Approximate bisimulations and state reduction of fuzzy automata under fuzzy similarity measures

Abstract For the purpose of reducing the number of states of fuzzy automata, we study approximate bisimulations of fuzzy automata under fuzzy similarity measures, which are induced by residuums of left-continuous t-norms. For a real number α ∈ [ 0 , 1 ] and a fuzzy similarity measure S, we define α-approximate bisimulation under S between two fuzzy automata and prove that the degree of similarity between them is more than or equal to α if there exists an α-approximate bisimulation between them. Furthermore, we put forward the notion of surjective functional α-approximate bisimulations between two fuzzy automata under S. Using the notion of surjective functional α-approximate bisimulations under S, the α-approximate bisimulations for a fuzzy automaton under S is defined as an equivalence relation with two additional properties. According to an α-approximate bisimulation for a fuzzy automaton, an aggregated fuzzy automaton has been constructed and the degree of similarity between them is also more than or equal to α. It is significant for us to find that there might not possess the greatest α-approximate bisimulation for a fuzzy automaton, and we provide a polynomial-time algorithm for computing all maximal α-approximate bisimulations. Finally, we provide the conditions of the existence of the greatest α-approximate bisimulation.

[1]  Yuxin Deng,et al.  Logical characterizations of simulation and bisimulation for fuzzy transition systems , 2016, Fuzzy Sets Syst..

[2]  Dan Meng,et al.  On the TL-transitivity of fuzzy similarity measures , 2017, Fuzzy Sets Syst..

[3]  Qian Wang,et al.  The universal fuzzy automaton , 2014, Fuzzy Sets Syst..

[4]  Yongming Li,et al.  Minimization of states in automata theory based on finite lattice-ordered monoids , 2007, Inf. Sci..

[5]  Mohammad Mehdi Zahedi,et al.  Similarity-based minimization of fuzzy tree automata , 2016 .

[6]  Zhihong Zou,et al.  Water quality evaluation of Haihe River with fuzzy similarity measure methods. , 2013, Journal of environmental sciences.

[7]  Ivana Jancic,et al.  Weak bisimulations for fuzzy automata , 2014, Fuzzy Sets Syst..

[8]  Witold Pedrycz,et al.  Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids , 2005, Fuzzy Sets Syst..

[9]  Andreas Maletti,et al.  Backward and forward bisimulation minimization of tree automata , 2009, Theor. Comput. Sci..

[10]  Peter Buchholz,et al.  Bisimulation relations for weighted automata , 2008, Theor. Comput. Sci..

[11]  Wenyi Zeng,et al.  Fuzzy risk analysis based on the similarity measure of generalized trapezoidal fuzzy numbers , 2017, J. Intell. Fuzzy Syst..

[12]  Nelma Moreira,et al.  Optimal state reductions of automata with partially specified behaviors , 2017, Theor. Comput. Sci..

[14]  Gianfranco Ciardo,et al.  An Efficient Fully Symbolic Bisimulation Algorithm for Non-Deterministic Systems , 2013, Int. J. Found. Comput. Sci..

[15]  Zorana Jancic,et al.  Computation of the greatest right and left invariant fuzzy quasi-orders and fuzzy equivalences , 2017, Fuzzy Sets Syst..

[16]  Miroslav Ciric,et al.  Bisimulations for fuzzy automata , 2011, Fuzzy Sets Syst..

[17]  Etienne E. Kerre,et al.  Bisimulations for Fuzzy-Transition Systems , 2010, IEEE Transactions on Fuzzy Systems.

[18]  Chao Yang,et al.  Approximate bisimulation relations for fuzzy automata , 2018, Soft Comput..

[19]  Keyun Qin,et al.  Some new approaches to constructing similarity measures , 2014, Fuzzy Sets Syst..

[20]  Zhuming Bi,et al.  A Rough Programming Model Based on the Greatest Compatible Classes and Synthesis Effect , 2013 .

[21]  Inés Couso,et al.  Additive similarity and dissimilarity measures , 2017, Fuzzy Sets Syst..

[22]  M. Ćiri,et al.  Computation of the greatest simulations and bisimulations between fuzzy automata , 2012 .

[23]  Bao Qing Hu,et al.  Aggregation distance measure and its induced similarity measure between intuitionistic fuzzy sets , 2015, Pattern Recognit. Lett..

