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Y.’) Lin 1 and Churn-Jung Liau 2 1 Department of Computer Science San Jose State University San Jose, CA 95192 tylin@cs.sjsu.edu 2 Institute of Information Science Academia Sinica, Taipei 115, Taiwan liaucj@iis.sinica.edu.tw Summary. This chapter gives an overview and refinement of recent works on binary granular computing. For comparison and contrasting, granulation and partition are examined in parallel from the prospect of rough Set theory (RST).The key strength of RST is its capability in representing and processing knowledge in table formats. Even though such capabilities, for general granulation, are not available, this chapter illustrates and refines some such capability for binary granulation. In rough set theory, quotient sets, table representations, and concept hierarchy trees are all set theoretical, while in binary granulation, they are special kind of pretopological spaces, which is equivalent to a binary relation Here a pretopological space means a space that is equipped with a neighborhood system (NS). A NS is similar to the classical NS of a topological space, but without any axioms attached to it 3 . Key words: Granular computing, rough set, binary relation, equivalence relation 22.1 Introduction Though the label, granular computing is relatively recent, the notion of granulation has in fact been appeared, under different names, in many related fields, such as pro- gramming, divide and conquer, fuzzy and rough set theories, pretopological spaces, interval computing, quantization, data compression, chunking, cluster analysis, be- lief functions, machine learning, databases, and many others. In the past few years, we have seen a renewed and fast growing interest in Granular Computing (GrC). Many applications of granular computing have appeared in fields, such as medicine, economics, finance, business, environment, electrical and computer engineering, a number of sciences, software engineering, and information science. 3 This is an expansion of the article (Lin, 2005) in IEEE connections, the news letter of the IEEE Computational Intelligence Society O. Maimon, L. Rokach (eds.), Data Mining and Knowledge Discovery Handbook, 2nd ed., DOI 10.1007/978-0-387-09823-4_22, © Springer Science+Business Media, LLC 2010 446 Tsau Young (’T. Y.’) Lin and Churn-Jung Liau Granulation seems to be a natural problem-solving methodology deeply rooted in human thinking. Many daily ”things” have been routinely granulated into sub”things;” human body has been granulated into head, neck, and so forth; geo- graphic features into mountains, planes, and others. The notion is intrinsically fuzzy, vague and imprecise. Mathematicians idealized it into the notion of partitions, and developed it into a fundamental problem-solving methodology; it has played major roles throughout the entire history of mathematics. Nevertheless, the notion of partitions, which absolutely does not permit any over- lapping among its granules, seems to be too restrictive for real world problems. Even in natural science, classification does permit small degree of overlapping; there are beings that are both appropriate subjects of zoology and botany. A more general theory is needed. Based on Zadeh’s grand project on granular mathematics, during his sabbati- cal leave (l996/l997) at Berkeley, Lin focused on a subset of granular mathematics, which he called granular computing (Zadeh, 1998). To stimulate research on granu- lar computing, a special interest group, with T. Y. Lin as its Chair, was formed within BISC (Berkeley Initiative in Soft Computing). Since then, granular computing has evolved into an active research area, generating many articles, books and presen- tations at conferences, workshops and special sessions. This chapter is devoted to present some of such development over the past few years. There are two possible approaches: (1) One is starting from fuzzy side and mov- ing down, and (2) the other one is from extreme crisp side and moving up. In this chapter, we take the second approach incrementally. Recall that algebraically a parti- tion is an equivalence relation, so a natural next step is the binary granulation defined by a binary relation. For contrasting, we may call a partition A-granulation and the more general granulation B-granulation. 22.2 Naive Model for Problem Solving An obvious approach to a large-scaled computing problem is: (1) To divide the prob- lem into subtasks, might be point by point and level by level. (2) To elevate or abstract the problem into concept/knowledge spaces, could be in multilevels. (3) To integrate the solutions of subtasks and quotient tasks (knowledge spaces) of several levels 22.2.1 Information Granulations/Partitions In the first step, we select an appropriate system of granulation/partition so that only the summaries of granules/equivalence classes may enter into the higher level computing. The information in data space is transformed to a concept space, pos- sibly in levels, which may be locally at each point or globally at eh whole uni- verse (Lin, 2003b). Classically, we granulate by partitioning (no overlapping on gran- ules). Such examples are plentiful: in mathematics (quotient groups, quotient rings and etc. (Birkhoff and MacLane, 1977)), in theoretical computer science (divide-and- conquer (Aho et al., 1974)), in software engineering (the structural, object oriented, 22 Granular Computing and Rough Sets - An Incremental Development 447 and component based design and programming (Szyperski, 2002)), in artificial intel- ligence (Hobbs, 1985, Zhang and Zhang, 1992), in rough set theory (Pawlak, 1991) among others. However, these are all partition based, where no overlapping of gran- ules is permitted. As we have observed, even in biology, classification does allow some overlapping. The focus of this presentation will be on non-partition theory,but only in an epsilon step away from partitioning method. 