Introducing fuzzy reactive graphs: a simple application on biology | Soft Computing Skip to main content
Springer Nature Link
Log in

Introducing fuzzy reactive graphs: a simple application on biology

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, we propose a generalization for fuzzy graphs in order to model reactive systems with fuzziness. As we will show, the resulting fuzzy structure, called fuzzy reactive graphs (FRG), is able to model dynamical aspects of some entities which generally appear in: biology, computer science and some other fields. The dynamical aspect is captured by a transition function which updates the values of the graph after an edge has been crossed. The update process takes into account aggregation functions. The paper proposes a notion for bisimulation for such graphs and briefly shows how modal logic can be used to verify properties of systems modeled with FSGs. The paper closes with a toy example in the field of Biology.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. For clarity, we will omit the external parenthesis whenever it is possible, e.g. we will write \(\varphi \wedge \psi \) instead of \((\varphi \wedge \psi )\).

  2. First, observe that \(w' \in S^0[w]\) is equivalent to (\(w,w'\)) \(\in S^0\). The notation \(\underset{w'\in S^0[w]}{\mathbf {T}} \left( I\left( \mu (w,w'),\llbracket M^{(w,w', \mu (w,w')}, w' \vDash \frac{A}{F}\phi )\rrbracket \right) \right) \), in short \(\underset{w'\in S^0[w]}{\mathbf {T}}(f(w'))\) means the iterative application of a T-norm — which is a binary operation — on \(f(w')= I\left( \mu (w,w'),\llbracket M^{(w,w', \mu (w,w')}, w' \vDash \frac{A}{F}\varphi )\rrbracket \bigg ), \mathrm{for}\,w' \in S^0[w] \right. \). That is, for \(S^0[w]=\emptyset \), \(\underset{w'\in S^0[w]}{\mathbf {T}}(f(w'))=1;\) for \(S^0[w]=\{v\}\), \(\underset{w'\in S^0[w]}{\mathbf {T}}(f(w')) = f(v)\); for \(S^0[w]= \{v_1, v_2\}\), \(\underset{w'\in S^0[w]}{\mathbf {T}}(f(w'))= T(f(v_1), f(v_2))\); for \(S^0[w]= \{v_1, v_2, v_3\}\), \(\underset{w'\in S^0[w]}{\mathbf {T}}(f(w'))= T(f(v_1), T(f(v_2), f(v_3)))\) and so on. Note that there is no ambiguity in this notation, since a T-norm is commutative and associative we do not need to consider an order on \(S^0[w]\).

  3. See the previous footnote.

References

  • Andrade A, Trindade R, Maia D, Santiago RHN, Guerreiro A (2014) Analysing some r-implications and its application in fuzzy mathematical morphology. J Intell Fuzzy Syst 27(1):201–209

    Article  MathSciNet  Google Scholar 

  • Areces C, Fervari R, Hoffmann G (2014) Swap logic. Log J IGPL 22(2):309–332

    Article  MathSciNet  Google Scholar 

  • Areces C, Fervari R, Hoffmann G (2015) Relation-changing modal operators. Log J IGPL 23(4):601–627

    Article  MathSciNet  Google Scholar 

  • Baczyński M, Jayaram B (2008) Fuzzy implications. Studies in fuzziness and soft computing. Springer, Berlin

    MATH  Google Scholar 

  • Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners, volume 221 of Studies in Fuzziness and Soft Computing. Springer

  • Callejas Claudio, Marcos João, Callejas Bedregal Benjamín René (2012) On some subclasses of the fodor-roubens fuzzy bi-implication. In: Luke O, Ruy de Queiroz, eds Logic, Language, Information and Computation, pp 206–215, Berlin, Heidelberg. Springer Berlin Heidelberg

  • Cao Y, Sun SX, Wang H, Chen G (2012) A behavioral distance for fuzzy-transition systems. IEEE Trans Fuzzy Syst 21(4):735–747

    Article  Google Scholar 

  • Chaves M, Preto M (2013) Hierarchy of models: from qualitative to quantitative analysis of circadian rhythms in cyanobacteria. Chaos Interdiscip J Nonlinear Sci 23(2):025113

    Article  MathSciNet  Google Scholar 

  • Chou K-C (1989) Low-frequency resonance and cooperativity of hemoglobin. Trends Biochem Sci 14(6):212

    Article  Google Scholar 

  • Claudio C, João M, Benjamín B (2013) Actions of automorphisms on some classes of fuzzy bi-implications. Mathware Softcomput Mag 1(20):94–114

