Based on Smale's machines over the reals [9], we will formalize approximate computing in numerical analysis. Classical numerical computing generate and accumulate minute errors that lead to real errors; to avoid that error estimations in the forms of ε-δ have to be embedded in the scheme of computing. Such Fundamental Theorems are formulated and proved. By errors a pre-matured version that was withdrawn has appeared in IFSA/NAFIPS2019.

A "perfect" theory for new fuzzy sets, called Zadeh sets is outlined. Here are four observations, one theorem, and the proposed theory. (1) Fuzzy sets are not fuzzy - An ancient critique. (2) The theory of fuzzy sets does not support fuzzy control - Example 1. (3) The concept of fuzzy sets induced by mappings is not natural-Section II-A. (4) Zadeh sets should be context free - Section III-A. (5) Multi-valued modeling Theorem A real number can be modeled uniquely by a family of smooth(C^∞-differentiable) membership functions. By extending this idea, we propose: A nice Zadeh set is the unique family of nice membership functions on a nice universe U that characterizes a nice "real world " fuzzy aggregate (class, family, collection or set) where nice means smooth(C^∞-differentiable), continuous or set theoretical (a set theoretical membership function is the usual membership function defined by Zadeh.) Initial analysis indicates that such three types of nice Zadeh sets seem to have captured the correct concept of fuzziness. In addition, mathematically speaking, the theory forms naturally, behaves smoothly and three categories of nice Zadeh sets are concrete categories. So we conclude Z adeh set theory is "perfect." Perhaps, we should point out the 'classical' categories of fuzzy subsets are not concrete.

Achieving total information flow security is feasible through the use of formal mathematical models. A fundamental theorem of a topological theory of Discretionary Access Controls is presented: Let U be a set of UNIX users. The association from each data owner p to its data owners group F(p) defines a generalized topological space (U, F) and a binary relation BF. Thus, a DAC-system is information-flow-secure (via file movements): if and only if (U, F) is a finite topological space; if and only if (U, BF) is a reflexive and transitive graph. An important consequence is the following proved secure system in current UNIX: The information flow (via file movements) of UNIX generated by Read-access is information-flow-secure: if (U, F) is a finite topological space;if (U, BF) is a reflexive and transitive graph. This model is based on a modified security principle: Every data owner p has the discretionary authority (Individual Data Owner p's authority, or IDO(p)-authority) over who can access the data. The precise security policy is the integration of IDO(p)-security for All data owner p in All possible data states. Once granted access, information propagation can still be controlled by the topological structure.

Based on the concept of isomorphism of relations, a relation is turned into a simplicial complex, which is a combinatorial representation of a polyhedron. So frequent itemsets mining is transform turned into geometric traversal problem. By leveraging on geometric structure of simplicial complex, a very fast algorithm for traversal is found; it is based on a geometric concept, called sub-cone construction. It is not only very fast but also use much less memory than existing methods. For a real world medical database of 1257 columns with 65K rows, the new algorithm takes only 5 seconds to find all frequent itemsets, while FP-growth takes approximately 1000 seconds; so simplicial complex method is about 200 times faster. Moreover the memory usage is substantially less; see the data.

A Business Cloud is defined to be a collection of company datasets that are stored on the "Cloud". For simplicity, we have assumed: Each company only has one dataset. There are information flows among these datasets. Within such an environment Chinese Wall Security Policy (CWSP) is revisited. Based on the "physical" view of Brewer and Nash, the Chinese Wall policy that regulates "Conflict of Interest" binary Relation, denoted by CIR among company datasets, is investigated. CIR has found to be anti-reflexive, symmetric and anti-transitive, in other words, the compliment of CIR is an equivalence relation. The main theorem states that, from the owner (a company) view, a dataset under CWSP will neve flow into enemies hands (companies in conflicts). More generally, if CIR is anti-reflexive and symmetric, then there are Aggressive Chinese Wall that surrounds the company datasets of those companies who are in conflict, and a Conservative Chinese Wall that that surrounds the company datasets whose owner are friends; note that the set friends is a subset of "not in conflict". The main theorem states that no information of those company datasets that are stored within the Aggressive Chinese Wall may never penetrate the Wall, similar conclusion for Conservative Chinese Wall. However those company datasets that are outside these two walls may penetrate the two walls.

