Minimal Realization of an Input–Output L-fuzzy Multiset Behavior

Abstract

This research aims to introduce and study the minimal realization problem of a new class of automaton with input and output as multisets. We begin by introducing the concrete categories [Formula: see text] and [Formula: see text] of non-deterministic and [Formula: see text]-fuzzy multiset automata over category Set and determine the functorial relationship between them, in which [Formula: see text] is a distributive lattice. Next, we introduce other concrete categories [Formula: see text] and [Formula: see text] of [Formula: see text]-fuzzy (crisp deterministic [Formula: see text]-fuzzy) multiset automata with output over category Set and their input–output [Formula: see text]-fuzzy multiset behavior. Further, we minimize the introduced category [Formula: see text] using functorial maps. Finally, we introduce the concept of realization of an input–output [Formula: see text]-fuzzy multiset behavior and using Myhill–Nerode’s theory, we construct a minimal crisp deterministic [Formula: see text]-fuzzy multiset automata with output in Set which realizes the given input–output [Formula: see text]-fuzzy multiset behavior.

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

“…Multiset theory ( also known as theory of bags ( cf., Blizard [25], Petrson [27], Yager [28]) has frequent and successful applications in several dimensions of Computer Science such as Classical automata (cf., [32][33][34]), Fuzzy automata (cf., [40,41,43,49]), and their respective formal language (cf., [35,36,38,39,43,52,53,68]) and their respective Grammars (cf., [33,37,42,69,70]), Petri nets [71], Formal control [26], Data analysis [72], Formal power series (cf., [73])and Concurrency (cf., [74]), etc. Even the theory of multiset has successful applications in, DNA computing (cf., Pȃun, Rozenberg and Salomaa [75]), membrane computing [76], Mathematics & computing (cf., Syropoulos [57,[77][78][79]), decision making (cf., Paul, and John, [80]), data analysis (cf., Tauler, Maeder and Juan [72], Tauler [81]), chemical programming (cf., Banâtre, Fradet and Radenac [82]), neural network (cf., McGregor [83]), Mathematics and information systems (cf., [84,85]), decipherability of codes (cf., Blanchet-Sadri and ...…”