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Approximate Reasoning by Parts

An Introduction to Rough Mereology

verfasst von: Lech Polkowski

Verlag: Springer Berlin Heidelberg

Buchreihe : Intelligent Systems Reference Library

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The monograph offers a view on Rough Mereology, a tool for reasoning under uncertainty, which goes back to Mereology, formulated in terms of parts by Lesniewski, and borrows from Fuzzy Set Theory and Rough Set Theory ideas of the containment to a degree. The result is a theory based on the notion of a part to a degree.

One can invoke here a formula Rough: Rough Mereology : Mereology = Fuzzy Set Theory : Set Theory. As with Mereology, Rough Mereology finds important applications in problems of Spatial Reasoning, illustrated in this monograph with examples from Behavioral Robotics. Due to its involvement with concepts, Rough Mereology offers new approaches to Granular Computing, Classifier and Decision Synthesis, Logics for Information Systems, and are--formulation of well--known ideas of Neural Networks and Many Agent Systems. All these approaches are discussed in this monograph.

To make the exposition self--contained, underlying notions of Set Theory, Topology, and Deductive and Reductive Reasoning with emphasis on Rough and Fuzzy Set Theories along with a thorough exposition of Mereology both in Lesniewski and Whitehead--Leonard--Goodman--Clarke versions are discussed at length.

It is hoped that the monograph offers researchers in various areas of Artificial Intelligence a new tool to deal with analysis of relations among concepts.

Inhaltsverzeichnis

Frontmatter

On Concepts. Aristotelian and Set—Theoretic Approaches

Abstract

In this chapter the reader is introduced to the idea of a concept. Historically, two main approaches can be discerned toward formalization of this idea. First, a holistic approach by Aristotle, in which a concept is construed as a term (category) by which any object can be either styled or not. This fact of calling an object a by a name term A is denoted by the formula a is A. With those formulas, Aristotle built a complete theory of concepts, the Calculus of Syllogisms, briefly exposed in section 1.1, below.

The alternative approach emerged with Georg Cantor’s set theory in mid–19th century and its essence was a representation of a concept by its elements; the relation of being an element was expressed as with Aristotle by the predicate is written down as ε (the ’esti’ symbol): the formula aεA reads an object a is an element of the concept (set) A. We offer in section 1.2 and following, an outline of modern set theory, aimed at establishing a terminology and basic facts of set theory, which seem to be indispensable to any researcher in computer science.

Lech Polkowski

Topology of Concepts

Abstract

Topology is a theory of certain set structures which have been motivated by attempts to generalize geometric reasoning based on Euclidean distance invariants and replace it by more flexible schemes. As the notion of closeness, or, distance, permeates a plethora of reasoning schemes, topological structures are often in focus of a reasoner. In many schemes of reasoning one resorts to the idea of a neighbor with the assumption that reasonably selected neighbors of a given object preserve its properties to a satisfactory degree (look at methods based on the notion of the nearest neighbor, for instance). The notion of a neighbor as well as a collective notion of a neighborhood are studied by topology.

Basic topological notions were originally defined by means of a notion of distance which bridges geometry to more general topological structures. The analysis of properties of distance falls into theory of metric spaces, see Hausdorff [12] and Fréchet [9], with which we begin our exposition of fundamentals of topology.

Generalizations to abstract structures are discussed in following sections, with emphasis on basic properties of compactness, completeness, continuity, relations to algebraic structures, quotient structures, and hyperspaces.

As the last topic, an important notion of a Čech topology is discussed, which arises often in applications, where the underlying structure cannot satisfy all requirements for a topology; such is, e.g., the case of mereological structures in general.

Lech Polkowski

Reasoning Patterns of Deductive Reasoning

Abstract

Processes of reasoning are divided into two main types: deductive and reductive. Deductive reasoning begins with a set of premises and concludes with a set of inferences obtained by specified rules of deduction, whereas reductive reasoning tries to obtain a set of premises/causes for an observed set of facts. In this chapter, we present the reader with some basic schemes of deductive reasoning. We begin with sentential calculus (propositional logic) which sets the pattern of deductive reasoning and then we discuss many–valued propositional calculi, predicate calculus, and modal calculi. Analysis of these types of reasoning results in notions which are relevant also for reductive types of reasoning, in particular, for rough and fuzzy types of reasoning discussed in the next chapter as examples of reductive reasoning.

Lech Polkowski

Reductive Reasoning Rough and Fuzzy Sets as Frameworks for Reductive Reasoning

Abstract

Reductive reasoning, in particular inductive reasoning, Bocheński [9], Łukasiewicz [30], is concerned with finding a proper p satisfying a premise p⇒q for a given conclusion q. With some imprecision of language, one can say that its concern lies in finding a right cause for a given consequence. As such, inductive reasoning does encompass many areas of research like Machine Learning, see Mitchell [37], Pattern Recognition and Classification, see Duda et al. [16], Data Mining and Knowledge Discovery, see Kloesgen and Zytkow [26], all of which are concerned with a right interpretation of data and a generalization of findings from them. The matter of induction opens up an abyss of speculative theories, concerned with hypotheses making, verification and confirmation of them, means for establishing optimality criteria, consequence relations, non–monotonic reasoning etc. etc., see, e.g., Carnap [12], Popper [55], Hempel [22], Bochman [10].