The primary goal is to introduce a new methodology into approximate computing for big numerical data. First, an unpleasant phenomenon in large scaled numerical computing is observed: The probability of getting "correct approximations" is near zero for big numerical data. Second, a theorem that echoes this observation is proved: Perfect approximations are impossible in classical approach (Theorem 1). Using Codd's terms in relational databases, "time-varying" intervals are used to build the real number system (Theorem 2). If the effective computing procedure defines a continuous real-valued function P, then the P(approximate value) and P(true value) can be smaller than any given e. In other words, in this new approach, we find a perfect approximation (Theorem 3).

The use of Statistical Finite Automata (SFA) has been explored in the field of understanding the DNA sequences; many focus on local patterns, namely partial representations of DNA sequences. In this paper, we focus on global and complete representations to understand the patterns in whole DNA sequences. Obviously, DNA sequences are not random. Based on Kolmogorov complexity theory, there should be some simple Turing machines that write out such sequences; here simple means the complexity of the Turing machine is simpler than the data. The primary goal of this paper is to approximate such simple Turing machines by SFA. We use SFA, via ALERGIA algorithm (in the light granular computing), to capture and analyze the translation process (DNA to protein) based on amino acids' chemical property viz., polarity. This, in turn, enables the understanding of interspecies DNA comparisons and the creation of phylogeny - the `tree of life'.

A topological space is a “space”, where “near” makes sense; it is formally defined by the Topological Neighborhood System (TNS). Here, we explore the concept of “conflict” by the system of “Anti-TNS”; by that we mean a “mathematical structure” that consists of a set of “punctured” neighborhoods, namely, the center point p of all neighborhoods of TNS has been removed. “Conflicts” are important concepts in computer security. The primary results is the axiomatization of ATNS. The main results are surprising: The set of the axioms is the same as that of topological spaces. Similar results for pretopological spaces also are obtained.

This paper explains the mathematics of large scaled granular computing (GrC), augmented with a new Knowledge theory, by unifying rough set theories (RS) into one single concept, namely, neighborhood systems (NS). NS was first introduced in 1989 by T. Y. Lin to capture the concepts of “near” (topology) and “conflict” (security). Since 1996 when the term Granular Computing (GrC) was coined by T. Y. Lin to label Zadeh's vision, NS has been pushed into the “heart” of GrC. In 2011, LNS, the largest NS, was axiomatized; it implied that this set of axioms defines a new mathematics that realizes Zadeh's vision. The main messages are: this new mathematics is powerful and practical.

The primary goal of this paper is to present a new approximation theory. In classical rough sets, topological concepts, closure, and interior are expressible by elementary sets; hence, they are the elementary knowledge approximations. However, for general binary relations, the corresponding closure and interior cannot always be interpreted by elementary knowledge. The primary results of this paper are to show that the appropriate generalized closure and interior are central knowledge approximations and have many expected nice properties, such as the upper approximation contains lower approximations. Here the central knowledge approximations mean the approximations are expressible by centers sets and the center set denotes the set of points that regard a binary neighborhood of the point p (which is the set of points that are (right) related to p) as its neighborhood. Many examples are used to justify our new view.

Based on granular computing, information flows in Discretionary Access Control (DAC) are examined. DAC are classified in the following nested order: From general to specific, binary neighborhood systems(binary relations), topological spaces (reflexive and transitive relations) and clopen spaces (equivalence relations) in geometric (algebraic) terms. In security terms, the two smaller classes meet information flow security and Chinese wall security policy in respective order. Roughly, information flow security policy (IFSP) means any data can never flow or propagate into the enemy hands of the initial owner. Chinese wall security policy is IFSP, in which enemy is a symmetric relation.

A very general formal model is proposed, it formalize almost all the known examples.

This article is about the data analysis and mining aspects of granular computing (GrC) that includes rough set theory (RST). This subject involves four concepts, deductive data analysis and mining (DDAM), relational databases (RDB), rough set theory (RST) and the granular computing (GrC). The results are quite far reaching.

