Uma tradução de redes de Petri Fuzzy generalizadas para gramáticas de grafos com atributos

Abstract

Petri Nets (PN) and Graph Grammars (GG) are suitable formalism’s to specify concurrent and distributed systems. There are several studies that relate PN and GG, where GG is seen as a generalization of PN since it allows expressing more information. There are different types of PN, however, in this work, the focus is given to Fuzzy Petri Nets (FPN), more specifically to Generalized Fuzzy Petri Nets (GFPN). The objective of this work is to propose a translation from GFPN to Attributed Graph Grammars, that allows to describe the behavior of a fuzzy inference system using graph grammars. An Attributed Graph Grammar (AGG) is an extension of conventional GG that incorporates attributes with abstract data types in graph definitions. The translation consists of transforming the components of the GFPN into components of an AGG. The fuzzy operations of the net are defined by means of an algebraic specification associated with the AGG and the components of the grammar, that is, the type graph, the initial graph and the set of rules, are built from the structure of the net. According to the semantics of GFPN, two modes of transition triggering are considered. In the first mode, tokens are consumed from the preconditions of the triggered transition. In the second, tokens are preserved. These two behaviors generate two types of rules for the grammar obtained from the translation, which simulate the net’s trigger modes. In order to ensure that the AGG obtained from the translation maintains the same behavior as the translated GFPN, it is proven that the translation preserves the semantics of the GFPN, that is, all behavior observed in the nets must be observed in the grammars and vice versa. More specifically, if a state change occurs in the GFPN by triggering a transition and a new state is achieved, then this new state must also be achieved in the AGG if the rule, obtained in the translation from the transition triggered in the net, is applied and vice versa. The translation is expected to provide a new model for the fuzzy approach derived from the translation proposed here. The obtained AGG can be directly used to verify properties of fuzzy systems using tools such as Rodin’s theorem prover. In addition, it is expected that the model obtained by the translation will serve as an inspiration for the definition of a more flexible Fuzzy Graph Grammars, in which one can choose the desired type of fuzzy sets and operations.