[24]  Yin Jian-hong.  The generating algorithm of the greatest compatible classes in a compatible relation , 2004 .

[25]  Mohammed Falah Mohammed,et al.  SAIRF: A similarity approach for attack intention recognition using fuzzy min-max neural network , 2017, J. Comput. Sci..

[26]  Chao Yang,et al.  $\epsilon$-Bisimulation Relations for Fuzzy Automata , 2018, IEEE Transactions on Fuzzy Systems.

[27]  Yongming Li,et al.  Fuzzy alternating Büchi automata over distributive lattices , 2017, Int. J. Approx. Reason..

[28]  Yixiang Chen,et al.  Algorithmic and logical characterizations of bisimulations for non-deterministic fuzzy transition systems , 2018, Fuzzy Sets Syst..

[29]  Manfred Droste,et al.  Model checking of linear-time properties in multi-valued systems , 2012, Inf. Sci..

[30]  Bin Ma,et al.  On the similarity metric and the distance metric , 2009, Theor. Comput. Sci..

[31]  Yongming Li.  Quantitative model checking of linear-time properties based on generalized possibility measures , 2017, Fuzzy Sets Syst..

[32]  Ping Li,et al.  Nondeterministic fuzzy automata with membership values in complete residuated lattices , 2017, Int. J. Approx. Reason..

[33]  Yang Wen-wu.  Bisimulation Relations for Fuzzy Finite Automata , 2009 .

[34]  Yongming Li,et al.  Fuzzy alternating automata over distributive lattices , 2018, Inf. Sci..

[35]  Rituparna Chutia,et al.  Fuzzy risk analysis in poultry farming using a new similarity measure on generalized fuzzy numbers , 2018, Comput. Ind. Eng..

[36]  Pachaiyappan Muthukumar,et al.  A similarity measure of intuitionistic fuzzy soft sets and its application in medical diagnosis , 2016, Appl. Soft Comput..

[37]  Witold Pedrycz,et al.  Minimization of lattice finite automata and its application to the decomposition of lattice languages , 2007, Fuzzy Sets Syst..

[38]  Sanjiang Li,et al.  Computation tree logic model checking based on multi-valued possibility measures , 2019, Inf. Sci..

[39]  Rui Zhao,et al.  Existence and Verification for Decentralized Nondeterministic Discrete‐Event Systems Under Bisimulation Equivalence , 2016 .

[40]  Yongming Li,et al.  Model Checking of Linear-Time Properties Based on Possibility Measure , 2012, IEEE Transactions on Fuzzy Systems.

[41]  Radim Belohlávek,et al.  On Approximate Minimization of Fuzzy Automata , 2009, J. Multiple Valued Log. Soft Comput..

[42]  Mohammad Mehdi Zahedi,et al.  Bisimulation for BL-general fuzzy automata , 2016 .

[43]  Agostino Dovier,et al.  An efficient algorithm for computing bisimulation equivalence , 2004, Theor. Comput. Sci..

[44]  Daowen Qiu,et al.  On the State Minimization of Fuzzy Automata , 2015, IEEE Transactions on Fuzzy Systems.

[45]  Yongming Li,et al.  Approximation and robustness of fuzzy finite automata , 2008, Int. J. Approx. Reason..

[46]  Chien-Chang Chou,et al.  A generalized similarity measure for fuzzy numbers , 2016, J. Intell. Fuzzy Syst..

[47]  Yongming Li,et al.  Quantitative Computation Tree Logic Model Checking Based on Generalized Possibility Measures , 2014, IEEE Transactions on Fuzzy Systems.

[48]  Weilin Deng,et al.  Supervisory Control of Fuzzy Discrete-Event Systems for Simulation Equivalence , 2015, IEEE Transactions on Fuzzy Systems.

[49]  Miroslav Ciric,et al.  Bisimulations for weighted automata over an additively idempotent semiring , 2014, Theor. Comput. Sci..

[50]  Hongyan Xing,et al.  Analysis and control of fuzzy discrete event systems using bisimulation equivalence , 2012, Theor. Comput. Sci..

[51]  Tatjana Petkovic,et al.  Fuzzy relation equations and reduction of fuzzy automata , 2010, J. Comput. Syst. Sci..

[52]  Shyamal Kumar Mondal,et al.  Fuzzy risk analysis using area and height based similarity measure on generalized trapezoidal fuzzy numbers and its application , 2015, Appl. Soft Comput..