Current state of granular computing (GrC) is summarized. The historical development of infinitesimal granules is reviewed for future lessons. A category theory based GrC model, that nearly realizes (we believe it will realize) all classical examples of GrC, is re-examined; all early models are derived. Some possible future directions are proposed.

In this paper, foundational issues on ldquodivide/granulate and conquerrdquo are raised and formalized. Namely, could two distinct problems be divided/granulated into the same set of subproblems? Or equivalently, could the same set of subproblems have more than one integrated solutions? To answer these questions, the concept of extension functors in homological algebra is imported. Informally speaking, the ldquomain programrdquo (the high level instructions), called quotient structure, may integrate the ldquoreturns of subroutine calls,rdquo called sub-structure to more than one integrated solutions of the original problem. The extension functor keeps such solution count.

Granular computing is an emerging technology. This paper summarizes its applications to the Web. A high frequent co-occurring ordered set of keywords is called a keyword set;it represents some concept in the given document set. These concepts forms a simplicial complex of concepts,which is regarded as a knowledge base for a new search engine.The return of a query will be organized ino meaningful clusters. A fundamental difference of this approach from semantic web is that the search engine carries the semantic of a Website, instead of the website has to be organized in the sense of semantic Web, namely, the semantics is machine readable.

The papers in this special section focus on granular computing, defined by Lotfi A. Zadeh as "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."

In this paper, Chinese wall security policy (CWSP) was re-examined via social networks and granular computing. A short proof of the strong Chinese wall security theorem was presented.

Granular computing on partitions (RST), coverings (GrCC) and pre-topology (neighborhood systems (NS)) are examined: 1) the order of generality is RST, GrCC, and then NS. 2) The quotient structures: In RST, it is the quotient set. In GrCC, it is a simplicial complex (in combinatorial topology). For NS, the structure has no known description. 3) A new and traditional approximation theories are presented: For both theories, RST is a topological space generated by a partition, called a clopen space. For NS, the classical one is a pre-topological space, a generalized topological spaces. For new one, it is a topological space. For GrCC, there are two possibilities. One is a special case of NS, which is the topological space generated by the covering. There is another topological space, the topology is generated by the finite intersections of the members of a covering The first one treats covering as a base, the second one as a subbase. 4) Knowledge representations in RST are symbol-valued systems. In GrCC, they are expression-valued systems. In NS, they are multi-valued system. 5) For RST, GrCC and Symmetric NS, the representation theories are complete in the sense that granular models can be recaptured fully by knowledge representations.

Granular Computing is a new computing paradigm. The panel will focus on its potentials on E-security and web intelligence among others. One of the main component of granular computing is "Granulate and Conquer," that lies in the heart of this conference. Fruitful discussions and conclusions will be expected.

What is granular computing? As the author have said in (T.Y. Lin, 2006): There are no mathematically valid formal definitions yet. Informally, any computing theory/technology that involves elements and granules (generalized subsets) may be called granular computing (GrC). Intuitively, elements are the data, and granules are the basic knowledge. So granular computing includes data and knowledge computing/engineering, data mining, knowledge discovery, learning and the uncertainty management (granules of no knowledge). We may lump these subfields into AI-engineering. Here granules are interpreted as generalized subsets that include classical subsets, fuzzy subsets, and sets with neighborhood systems (e.g., the alpha cut as neighborhood systems of the cores). As fuzzy sets are defined by membership functions which are bounded real-valued functions, so we will extend further to all functions, even to generalized functions (e.g. Dirac functions), measures/probabilities, generalized measures (e.g. belief functions)

In 1989, Brewer and Nash (BN) presented a fascinating idea, called Chinese wall security policy model, for commercial security. Their idea was based on the analysis of the notion, conflict of interest binary relation (CIR). Unfortunately, their formalization did not fully catch the appropriate properties of CIR. In this paper, we present a theory based on granulation that has captured the essence of BN's intuitive idea. The results are more than the Chinese wall models: malicious Trojan horses in certain DAC model (discretionary access control) can be controlled or confined.