Our purpose in this chapter is humble; we wish to give an insight into two paradigms intended for inductive reasoning and producing decision rules from data: rough set theory and fuzzy set theory.

We pay attention to structure and basic tools of these paradigms; rough sets are interesting for us, as forthcoming exposition of rough mereology borders on rough sets and uses knowledge representation in the form of information and decision systems as studied in rough set theory. Fuzzy set theory, as already observed in Introduction, is to rough mereology as set theory is to mereology, a guiding motive; in addition, main tools of fuzzy set theory: t–norms and residual implications are also of fundamental importance to rough mereology, as demonstrated in following chapters.

Lech Polkowski

Mereology

Abstract

Mereology emerged in the beginning of XXth century due to independent efforts of S. Leśniewski and A. N. Whitehead. In the scheme of Leśniewski, the predicate of being a part was taken as the primitive notion whereas in the development of Whitehead’s ideas the primitive notion was adopted as the predicate of being connected. Mereology presents an alternative, holistic, approach to concepts which is especially suited to reasoning about extensional objects, e.g., spacial ones as witnessed, e.g., by the Tarski axiomatization of geometry of solids, or recent applications to geometric information systems or analysis of statements about spatial objects and relations in natural language.

Lech Polkowski

Rough Mereology

Abstract

A scheme of mereology, introduced into a collection of objects, see Ch. 5, sets an exact hierarchy of objects of which some are (exact) parts of others; to ascertain whether an object is an exact part of some other object is in practical cases often difficult if possible at all, e.g., a robot sensing the environment by means of a camera or a laser range sensor, cannot exactly perceive obstacles or navigation beacons. Such evaluation can be done approximately only and one can discuss such situations up to a degree of certainty only. Thus, one departs from the exact reasoning scheme given by decomposition into parts to a scheme which approximates the exact scheme but does not observe it exactly.

Lech Polkowski

Reasoning with Rough Inclusions: Granular Computing, Granular Logics, Perception Calculus, Cognitive and MAS Reasoning

Abstract

Rough mereology allows for a plethora of applications in various reasoning schemes due to universality of its primitive predicate of a part to a degree. We have already stressed that by its nature, rough mereology is especially suited to reasoning with collective concepts like geometric figures or solids, or, concepts learned by machine learning methods, i.e., with collective concepts. Those applications are presented in Ch. 8 and Ch. 9. In this chapter, we begin this discussion with a formal approach to the problem of granulation of knowledge and then we examine rough mereological logics: from our results in Ch. 6 it follows that representing implication with a rough inclusion μ leads to logics which extend and generalize fuzzy logics. As an application, we propose a formal rendering of the idea of perception calculus, due to Zadeh [67]. We apply rough mereological schemes to reasoning by multi–agent (MAS) systems, and finally we present a rough mereological variant of cognitive reasoning in neural–like systems.

Lech Polkowski

Reasoning by Rough Mereology in Problems of Behavioral Robotics

Abstract

In Ch. 6, we have developed basic notions and propositions of rough mereogeometry and rough mereotopology. We have stressed that by its nature, rough mereology does address collective concepts, relations among which are expressed by partial containment rendered as the predicate of a part to a degree. Behavioral robotics falls into this province, as usually robots as well as obstacles and other environmental objects are modeled as figures or solids. In this chapter, we discuss planning and navigation problems for mobile autonomous robots and their formations. In particular, we give a formal definition of a robot formation based on the betweenness relation, cf., Ch. 6., sect. 10. First, we introduce the subject of planning in robotics.

Lech Polkowski

Rough Mereological Calculus of Granules in Decision and Classification Problems

Abstract

The idea of mereological granulation of knowledge, proposed and presented in detail in Ch. 7, sect. 3, finds an effective application in problems of synthesis of classifiers from data tables. This application consists in granulation of data at preprocessing stage in the process of synthesis: after granulation, a new data set is constructed, called a granular reflection, to which various strategies for rule synthesis can be applied. This application can be regarded as a process of filtration of data, aimed at reducing noise immanent to data. This chapter presents this application.

Lech Polkowski

Backmatter

Titel: Approximate Reasoning by Parts
verfasst von: Lech Polkowski
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-22279-5
Print ISBN: 978-3-642-22278-8
DOI: https://doi.org/10.1007/978-3-642-22279-5

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Inhaltsverzeichnis

Frontmatter

On Concepts. Aristotelian and Set—Theoretic Approaches

Topology of Concepts

Reasoning Patterns of Deductive Reasoning

Reductive Reasoning Rough and Fuzzy Sets as Frameworks for Reductive Reasoning

Mereology

Rough Mereology

Reasoning with Rough Inclusions: Granular Computing, Granular Logics, Perception Calculus, Cognitive and MAS Reasoning

Reasoning by Rough Mereology in Problems of Behavioral Robotics

Rough Mereological Calculus of Granules in Decision and Classification Problems

Backmatter

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