Operators of Approximations and Approximate Power Set Spaces

Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.

Notes on Rough Set Approximations and Associated Measures

We review and compare two definitions of rough set approximations.One is defined by a pair of sets in the universe and the other by a pair of sets in the quotient universe.The latter definition,although less studied,is semantically superior for interpreting rule induction and is closely related to granularity switching in granular computing.Numerical measures about the accuracy and quality of approximations are examined.Several semantics difficulties are commented.

L_p(1≤p≤∞) APPROXIMATION BY AN n-DIMENSIONAL SUBSPACE ON A SET OF n+1 POINTS

Some new characterizations and immediate explicit expressions of best L(1≤p≤∞) approximation and their deviations by an n-dimensional subspace on a set of n+1 points are given.

Approximation operators based on vague relations and roughness measures of vague sets

Rough set theory and vague set theory are powerful tools for managing uncertain, incomplete and imprecise information. This paper extends the rough vague set model based on equivalence relations and the rough fuzzy set model based on fuzzy relations to vague sets. We mainly focus on the lower and upper approxima- tion operators of vague sets based on vague relations, and investigate the basic properties of approximation opera- tors on vague sets. Specially, we give some essential characterizations of the lower and upper approximation operators generated by reflexive, symmetric, and transi- tive vague relations. Finally, we structure a parameterized roughness measure of vague sets and similarity measure methods between two rough vague sets, and obtain some properties of the roughness measure and similarity measures. We also give some valuable counterexamples and point out some false properties of the roughness measure in the paper of Wang et al.

Uncertainty Representation of Ocean Fronts Based on Fuzzy-Rough Set Theory

Analysis of ocean fronts＇ uncertainties indicates that they result from indiscemibility of their spatial position and fuzziness of their intensity. In view of this, a flow hierarchy for uncertainty representation of ocean fronts is proposed on the basis of fuzzy-rough set theory. Firstly, raster scanning and blurring are carried out on an ocean front, and the upper and lower approximate sets, the indiscernible relation in fuzzy-rough theories and related operators in fuzzy set theories are adopted to represent its uncertainties, then they are classified into three sets： with members one hundred percent belonging to the ocean front, belonging to the ocean front＇s edge and definitely not belonging to the ocean front. Finally, the approximate precision and roughness degree are utilized to evaluate the ocean front＇s degree of uncertainties and the precision of the representation. It has been proven that the method is not only capable of representing ocean fronts＇ uncertainties, but also provides a new theory and method for uncertainty representation of other oceanic phenomena.

Attribute reduction based on fuzziness of approximation set in multi-granulation spaces

Rough set theory is an important tool to solve uncertain problems. Attribute reduction, as one of the core issues of rough set theory, has been proven to be an effective method for knowledge acquisition. Most of heuristic attribute reduction algorithms usually keep the positive region of a target set unchanged and ignore boundary region information. So, how to acquire knowledge from the boundary region of a target set in a multi-granulation space is an interesting issue. In this paper, a new concept, fuzziness of an approximation set of rough set is put forward firstly. Then the change rules of fuzziness in changing granularity spaces are analyzed. Finally, a new algorithm for attribute reduction based on the fuzziness of 0.5-approximation set is presented. Several experimental results show that the attribute reduction by the proposed method has relative better classification characteristics compared with various classification algorithms.

Approximate Self-Similarity of a Class of Cookie-Cutter-Like Sets

According to the bounded variation theory and the bounded distortion property of cookie-cutter-like （CCL） sets, the approximate self-similarity of cookie-cutter-like sets satisfying certain conditions is studied. Based on the mean value theorem, it is proved that a class of special cookie-cutter-like sets is approximately self-similar. The results obtained in this paper extend the corresponding results that have already existed. K

Some Structural Properties of SAT

The following four conjectures about structural properties of SAT are studied in this paper. (1) SAT ∈ PSPARSEnNP; (2) SAT ∈ SRTDtt; (3) SAT ∈ PttbAPP; (4) FPttSAT = FPlogSAT. It is proved that some pairs of these conjectures imply P = NP, for example, if SAT E pSPARsEnNP and SAT 6 PttbAPP, or if SAT E SRTDtt and SAT E PttbAPP, then P = NP. This improves previous results in literature.