A text/web document is a knowledge representation of a human idea (a structured set of thoughts). This paper refines TFIDF and Extended TFIDF(ETFIDF)[16]; These values really measures the co-occurrences of tokens. The ETFID captures the semantic more accurately. Tokens with high TFIDF values are called Keywords. The sets of (n+1) Co-occurring keywords with High ETFIDF are called n-granules. The collection of keywords and n-granules can be interpreted geometrically; they form a non-closed simplicial complex. The corresponding non-closed polyhedron is called Latent Semantic Space(LSS). LSS is a geometric knowledge base that provides the semantic to search engine

Granulation is a natural problem-solving methodology deeply rooted in human thinking; it is intrinsically fuzzy, vague and imprecise. Mathematicians idealized it to partition, and developed into a fundamental problem solving methodology. Granulation and partition are examined in parallel from the prospect of problem solving. In partition theory, knowledge processing are transformed into table or tree processing. For general granulation such transformation is not there. In this paper, we take a new fresh look at previous results (Lin, 1998a), including the recent applications in computer security (Chinese Wall Security Policy model) from this prospect; the knowledge processing have been transformed to table and tree processing in the (pre-)topological setting

What should be the correct perception of a fuzzy set? We believe a fuzzy set should be able to tolerate "small amounts" of "physical" perturbations. There are three approaches: roughly, type II fuzzifies the grades, Lin fuzzifies the membership functions, and Thiele captures Kripke style semantics. However, it is not immediate these approaches could tolerate "physical" perturbation. In this paper, we propose a homotopy approach that clearly relates the notion of qualitative fuzzy set with "physical" deformation.

Using computing with words as a representation theory of Lin-Zadeh's notion of granular computing, the bitmap indexes of relational tables is formalized and extended. We call it a granular data model (GDM). If all granulations are partitions, a GDM is reduced to a classical relation in relational data model. Based on GDM, semantically rich rules can be mined. The underlying theme of this paper is computing with words; data mining on GDM is one form of computing with words.

The main theorem is: generalized associations of a relational table can be found by a finite set of linear inequalities within polynomial time. It is derived from the following three results, which were established in ICDMO'02 and are re-developed here. They are: (1) isomorphic theorem: isomorphic relations have isomorphic patterns. Such an isomorphism classifies relational tables into isomorphic classes. (2) A variant of the classical bitmaps indexes uniquely exists in each isomorphic class. We take it as the canonical model of the class. (3) All possible attributes/features can be generated by a generalized procedure of the classical AOG (attribute oriented generalization). Then, (4) the main theorem for canonical model is established. By isomorphism theorem, we had the final result (5).

Automated discovery and extraction of biological knowledge from biomedical web documents has become essential because of the enormous amount of biomedical literature published each year. In this paper we present an ontology-based scalable and portable information extraction system to automatically extract biological knowledge from huge collection of online biomedical web documents. Our method integrates ontology-based semantic tagging, information extraction and data mining together, automatically learns the patterns based on a few user seed tuples, and then extract new tuples from the biomedical web documents based on the discovered patterns. A novel system SPIE (Scalable and Portable Information Extraction) is implemented and tested on the PuBMed to find the chromatin protein-protein interaction and the experimental results indicate our approach is very effective in extracting biological knowledge from huge collection of biomedical web documents.

Classical MAC and DAC are reexamined in the light of e-government. In this paper, we focus on the information flows, which have been addressed by classical MAC, in my view, somewhat inadequately, and never been addressed by classical DAC.

Mining association rules from databases is a time-consuming process. Finding the large item set fast is the crucial step in the association rule algorithm. In this paper we present a fast association rule algorithm (Bit-AssoRule) based on granular computing. Our Bit-AssocRule doesn't follow the generation-and-test strategy of Apriori algorithm and adopts the divide-and-conquer strategy, thus avoids the time-consuming table scan to rind and prune the itemsets, all the operations of finding large itemsets from the datasets are the fast bit operations based on its corresponding granular. The experimental result of our Bit-AssocRule algorithm with Apriori, AprioriTid and AprioirHybrid algorithms shows Bit-AssocRule is 2 to 3 orders of magnitudes faster. Our research indicates that bitmap and granular computing can greatly improve the performance of association rule algorithm, and are very promising for data mining applications.

