Skip to main content
SpringerLink
Log in

Characterization of lattice-valued multiset finite automata

  • Original Paper
  • Published:
Granular Computing Aims and scope Submit manuscript

Abstract

This work aims to characterize a new class of automaton with input as multisets. First, we introduce two finite monoids through different congruence relations on multiset associated with lattice-valued multiset finite automata and show that they are isomorphic to each other. Next, we present the quotient structure of lattice-valued multiset finite automata by defining an admissible relation on the set of states of a given lattice-valued multiset finite automata. Then we show that there is an isomorphism between lattice-valued multiset finite automata and the quotient structure of another lattice-valued multiset finite automata. Finally, we introduce the concept of reachability, observability (coreachability), and response maps of lattice-valued multiset finite recognizer. Interestingly, we show that the lattice-valued response map of a lattice-valued multiset finite recognizer leads us to provide a characterization of a lattice-valued multiset regular language.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. For more detail on category theory we refer the work of Adámek et al. (1990).

References

  • Abolpour K, Zahedi M (2012) BL-general fuzzy automata and accept behavior. J Appl Math Comput 38(1–2):103–118

    Article  MathSciNet  MATH  Google Scholar 

  • Adámek J, Herrlich H, Strecker G (1990) Abstract and concrete categories—the joy of cats

  • Blanchet-Sadri F, Morgan C (2001) Multiset and set decipherable codes. Comput Math Appl 41(10–11):1257–1262

    Article  MathSciNet  MATH  Google Scholar 

  • Blizard WD et al (1988) Multiset theory. Notre Dame J Formal Logic 30(1):36–66

    Article  MathSciNet  MATH  Google Scholar 

  • Cavaliere M, Freund R, Oswald M et al (2007) Multiset random context grammars, checkers, and transducers. Theor Comput Sci 372(2–3):136–151

    Article  MathSciNet  MATH  Google Scholar 

  • Cerf V, Fernandez E, Gostelow K et al (1971) Formal control and low properties of a model of computation, report eng 7178. Computer Science Department, University of California, Los Angeles, CA, December p 81

  • Cheng Y, Zhao F, Zhang Q et al (2021) A survey on granular computing and its uncertainty measure from the perspective of rough set theory. Granul Comput 6(1):3–17

    Article  Google Scholar 

  • Ciobanu G, Gontineac M (2006) Mealy multiset automata. Int J Found Comput Sci 17(01):111–126

    Article  MathSciNet  MATH  Google Scholar 

  • Ciobanu G, Gontineac VM (2005) Algebraic and coalgebraic aspects of membrane computing. In: International workshop on membrane computing. Springer, pp 181–198

  • Csuhaj-Varjú E, Martin-Vide C, Mitrana V (2000) Multiset automata. In: Workshop on membrane computing. Springer, pp 69–83

  • Dubey M, Tiwari S (2019) The relationship among fuzzy languages, upper sets and fuzzy ordered monoids. New Math Nat Comput 15(02):361–372

    Article  Google Scholar 

  • Dubey M, Tiwari S, Šostak A (2020) Categories of quantale-valued fuzzy automata: determinization and minimization. J Appl Math Comput 63(1):771–785

    Article  MathSciNet  MATH  Google Scholar 

  • Dubey M, Tiwari S, Kavikumar J (2021) On the categories of \({L}\)-valued and \(Q\)-valued deterministic fuzzy automata. New Math Nat Comput 1–22

  • Gautam V (2021) \(l\)-valued multiset automata and \(l\)-valued multiset languages. New Math Nat Comput 17(01):175–189

    Google Scholar 

  • Ignjatović J, Ćirić M, Bogdanović S (2008) Determinization of fuzzy automata with membership values in complete residuated lattices. Inf Sci 178(1):164–180

    Article  MathSciNet  MATH  Google Scholar 

  • Ignjatović J, Ćirić M, Bogdanović S et al (2010) Myhill–Nerode type theory for fuzzy languages and automata. Fuzzy Sets Syst 161(9):1288–1324

    Article  MathSciNet  MATH  Google Scholar 

  • Jin J, Li Q, Li Y (2013) Algebraic properties of L-fuzzy finite automata. Inf Sci 234:182–202

    Article  MathSciNet  MATH  Google Scholar 

  • Jun YB (2007) Quotient structures of intuitionistic fuzzy finite state machines. Inf Sci 177(22):4977–4986

    Article  MathSciNet  MATH  Google Scholar 

  • Kudlek M, Totzke P, Zetzsche G (2009) Multiset pushdown automata. Fund Inform 93(1–3):221–233

    MathSciNet  MATH  Google Scholar 

  • Kudlek M, Martín-Vide C, PĂun G (2000) Toward a formal macroset theory. In: Workshop on membrane computing. Springer, pp 123–133

  • Li Y (2008) Approximation and robustness of fuzzy finite automata. Int J Approx Reason 47(2):247–257

