格值有限状态机的弱交换性
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摘要
提出了格值有限状态机可弱交换的概念和格值变换半群可弱交换的概念,用矩阵和半群对格值有限状态机的弱交换性进行刻画,得出了格值有限状态机可弱交换的几个等价条件,并找出了格值有限状态机的弱交换性与其伴随第二类格值变换半群弱交换性的关系。此外,研究了格值有限状态机的满直积、限制直积、级联积、圈积及和的弱交换性,并研究了格值有限状态机伴随第二类格值随变换半群的全直积和限制直积的弱交换性,得出了其全直积和限制直积可弱交换的充分条件。
In this paper, the definition of weak commutativity of lattice-valued finite state machine and the definition of weak commutativity of lattice-valued transformation semigroup are given. Weak commutativity of lattice-valued finite state machine is characterized by using matrix and semigroup, and some equivalent conditions of weak commutativity of lattice-valued finite state machine are obtained. The relation of weak commutativity of lattice-valued finite state machine between weak commutativity of second class lattice-valued transformation semigroup is also found out. Moreover, weak commutativity of full direct product, restricted direct product, cascade product, wreath product and sum of lattice-valued finite state machines are studied, weak commutativity of full direct product and restricted direct product of second class lattice-valued transformation semigroups associated with lattice-valued finite state machines are also studied, and a sufficient condition is obtained.
In this paper, the definition of weak commutativity of lattice-valued finite state machine and the definition of weak commutativity of lattice-valued transformation semigroup are given. Weak commutativity of lattice-valued finite state machine is characterized by using matrix and semigroup, and some equivalent conditions of weak commutativity of lattice-valued finite state machine are obtained. The relation of weak commutativity of lattice-valued finite state machine between weak commutativity of second class lattice-valued transformation semigroup is also found out. Moreover, weak commutativity of full direct product, restricted direct product, cascade product, wreath product and sum of lattice-valued finite state machines are studied, weak commutativity of full direct product and restricted direct product of second class lattice-valued transformation semigroups associated with lattice-valued finite state machines are also studied, and a sufficient condition is obtained.
引文
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[25] 黄飞丹,邓泽喜.量子自动机的交换性[J].计算机工程与应用,2016,52(20):58～63.
[26] 黄飞丹,杨京开,邓泽喜.格值有限状态机的交换性[J].模糊系统与数学,2015,29(6):145～153.
[27] Huang F D,Yang J K,Zhai Y,Li X J.Weak commutativity of fuzzy finite state machines[C]//13th International Conference on Natural Computation,Fuzzy Systems and Knowledge Discovery(ICNC-FSKD),2017:1188～1196.
[2] Mordeson J N,Malik D S.Fuzzy automata and languages:Theory and applications[M].Boca Raton,London,New York,Washington,D.C.:Chapman and Hall/CRC,2002.
[3] Basak N C,Gupta A.On quotient machines of a fuzzy automaton and the minimal machine[J].Fuzzy Sets and Systems,2002,125:223～229.
[4] Cheng W,Mo Z W.Minimization algorithm of fuzzy fnite automata[J].Fuzzy Sets and Systems,2004,114:439～448.
[5] Petkovic T.Congruences and homomorphisms of fuzzy automata[J].Fuzzy Sets and Systems,2006,157:444～458.
[6] Qiu D W.Characterizations of fuzzy finite automata[J].Fuzzy Sets and Systems,2004,141:391～414.
[7] Li Z H,Li P,Li Y M.The relationships among several types of fuzzy automata[J].Information Sciences,2006,176:2208～2226.
[8] Sharma B K,Tiwari S P,Sharan S.On algebraic study of fuzzy multiset finite automata[J].Fuzzy Information and Engineering,2016,8(3):315～327.
[9] Jancic Z,et al.Further improvements of determinization methods for fuzzy finite automata[J].Fuzzy Sets and Systems,2016,301:79～102.
[10] Qiu D W.Automata theory based on complete residuated lattice-valued logic[J].Science in China,Series F,2001,44(6):419～429.
[11] 李永明.格值自动机及其语言[J].陕西师范大学学报(自然科学版),2003,31(4):1～6.
[12] 李平,李永明.几类格值自动机的关系[J].模糊系统与数学,2005,19(3):96～100.
[13] Li Y M,Pedryczc W.Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoids[J].Fuzzy Sets and Systems,2005,156:68～92.
[14] 雷红轩,潘超.格值有限自动机及其性质[J].内江师范学院学报,2006,21(4):9～12.
[15] Li Y M.A categorical approach to lattice-valued fuzzy automata[J].Fuzzy Sets and Systems,2006,157:855 ～864.
[16] Lei H X,Li Y M.Minimization of states in automata theory based on finite lattice-ordered monoids[J].Information Sciences,2007,177:1413～1421.
[17] 韩召伟,李永明.格值Mealy自动机的同余和同态[J].模糊系统与数学,2007,21(2):53～64.
[18] 刘军,莫智文.格值有限自动机的乘积[J].高校应用数学学报,2009,24(1):121～126.
[19] Gómez M,Lizasoain I,Moreno C.Lattice-valued finite state machines and lattice-valued transformation semigroups[J].Fuzzy Sets and Systems,2012,208:1～21.
[20] Tang J G,Luo M K,Tang J.Results on the use of category theory for the study of lattice-valued finite state machines[J].Information Sciences,2014,288:279～289.
[21] Malik D S,Mordeson J N,Sen M K.The cartesian composition of fuzzy finite state machines[J].Kybernetes,1995,24(4):98～110.
[22] 谢正卫,翟莹,邓培民,易忠.概率有限状态自动机的代数性质[J].计算机研究与发展,2013,50(12):2691～2698.
[23] 谢正卫,翟莹,黄飞丹,邓培民,易忠.两类模糊有限状态机积的交换性[J].计算机研究与发展,2014,51(9):2130～2136.
[24] 谢正卫.概率有限自动机的交换性[J].江苏理工学院学报,2014,20(6):21～26.
[25] 黄飞丹,邓泽喜.量子自动机的交换性[J].计算机工程与应用,2016,52(20):58～63.
[26] 黄飞丹,杨京开,邓泽喜.格值有限状态机的交换性[J].模糊系统与数学,2015,29(6):145～153.
[27] Huang F D,Yang J K,Zhai Y,Li X J.Weak commutativity of fuzzy finite state machines[C]//13th International Conference on Natural Computation,Fuzzy Systems and Knowledge Discovery(ICNC-FSKD),2017:1188～1196.