In this paper, we discuss some new bitmap techniques for optimizing decision support queries and association rule algorithm. We first show how to use a new type of predefined bitmap join index (prejoin/spl I.bar/bitmap/spl I.bar/index) to efficiently execute complex decision support queries with multiple outer join operations involved and push the outer join operations from the data flow level to the bitmap level and achieve significant performance gain. Then we discuss a bitmap based association rule algorithm. Our bitmap based association rule algorithm Bit-AssocRule doesn't follow the generation-and-test strategy of a priori algorithm and adopts the divide-and-conquer strategy, thus avoids the time-consuming table scan to find and prune the item sets, all the operations of finding large item sets from the datasets are the fast bit operations. The experimental results show Bit-AssocRule is 2 to 3 orders of magnitude faster than a priori and a priori hybrid algorithms. Our results indicate that bitmap techniques can greatly improve the performance of decision support queries and association rule algorithm, and bitmap techniques are very promising for the decision support query optimization and data mining applications.

The notion of neighborhood systems (NS), which is an abstraction of "near" or "negligible distance," is used to model information granulation. A granule at a point (object) p consists of a clump (crisp/fuzzy set) of data that are related or near the point p. Mathematically, NS, by its definition, includes topological spaces (TS) and rough sets (RS) as special cases. The deeper part of NS is its ability in representing the context dependent notion of near-ness or approximation (via small granule). We apply the idea to formulate the notion of qualitative fuzzy sets.

In 1989, Brewer and Nash presented a Chinese Wall Security Policy model (CWSP); the model exited the whole conference. The intuition is to place Chinese Wall on the boundary of the conflict of interests classes so that information flow among competing companies cannot occur. Unfortunately, later in the same year we pointed out their erroneous assumption that the conflict of interests relation (CIR) is an equivalence relation; and presented an aggressive Chinese Wall Security Policy model (ACWSP) that was based on an analysis of CIR as a binary relation. Based on our relative recent observation that each binary relation induces an equivalence relation, we show that ACWSP can be strengthened by the induced equivalence relation; intuitively the walls are placed on the boundary of induced equivalence classes.

Modeling in data mining has not been fully explored; so data mining is often regarded as a set of added operations in database systems. The consequence is that many notions are overloaded. Some terms often have different semantics. This paper continues previous effort to explore and reiterate some fundamental issues in data mining.

This workshop introduces the foundations of data mining in the context of granular and rough computing. Unlike conventional data mining research, this workshop focuses on deep reflection into the meaning of patterns obtained from data. Since patterns are embedded in sets of real world objects, algebraic or geometric views on sets may play important roles. For this purpose, granular and rough computing are most important methodologies. In other words, granular and rough computing gives one of the best tools to analyze algebraic or geometric characteristics of patterns and data.

Though traditional neural networks and fuzzy logic are powerful universal approximators, however without some refinements, they may not, in general, be good approximators for adaptive systems. By extending fuzzy sets to qualitative fuzzy sets, fuzzy logic may become universal approximators for adaptive systems. Similar considerations can be extended to neural networks.

It is well known that relational theory carries very little semantic. To mine deeper semantics, additional modeling is necessary. In fact, some "pure" association rules are found to exist even in randomly generated data. We consider a relational database in which every attribute value has some additional information, such as price, fuzzy degree, neighborhood, or security compartment and levels. Two types of additions are considered: one is structure added, the other is valued-added. Somewhat a surprise, the additional cost in semantics checking is found to be very well compensated by the pruning of non-semantic rules.

With explosive growth of e-Business on the Internet, and wireless networks, users face more and more challenging network-based application problems in terms of intelligent e-Applications. To increase the QoI (Quality of Intelligence) of e-Business, we propose a new research area called Computational Web Intelligence (CWI) based on both computational intelligence (CI) and Web technology (WT). Generally, the intelligent e-brainware using CWI techniques plays an important role in smart e-Business. Fundamental concepts, basic methods, major applications and future trends of CWI are described to briefly show a general framework of CWI from different aspects.

A "correct" selection of attributes (features) is vital in data mining. As a first step, this paper constructs all possible attributes of a given relation. The results are based on the observations that each relation is isomorphic to a unique abstract relation, called a canonical model. The complete set of attributes of the canonical model is, then, constructed. Any attribute of a relation can be interpreted (via isomorphism) from such a complete set.

