Categorical approach to the construction of fuzzy graph grammars

Abstract

The categorical approach is proposed to the formalization of fuzzy graph grammars obtained as a result of generalization of sequential graph grammars. This approach takes into consideration the basic types of fuzziness that arise in constructing categories of fuzzy objects and describing transformations of fuzzy graphs generated by fuzzy sets. All the problems of undecidability that are well known for Chomsky grammars are proved to hold true for fuzzy graph grammars.

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.…”

Transformations of fuzzy graphs specified by FD-grammars

“…In this section, we consider concepts that continue the ideas of the original categorical approach to fuzzy graph transformation introduced in [6,7]. In contrast to [6], we define productions over fuzzy graphs by two morphisms that correspond to eliminating and adding elements of a graph, which makes it possible to characterize more precisely the applicability conditions for such productions.…”

“…In contrast to [6], we define productions over fuzzy graphs by two morphisms that correspond to eliminating and adding elements of a graph, which makes it possible to characterize more precisely the applicability conditions for such productions. See [13,14] for the other concepts of the theory of categories, including definitions of pushout and colimit.…”

“…The authors showed in [6] that this interval of truth values for edges and nodes of the fuzzy graph G can be generalized to a complete lattice of objects of category Fuzz( ) Q , where Q is a partially ordered support set. Eliminating truth values of nodes and edges of G, we obtain a nonfuzzy graph called the support graph of the fuzzy graph G.…”

“…To develop such techniques, fuzzy graphs can be used that model complex relationships of objects inside local components of a software system, which occur, for example, in simulating software architecture [5]. Transformations of fuzzy graphs are defined at this level to specify changes in relations among objects [6][7][8][9].…”