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.
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Notes
For clarity, we will omit the external parenthesis whenever it is possible, e.g. we will write \(\varphi \wedge \psi \) instead of \((\varphi \wedge \psi )\).
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 \(S^0[w]\).
See the previous footnote.
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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.
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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
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DOI: https://doi.org/10.1007/s00500-020-05353-1