Lech Polkowski

Rough mereology: A new paradigm for approximate reasoning

Abstract We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature: Dempster-Shafer theory, bayesian-based reasoning, belief networks, fuzzy logics, etc. We propose rough mereology as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes approximate proofs understood as schemes constructed to support our assertions about the world on the basis of our incomplete or uncertain knowledge.

Towards adaptive calculus of granules

An importance of the idea of granularity of knowledge for approximate reasoning has been stressed in Pawlak (1997) and Zadeh (1966, 1997). We address here the problem of synthesis of adaptive decision algorithms and we propose an approach to this problem based on the notion of a granule which we develop in the framework of rough mereology. This framework does encompass both rough and fuzzy set theories. Our approach may be applied in the problems of approximate synthesis of complex objects (solutions) in distributed systems of intelligent agents.

Rough Mereological Calculi of Granules: A Rough Set Approach To Computation

Rough Mereology is a paradigm allowing for a synthesis of main ideas of two potent paradigms for reasoning under uncertainty: Fuzzy Set Theory and Rough Set Theory. Approximate reasoning is based in this paradigm on the predicate of being a part to a degree. We present applications of Rough Mereology to the important theoretical idea put forth by Lotfi Zadeh (1996, 1997), i.e., Granularity of Knowledge: We define granules of knowledge by means of the operator of mereological class and we extend the idea of a granule over complex objects like decision rules as well as decision algorithms. We apply these notions and methods in the distributed environment discussing complex problems of knowledge and granule fusion. We express the mechanism of complex granule formation by means of a formal grammar called Synthesis Grammar defined over granules of knowledge, granules of classifying rules, or over granules of classifying algorithms. We finally propose hybrid rough‐neural schemes bridging rough and neural computations.

Rough mereological foundations for design, analysis, synthesis, and control in distributed systems

Abstract We propose a unified formal treatment of problems of design, analysis, synthesis and control in distributed systems of intelligent agents. Our approach is rooted in rough set theory and we propose rough mereology as a foundational basis for our approach.

Rough-Neuro Computing

We outline a rough-neuro computing model as a basis for granular computing. Our approach is based on rough sets, rough mereology and information granule calculus.

Synthesis of Decision Systems from Data Tables

We discuss two basic questions related to the synthesis of decision algorithms. The first question can be formulated as follows: what strategies can be used in order to discover the decision rules from experimental data? Answering this question, we propose to build these strategies on the basis of rough set methods and Boolean reasoning techniques. We present some applications of these methods for extracting decision rules from decision tables used to represent experimental data. The second question can be formulated as follows: what is a general framework for approximate reasoning in distributed systems? Answering this question, we assume that distributed systems are organized on rough mereological principles in order to assembly (construct) complex objects satisfying a given specification in a satisfactory degree. We discuss how this approach can be used for building the foundations for approximate reasoning. Our approach is based on rough mereology, the recently developed extension of mereology of Lesniewski.

Decision Algorithms: a Survey of Rough Set - Theoretic Methods

In this paper we present some strategies for synthesis of decision algorithms studied by us. These strategies are used by systems of communicating agents and lead from the original (input) data table to a decision algorithm. The agents are working with parts of data and they compete for the decision algorithm with the best quality of object classification. We give examples of techniques for searching for new features and we discuss some adaptive strategies based on the rough set approach for the construction of a decision algorithm from a data table. We also discuss a strategy of clustering by tolerance.

Rough mereology in information systems. A case study: qualitative spatial reasoning

Rough Mereology has been proposed as a paradigm for approximate reasoning in complex information systems [65], [66], [67], [68], [76]. Its primitive notion is that of a rough inclusion functor which gives for any two entities of discourse the degree in which one of them is a part of the other. Rough Mereology may be regarded as an extension of Rough Set Theory as it proposes to argue in terms of similarity relations induced from a rough inclusion instead of reasoning in terms of indiscernibility relations (cf. Chapter 1); it also proposes an extension of Mereology as it replaces the mereological primitive functor of being a part with a more general functor of being a part in a degree. Rough Mereology has deep relations to Fuzzy Set Theory as it proposes to study the properties of partial containment which is also the fundamental subject of study for Fuzzy Set Theory.

Approximate Reasoning by Parts

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.

A rough set paradigm for unifying rough set theory and fuzzy set theory

In this plenary address, we would like to discuss rough inclusions defined in Rough Mereology, a joint idea with A. Skowron, as a basis for common models for rough as well as fuzzy set theories. We would like to justify the point of view that tolerance (or, similarity) is the leading motif common to both theories and in this area paths between the two lie.