Transformations of fuzzy graphs specified by FD-grammars

Abstract

The categorical approach is proposed to formalize transformations of FD-graphs that consist of networks of distributed components whose nodes are specified by fuzzy graphs. Necessary and sufficient conditions are formally defined for FD-graph transformations that do not violate structure integrity and can be constructed componentwise. FD-grammars that generalize fuzzy graph grammars are proposed to describe the admissible transformations of FD-graphs.

Transformational approach to the development of software architectures on the basis of fuzzy graph models

“…To formalize fuzzy graph transformation systems (FGTSs) [13,14] that generalize sequential graph transformation systems (graph grammars) and take into account basic kinds of fuzziness that arise in constructing basic categories of fuzzy objects and in describing transformations of fuzzy graphs generated by fuzzy sets, category-theoretic approach is used.…”

“…Therefore, to formalize the fuzziness of basic concepts used in fuzzy graphs [8,13,14], the category approach is applied as a mathematical subdiscipline studying the most general properties of relations that exist between mathematical objects and that are independent of internal object structures.…”

Category-theoretic formalization in designing fuzzy intelligent systems

“…The category-theoretic approach can be used to formalize FGTSs, which are a generalization of sequential GTS and take into account the main forms of fuzziness that occur both in constructing basis categories of fuzzy objects and in describing the transformations of fuzzy graphs generated by fuzzy sets [7][8][9]. To this end, we will use the "pushout" structure, which is a category generalization of the union of two objects with allowance for the common subobject.…”

“…Based on the category approach, transformations of FD-graphs [9] are formalized as structured component transformations of fuzzy graphs. The necessary and sufficient conditions for implementing transformations of FD-graphs are formally defined; they do not violate the structure integrity and allows performing a constructive (componentwise) construction of a pushout in the FDGraph category.…”

“…In turn, distributed FGTSs are a generalization of FGTSs, which allows describing feasible transformations of FD-graphs [9]. Changing the form of a graph during a transformation step makes it possible to model dynamic network structures of components that specify the architecture of the fuzzy system.…”