contraction inference corresponds to the basic operation of belief contraction. This augments the idea that belief change and nonmonotonic inference are “two sides of the same coin” and extends it to belief contractions.
Alexander Bochman

Belief Change

Frontmatter
11. Belief Change and Its Problems
Abstract
We have seen in Part Two that epistemic states provide a powerful and versatile representation for nonmonotonic reasoning. In this part we are going to show that epistemic states also constitute a very useful tool for representing and studying belief change. To begin with, we will briefly describe in this chapter the main approaches to representing belief change, as well as common problems arising with these representations. Then we will indicate how these problems can be resolved by taking epistemic states as an alternative background for belief change processes. We will also sketch new opportunities and directions of research about belief change arising in the suggested framework.
Alexander Bochman
12. Contractions of Epistemic States
Abstract
As we argued in the preceding chapter, belief change operations should be defined primarily as operations on epistemic states; though any epistemic state has an associated belief set, changes to the latter will be determined by changes made to the underlying epistemic state. In this chapter we will describe this process for the case of contractions.
Alexander Bochman
13. Merge and Expansion
Abstract
Whereas the general purpose of the contraction operation amounts to removing information (that is no longer believed), the operation of expansion consists in adding information to epistemic states. In the AGM theory this is achieved through a straightforward addition of the new proposition to the belief set, while in the base approach the new proposition is added directly to the base.
Alexander Bochman
14. Compound and Derived Changes
Abstract
In this last chapter we will study some compound operations that are definable in our framework, the most important among them being the revision operation. Also, we will briefly discuss changes in inference and entrenchment relations that are generated by expansions and revisions of epistemic states. Finally, we will single out sets of epistemic states that form ‘natural classes’ with respect to the main operations discussed in the previous chapters.
Alexander Bochman
Backmatter
Metadaten
Titel
A Logical Theory of Nonmonotonic Inference and Belief Change
verfasst von
Dr. Alexander Bochman
Copyright-Jahr
2001
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-04560-2
Print ISBN
978-3-642-07516-2
DOI
https://doi.org/10.1007/978-3-662-04560-2