    Article  MathSciNet  MATH  Google Scholar 

  • Li Y, Pedrycz W (2005) Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids. Fuzzy Sets Syst 156(1):68–92

    Article  MathSciNet  MATH  Google Scholar 

  • Li Z, Li P, Li Y (2006) The relationships among several types of fuzzy automata. Inf Sci 176(15):2208–2226

    Article  MathSciNet  MATH  Google Scholar 

  • Mao H, Hu M, Yao Y (2021) Algebraic approaches to granular computing. Granul Comput 6(1):119–131

    Article  Google Scholar 

  • McGregor S (2008) Further results in multiset processing with neural networks. Neural Netw 21(6):830–837

    Article  Google Scholar 

  • Močkoř J (2019) Monads and a common framework for fuzzy type automata. Int J Gen Syst 48(4):406–442

    Article  MathSciNet  Google Scholar 

  • Mordeson JN, Malik DS (2002) Fuzzy automata and languages: theory and applications. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  • Murthy VK, Krishnamurthy E (1995) Probabilistic parallel programming based on multiset transformation. Futur Gener Comput Syst 11(3):283–293

    Article  Google Scholar 

  • Pal P, Tiwari S (2020) Category of L-valued multiset automata and Brzozowski’s algorithm. New Math Nat Comput 16(03):481–496

    Article  Google Scholar 

  • Petković T (2006) Congruences and homomorphisms of fuzzy automata. Fuzzy Sets Syst 157(3):444–458

    Article  MathSciNet  MATH  Google Scholar 

  • Qiu D (2004) Characterizations of fuzzy finite automata. Fuzzy Sets Syst 141(3):391–414

    Article  MathSciNet  MATH  Google Scholar 

  • Santos ES (1968) Maximin automata. Inf Control 13(4):363–377

    Article  MathSciNet  MATH  Google Scholar 

  • Sharma B, Syropoulos A, Tiwari S (2016) On fuzzy multiset regular grammars. Ann Fuzzy Math Inform 12(5):617–639

    MathSciNet  MATH  Google Scholar 

  • Sharma B, Tiwari S, Sharan S (2016) On algebraic study of fuzzy multiset finite automata. Fuzzy Inf Eng 8(3):315–327

    Article  MathSciNet  Google Scholar 

  • Sharma B, Gautam V, Tiwari S et al (2017) On fuzzy multiset regular languages. J Fuzzy Set Valued Anal 1:50–61

    Article  Google Scholar 

  • Singh S, Tiwari S (2021) On the category of L-fuzzy automata, coalgebras and dialgebras. Fuzzy Sets Syst 420:1–28

    Article  MathSciNet  Google Scholar 

  • Tauler R, Maeder M, De Juan A (2020) Multiset data analysis: extended multivariate curve resolution

  • Tiwari S, Gautam V, Dubey M (2016) On fuzzy multiset automata. J Appl Math Comput 51(1–2):643–657

    Article  MathSciNet  MATH  Google Scholar 

  • Tiwari S, Yadav VK, Dubey M (2016) Minimal realization for fuzzy behaviour: a bicategory-theoretic approach. J Intell Fuzzy Syst 30(2):1057–1065

    Article  MATH  Google Scholar 

  • Wang Y, Li Y (2018) Minimization of lattice multiset finite automata. J Intell Fuzzy Syst 35(1):627–637

    Article  Google Scholar 

  • Wang J, Yin M, Gu W (2013) Fuzzy multiset finite automata and their languages. Soft Comput 17(3):381–390

    Article  MATH  Google Scholar 

  • Wee WG, Fu KS (1969) A formulation of fuzzy automata and its application as a model of learning systems. IEEE Trans Syst Sci Cybern 5(3):215–223

    Article  MATH  Google Scholar 

  • Wu QE, Wang T, Huang YX et al (2009) State fusion of fuzzy automata with application on target tracking. Comput Math Appl 57(6):949–960

    Article  MATH  Google Scholar 

  • Yager RR (1986) On the theory of bags. Int J Gen Syst 13(1):23–37

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Editor-in-Chief and the anonymous Referee(s) for their valuable comments and suggestions for improving the paper. The work of second author is supported through the ERDF/ESF project AI-Met4AI No. CZ.02.1.01/0.0/0.0/17\(\_{0}\) 49/0008414.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, 303007, Rajasthan, India

    M. K. Dubey & Mallika Dhingra

  2. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, 632014, Tamil Nadu, India

    Anand P. Singh

Authors
  1. M. K. Dubey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anand P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mallika Dhingra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. K. Dubey.

Ethics declarations

Conflict of interest

Authors declare that there is no conflict of interest.

Ethical approval

This article does not contain any studies with human or animals participants.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dubey, M.K., Singh, A.P. & Dhingra, M. Characterization of lattice-valued multiset finite automata. Granul. Comput. 7, 821–836 (2022). https://doi.org/10.1007/s41066-021-00298-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41066-021-00298-8

Keywords

Search

Navigation