Attribute (feature) transformations on databases are examined from a data mining prospect. Theoretical examples from classical mathematics are used to illustrate the effects of the transformations: (1) Certain examples show that attribute transformations are the only means to bring out the patterns to visible states. (2) There are no apparent hints from the raw data as to what transformations could be employed; domain experts need to be in the loop of data mining. (3) All transformations can be expressed, or more precisely approximated, by polynomials. Such expressions provide some means for a brute force search (one degree at a time) of the transformations. A reasonable success in a linear case (degree one) is reported in Ref. 2. © 2002 John Wiley & Sons, Inc.

The effects of attribute transformations have been examined theoretically in part I of this article. This is part II, and its focus is on applications. Specific linear transformations, which have statistical meaning, are applied to a selected set of economic and stock market data. The data are selected from the computer, semiconductor, and semiconductor equipment industries. The main data mining tool is the rough set based software, DataLogic/R+, augmented with programs that perform linear transformations, concept generalization, and so on. Some useful “predictive” rules are discovered. Here, “predictive” is used in the sense that the logical patterns involve time elements. We should note that even in such simple cases, a trail-and-error approach is necessary for finding the right transformation. © 2002 John Wiley & Sons, Inc.

Index Miner is a Java-based data mining tool which contains the methodologies of classification, clustering, association rules, visualization, attribute transformation and feature selection. The paper briefly introduces this tool and discusses the methods that are implemented in this tool to show how to create and perform a learning process for data in order to mine the patterns that are hidden in the data.

As more complicated and complex situations in current markets, especially in the highly competitive electronic commerce environment emerge, proper customer services become more important than before. Therefore, data mining to assist customer relationship management (CRM) becomes a major topic in the Internet world. The article presents the use of rough sets to build up a one-to-one service Web site for electronic commerce service companies to understand customer desires and quickly respond to customer wants and needs. This makes these companies more competitive than their competitors. The article discusses the concepts and methods for building up a Web based one-to-one CRM system.

D.D.C. Brewer and M.J. Nash (IEEE Symposium on Security and Privacy, pp. 206-14, 1988) introduced Chinese Wall security policy models for commercial security. These were later updated by T.Y. Lin (Proc. 5th Aerospace Comput. Security Appl. Conf., pp. 286-93, 1989). Applying Z. Pawlak's (1997) idea of conflict analysis, this paper introduces a practical way of computing an extended model.

Earlier the authors have shown that rough sets can be characterized by six topological properties. In this paper, a new formal logic system based on such axioms is proposed. It will be called First-Order Logic for Rough Approximation or simply Rough

The main congestion problem for LAN/MAN interconnection via ATM lies in the possibility of congestion at the receiving gateway. The authors propose a buffer management scheme that stores packets in a buffer if there is a speed mismatch between the source gateways and the receiving gateway. If congestion occurs in the source gateway, it sends a source quench message to slow down the traffic from the host. They calculate the buffer size of the receiving gateway according to the bandwidth of the virtual paths, the number of virtual paths connected, the mean packet size, and the propagation delay of the turn-off signal. Their method can reduce the processing overhead of the gateways as well as achieve real-time control.>

Proposes a probabilistic type of measure theory which estimates the amount of security relevant information of every subset of a given aggregate. This probabilistic measure theory provides each application a means or mechanism to furnish the system a numerical measure to assist the system security officer for making decisions on releasing or downgrading the internal data of the aggregate. A scenario for solving the aggregation problems during the design phase is proposed. In developing this work, two guiding principles are applied: Minimum Aggregation Principle (MAP) and the Maximum Protection Principle (MPP). MAP keeps both the number of elements in an aggregation and the number of aggregates to a minimum, while MPP keeps the unnecessary risks to a minimum.>

In Brewer and Nash's Chinese Wall security policy model, there is a very strong implicit assumption that the 'conflict of interest' is an equivalence relation. It is called the BN-axiom. Such axiom is valid only for some very special circumstances. By modifying their formulation, a modified Brewer and Nash model without BN-axiom is defined. Such model turns out to be rather 'conservative' in the sense that the Chinese Walls are built within Chinese territory; it is not really a Chinese Wall model. Next, a new formal model is introduced in which Chinese Walls are built right on the boundary of China-an aggressive model for Chinese Wall security policy.>