Abstract
The coalgebraic method is of great significance to research in process algebra, modal logic, object-oriented design and component-based software engineering. In recent years, fuzzy control has been widely used in many fields, such as handwriting recognition and the control of robots or air conditioners. It is then an interesting topic to analyze the behavior of fuzzy automata from a coalgebraic point of view. This paper models different types of fuzzy automata as coalgebras with a monad structure capturing fuzzy behavior. Based on the coalgebraic models, we can define a notion of fuzzy language and consider several versions of bisimulation for fuzzy automata. A group of combinators is defined to compose fuzzy automata of two branches: state transition and output function. A case study illustrates the coalgebraic models proposed and their composition.
Highlights
Considering fuzzy output maps, we focus on three types of fuzzy automata: Fuzzy Moore Automata (FMrA), Fuzzy Mealy Automata (FMlA) and Fuzzy Unified Automata (FUA)
FI,O -coalgebras provide a universal framework for defining fuzzy language and bisimulation for different fuzzy automata while TI,O -coalgebras serve as a basis for composition calculi of fuzzy Mealy automata
We modeled different types of fuzzy automata as coalgebraic models with the same transition structure
Summary
A recent thesis [33] proposes a coalgebraic approach to fuzzy automata, which obtains the following results: (a) a coalgebraic definition of the fuzzy language recognized by a fuzzy automaton, (b) the definition of a functor describing the determinization process of a fuzzy automata via a generalization of the powerset construction, (c) a coalgebraic definition of bisimulation on fuzzy automata allowing the construction of a quotient fuzzy automaton It only considers the output as the current membership value for the current state. Explore the fuzzy-set monad to serve as the basis to a coalgebraic approach; Provide a coalgebraic framework for different types of fuzzy automata, where the notions of fuzzy language and bisimulation can be addressed; Define appropriate combinators for composing fuzzy automata from two branches:state transition and output function.
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