22.2.2 Knowledge Level Processing and Computing with Words The information in each granule is summarized and the original problem is re- expressed in terms of symbols, words, predicates or linguistic variables. Such re- expressing is often referred to as knowledge representations. Its processing has been termed computing with symbols (table processing, computing with words, knowl- edge level processing, even precisiated natural language, depending on the complex- ity of the representations. In this chapter, we are computing on the space of granules or ”quotient space.” in which each granule is represented by a word that carries different degree of seman- tics. For partition theory, the knowledge representation is in table format (Pawlak, 1991) and its computation is syntactic in nature. For binary granulation, that we have focused here, is semantic oriented. We expand and streamline the previous works (Lin, 1998a,Lin, 1998b,Lin, 2000); the main idea is to transfer the computing with words into computing with symbols. Loosely speaking computing with symbols or symbolic computing is an “ax- iomatic” Computing: all rules of computing symbols are determined by the axioms. The computation follows the formal specifications. Such computing occurs only in an ideal situation. In many real world applications, unfortunately, such as non-linear computing, the formal specifications are often unavailable. So computing with words are needed; it can be processed informally. Semantics of words often may not be completely or precisely formalized. Their semantic computing is often carried out in the systems with human helps (the semantics of symbols are not implemented). Human enforced semantic computing are common in data processing environment. 22.2.3 Information Integration and Approximation Theory Most applications require the solutions be presented in the same level as input data. So the solutions often need to be integrated from subtasks (solutions in granules) and quotient tasks (solutions in the spaces of granules). For some applications, such as Data Mining and some rough set theory, are aimed at high level information; in such cases this step can be skipped. In general, the integration is not easy. In partition world, many theories have been developed in mathematics; e. g., extension functors. The approximation theory of pretopological spaces and rough set theory can be regarded as in this step. 448 Tsau Young (’T. Y.’) Lin and Churn-Jung Liau 22.3 A Geometric Models of Information Granulations For understanding the general idea, in this section, we recall and refine a previous formalization in (Lin, 1998a). The goal is to formalize Zadeh’s informal notion of granulation mathematically. As original thesis is informal, the best we could do is to present, hopefully, con- vincing arguments. We believe our formal theory is very close to the informal one. According to Zadeh (1996): Information granulation involves partitioning a class of objects(points) into granules, with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity or functionality. We will literally take Zadeh’s informal words as a formal definition of granula- tion. We observe that: 1. A granule is a group of objects that are draw together (by indistinguishability, similarity or functionality). The phrase ”drawn together” implicitly implies certain level of symmetry among the objects in a granule. Namely, if p is drawn towards q, then q is also drawn towards p. Such symmetry, we believe, is imposed by imprecise-ness of natural language. To avoid such an implications, we will rephrase it to ”drawn towards an object p,” so that it is clear the reverse may or may not be true. So we have first revision: 2. A granule is a group B(p) of objects that are draw toward an object p. Here p varies through every object in the universe. 3. Such an association between object p and a granule B(p) induces a map from the object space to power set of object space. This map has been called a binary granulation (BG). 4. Geometric View: We may use geometric terminology and refer to the granule as a neighborhood of p, and the collection {B(p)} a binary neighborhood system (BNS). It is possible that B(p) is an empty set. In this case we will simply say p has no neighborhood (abuse of language; to be very correct, we should say p has an empty neighbor- hood). Also it is possible that different points may have the same neighborhood (granule) B(p)=B(q). The set of all q, where B(q) is equal to B(p), is called the centers C(p) of B( p). 5. Algebraic View: Consider the set R = {(p, u)}, where u in B(p) and p in U. It is clear that R is a subset of U ×U, hence defines a binary relation (BR), and vice versa. Proposition 1 A binary neighborhood system (BNS), A binary granulation (BG), and a binary relation (BR) are equivalent. From the analysis given above, we propose the following mathematical model for information granulation. 22 Granular Computing and Rough Sets - An Incremental Development 449 Definition 1 By a (single level) information granulation defined on a set U we mean a binary granulation (binary neighborhood system, binary relation) defined on U. Let us goes a little bit further. Note that the binary relation is a mathematical expression of Zadeh’s ”indistinguishability, similarity or functionality.” We abstract the three properties into a list of abstract binary relations {B j | j run through some index set }, where each B j is a binary relation. Note that at each point p, each B j induces a neighborhood B j (p). Some may be empty, or identical. By removing empty set and duplications, the family have been we re-indexed N i (p). As in the single level case, we will define directly the granulation N : U → 2 2 U ; p →{B i (p) | i run through some index set }. The collection {B i (p)} is called a neighborhood system(NS)or (LNS); the latter one is used to distinguish itself from the neighborhood system (TNS) of a topological space (Lin, 1989a, Lin, 1992). Definition 2 By a local multi-level information granulation defined on U, we mean a neighborhood system (NS) is defined on U. By a global multi-level information granulation defined on U, we mean a set of BG is defined on U. All notions can be fuzzified. The right way to look at this section is to assume implicitly there is a modifier ”crisp/fuzzy” to all notions presented above. 22.4 Information Granulations/Partitions Technically, granular computing is actually computing with constraints. Especially in “infinite world”, granulation is often given in terms of constraints. In this chapter, we concerns primarily with constraints that are mathematically represented as binary relations 22.4.1 Equivalence Relations(Partitions) Partition is a decomposition of the universe into a family of disjoint subsets. They are called equivalence classes, because a partition induces an equivalence relation and vice versa. In this chapter, we will view the equivalence class in a special way. Let A ⊆U ×U be an equivalence relation (a reflexive, symmetric and transitive binary relation). For each p, let A p = {v ∈U : pAv} (22.1) A p is the equivalence class containing p, and will be called A-granule for the purpose of contrasting with general cases. 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