A General Categorical Framework of Minimal Realization Theory for a Fuzzy Multiset Language

Abstract

This paper is to study the minimal realization theory for a fuzzy multiset language in the framework of category theory, which has already provided the tools and techniques for the advancement of several features of theoretical computer science. Specifically, by using the well-known categorical concepts, it is shown herein that there is a minimal realization (called the Nerode realization) for each fuzzy multiset language, and all minimal realizations for a given fuzzy multiset language are isomorphic to it.

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

“…The FMRG, FMFA and FMLs have been studied by Wang, Yin and Gu [37] and by Sharma, Syropoulos and Tiwari [42], Sharma, Gautam, Tiwari and Bhattacherjee [97]. The category of lattice-valued FMFA and minimal realization of L-valued multiset languages by Brozozowski's algorithm were discussed by Pal and Tiwari [44], while the minimal realization of FMLs in the general categorical framework was recently studied by Yadav and Tiwari [47]. The most recent articles dealing with FMFA and showing the importance of multiset theory in theoretical computer science are due to Dhingra et.…”

“…The topological concepts already discussed in the case of (classical/fuzzy) automata (cf., [98][99][100]), the concepts of products and generalized products are well studied in the case of classical/fuzzy automata are remained to be explored in case of fuzzy multiset finite automata, we have been worked on these problems and ready to submit the related manuscripts. Other directions of future scope of study done in this paper are to study minimal realization of fuzzy multiset finite automata, where membership structure of fuzzy sets may be algebraic structures different from [0, 1] and distributive lattices keeping in the mind the fact that the nature of input sets (crisp set [19,20], fuzzy sets [101], multisets [47,51]) and structure of membership values ([0,1][20], poset, distributive lattice [102], residuated lattice [103,104], LSET [47]) of fuzzy automata play a very important role in characterization of various concepts in different versions of fuzzy automata, i.e., the properties of fuzzy automata which hold with one membership structure of fuzzy sets may not hold with other membership structures of fuzzy sets, e.g., categorical characterizations of concepts associated with fuzzy multiset finite automata studied in sections 5 and onwards of [47] do not simply holds if we change membership structure of fuzzy sets from LSET to any one of the structures [0, 1], arbitrary sets, posets, distributive lattice or complete residuated lattices because of role of functor U defined in proposition 10 of [47]. The relationship of categorical concepts with automata theory (cf., [62,[105][106][107][108][109][110]) and partial order sets [105]) are well known, such study may be carried out in case of FMFA and posets/lattice structures associated with FMFA introduced in this paper.…”

“…Furthermore, Wang and Li [43] studied the lattice-valued FMFA minimization problem and Pal and Tiwari [44] considered Brazozowaski's algorithm and the categorical approaches for the same, while Singh, Dubey and Perfilieva [45] discussed quotient structures of FMFA, and Gautam [46] studied l-valued FMFA and l-valued FMLs. Recently Yadav and Tiwari [47] provide a general categorical framework of minimal realization of fuzzy multiset languages. The most recent contributions dealing with theoretical aspects of FMFA theory are due to Shamsizadeh and Zahedi [48], Dhingra, Dubey and Jacob [49], Kaur et.…”