Granular computing based on fuzzy and tolerance relations

Granular computing (GC) as a new field has grown rapidly since the term was proposed. There are two basic concepts underlying GC, i.e., fuzziness/fuzzification and granularity/granulation. Fuzzy set theory is an effective tool to deal with the fuzzification problem. There has been a large number of works on this aspect such as computing with words. The difficulty is in the concept of granulation. There was only an informal definition proposed by Zadeh for the concept. So far only a few works addressed the problem such as rough set and quotient space theories. In these works, the granulation is mainly defined by equivalence relations, i.e., a partition model. For example, in quotient space theory, a problem is represented by a triplet (X,F,T), where X -the universe with the finest grain-size, F -the attribute of X, and T- the structure of X. When we view the same problem at a coarser grain size, we have a coarse-grained universe denoted by [X]. Then we have a new representation ([X],[F],[T]) of the problem. The coarse universe [X] is defined by an equivalence relation R on X. Then, representation ([X],[F],[T]) is called a quotient space of(X,F,T), where [X] -the quotient set of X, [F] -the quotient attribute of F, and [T] -the quotient structure of T. Obviously, the set of representations of a problem at different granularities composes a complete semi-order lattice. But in many real applications, we must deal with non-partition models such as tolerance relations (or similarity relations, neighboring relations). In the talk, we will discuss the granular computing based on fuzzy and tolerance relations from the quotient space theory point of view. We focus on the connections among different grain-size worlds, especially, the two basic properties between quotient spaces, i.e., falsity and truth preserving properties. We show that these two basic properties still hold in fuzzy and tolerance worlds. Using these properties, the computational complexity can be reduced either in problem solving or machine learning. We will extend the quotient space theory based multi-granular computing from crispy world to fuzzy and tolerance worlds. The optimal path finding in complex networks is given as an example to show the application of the theoretical results.