    Google Scholar 

  • Costa Valdigleis S, Farias Antonio DS, Bedregal B, Santiago RHN, de Canuto P, Anne M (2018) Combining multiple algorithms in classifier ensembles using generalized mixture functions. Neurocomputing 313:402–414

    Article  Google Scholar 

  • Cruz A, Bedregal BC, Santiago RHN (2018) On the characterizations of fuzzy implications satisfying i(x, i(y, z))=i(i(x, y), i(x, z)). Int J Approx Reason 93:261–276

    Article  MathSciNet  Google Scholar 

  • De Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comput Biol 9(1):67–103

    Article  MathSciNet  Google Scholar 

  • Farias ADS, Santiago RHN, Bedregal B (2016) Some properties of generalized mixture functions. pages 288–293. cited By 5

  • Figueiredo D, Barbosa LS (2019) Reactive models for biological regulatory networks. In: Chaves M, Martins MA (eds) Molecular logic and computational synthetic biology. Springer, Cham, pp 74-88

  • Figueiredo D, Rocha E, Chaves M, Martins MA (2019) rPrism—a software for reactive weighted state transition models. In: 6th International Workshop on Hybrid Systems and Biology, pp 165–174. Springer

  • Fodor JC, Roubens MR (1994) Fuzzy preference modelling and multicriteria decision support. Theory and Decision Library D. Springer, Netherlands

    Book  Google Scholar 

  • Gabbay DM, Marcelino S (2009) Modal logics of reactive frames. Stud Log 93(2–3):405

    Article  MathSciNet  Google Scholar 

  • Gabbay D, Marcelino S (2012) Global view on reactivity: switch graphs and their logics. Ann Math Artif Intell 66(1–4):131–162

    Article  MathSciNet  Google Scholar 

  • Jain M, Madeira A, Martins MA (2020) A fuzzy modal logic for fuzzy transition systems. Electron Notes Theor Comput Sci 348:85–103

    Article  Google Scholar 

  • Klement EP, Mesiar R, Pap E (2013) Triangular norms. Trends in logic. Springer, Netherlands

    MATH  Google Scholar 

  • Lee KH (2004) First course on fuzzy theory and applications. Springer, Berlin

  • Lee KH (2006) First course on fuzzy theory and applications. Advances in intelligent and soft computing. Springer, Berlin

    Google Scholar 

  • Megginson HJ, Laing P, Mednick SA (1996) Rhesus incompatibility as a risk factor for schizophrenia in male adults. Arch Gen Psych 53(1):19–24

    Article  Google Scholar 

  • Pinheiro J, Bedregal B, Santiago RHN, Santos H (2018) A study of (t, n)-implications and its use to construct a new class of fuzzy subsethood measure. Int J Approx Reason 97:1–16

    Article  MathSciNet  Google Scholar 

  • Pinheiro J, Bedregal B, Santiago RHN, Santos H (2017) (t, n)-implications. In: 2017 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp 1–6

  • Reiser RHS, Bedregal B, Santiago RHN, Amaral MD (2013) Canonical representation of the yager’s classes of fuzzy implications. Comput Appl Math 32(3):401–412

    Article  MathSciNet  Google Scholar 

  • van Benthem J (2005) An essay on sabotage and obstruction. In: Mechanizing mathematical reasoning, pp 268–276. Springer

Download references

Acknowledgements

This study was funded by National Council for Scientific and Technological Development (CNPq) within the Project 312053/2018-5, by Coordination for the Improvement of Higher Education Personnel (CAPES) within the project Capes-Print 88887.363001/2019-00, ERDF - The European Regional Development Fund through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project POCI-01-0145-FEDER-030947 and project with reference UIDB/04106/2020 and UIDP/04106/2020 at CIDMA.

Author information

Authors and Affiliations

  1. Departamento de Informática e Matemática Aplicada — DIMAp, Universidade Federal do Rio Grande do Norte — UFRN, Natal, Brazil

    Regivan Santiago

  2. CIDMA and Department of Mathematics, Universidade de Aveiro, Aveiro, Portugal

    Manuel A. Martins & Daniel Figueiredo

Authors
  1. Regivan Santiago
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Manuel A. Martins
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Daniel Figueiredo
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Regivan Santiago.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Communicated by Tomas Veloz.

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

Santiago, R., Martins, M.A. & Figueiredo, D. Introducing fuzzy reactive graphs: a simple application on biology. Soft Comput 25, 6759–6774 (2021). https://doi.org/10.1007/s00500-020-05353-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-020-05353-1

Keywords