Construction of a minimal realization and monoid for a fuzzy language: a categorical approach

Tiwari, S. P.; Yadav, Vijay K.; Singh, Anupam

doi:10.1007/s12190-014-0782-5

Cited by 26 publications

(4 citation statements)

References 17 publications

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“…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.…”

Section: Discussionmentioning

confidence: 99%

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

Ruhela,

Syropoulos,

Yadav

et al. 2023

Preprint

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The aim of this work is to broaden the theory of computation by making use of fuzzy sets and multisets, which are generalization of ordinary sets. We study fuzzy multiset finite automata (FMFA), which are characterized by the incorporation of vagueness when it comes to the choice of the next state and multiple occurrences of the same input symbol. The resulting automaton is a genaralization of both classical automata and fuzzy automata. We discuss the algebraic characteristics of FMFA sub-automata—separatedness, strongly connectedness, cyclic, directable, and triangular FMFA in terms of equivalence classes induced by an equivalence relation defined on state set of FMFA. Intrestingly, for a given FMFA, a new FMFA can be constructed that is a homomorphic image of the original FMFA. We define different posets structures that can be associated with a given FMFA and show that some of them are upper semilattices and discuss the inter-relationship of a given poset, a finite upper semilattice (FUSL), a FMFA and a posets/FUSL associated with a given FMFA. Finally, we introduce the concept of decomposition of a FMFA in two different ways and characterize the strongly connected, triangular and directable FMFA.

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Section: Discussionmentioning

confidence: 99%

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

Ruhela,

Syropoulos,

Yadav

et al. 2023

Preprint

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“…The algebraic aspects of fuzzy automata and languages have been studied in [32,51]. The minimal realization problem of fuzzy languages has been studied algebraically in [21], by category-theoretic approach in [50,53,55], and in bicategory theoretic setting in [52,54,63] to brings closer the gap between classical automata theory and natural languages. The fuzzy automata and languages have been shown helpful in many applications like supervisory control [43], learning systems [60], heart problem deduction [8].…”

Section: Introductionmentioning

confidence: 99%

Generalized rough and fuzzy rough automata for semantic computing

Yadav

Tiwari

Kumari

et al. 2022

Int. J. Mach. Learn. & Cyber.

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The classical automata, fuzzy finite automata, and rough finite state automata are some formal models of computing used to perform the task of computation and are considered to be the input device. These computational models are valid only for fixed input alphabets for which they are defined and, therefore, are less user-friendly and have limited applications. The semantic computing techniques provide a way to redefine them to improve their scope and applicability. In this paper, the concept of semantically equivalent concepts and semantically related concepts in information about real-world applications datasets are used to introduce and study two new formal models of computations with semantic computing (SC), namely, a rough finite-state automaton for SC and a fuzzy finite rough automaton for SC as extensions of rough finite-state automaton and fuzzy finite-state automaton, respectively, in two different ways. The traditional rough finite-state automata can not deal with situations when external alphabet or semantically equivalent concepts are given as inputs. The proposed rough finite-state automaton for SC can handle such situations and accept such inputs and is shown to have successful real-world applications. Similarly, a fuzzy finite rough automaton corresponding to a fuzzy automaton is also failed to process input alphabet different from their input alphabet, the proposed fuzzy finite rough automaton for SC corresponding to a given fuzzy finite automaton is capable of processing semantically related input, and external input alphabet information from the dataset obtained by real-world applications and provide better user experience and applicability as compared to classical fuzzy finite rough automaton.

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“…Wechler [34], initiated the study of fuzzy automata and fuzzy languages taking membership values in structured sets. Following Wechler [34] initiative, fuzzy automata and fuzzy languages having structure for membership values of a fuzzy set in arbitrary set [35], bipointed sets [35,36], poset [37,38], locally nite complete lattice [39], lattices [40,41], lattice order monoid [42,43], complete residuated lattices [35,[44][45][46], have been studied in di erent directions from theoretical and application point of view and became an important tool for reducing the gap between formal languages and natural languages.…”

Section: Introductionmentioning

confidence: 99%

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

Yadav

Tiwari

2022

Mathematical Problems in Engineering

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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.

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Construction of a minimal realization and monoid for a fuzzy language: a categorical approach

Cited by 26 publications

References 17 publications

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

Fuzzy Multiset Finite Automata: Some Algebraic and Lattice Theoretic Characterizations

Generalized rough and fuzzy rough automata for semantic computing

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

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