Fuzzy finite automata and fuzzy monadic second-order logic

We introduce fuzzy monadic second-order (LMSO-) logic and prove that the behaviours of fuzzy finite automata with membership values in an MV-algebra are precisely the fuzzy languages definable with sentences of our LMSO logic. This generalizes Buchipsilas and Elgotpsilas fundamental theorems to fuzzy logic setting. We also consider fuzzy first-order logic and show that star-free fuzzy languages and aperiodic fuzzy languages introduced here coincide with the fuzzy first-order definable ones.

[1]  Paul Gastin,et al.  Weighted automata and weighted logics , 2005, Theor. Comput. Sci..

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

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

[4]  J. Büchi.  Weak Second‐Order Arithmetic and Finite Automata , 1960 .

[5]  Yongming Li,et al.  Approximation and universality of fuzzy Turing machines , 2008, Science in China Series F: Information Sciences.

[6]  C. C. Elgot.  Decision problems of finite automata design and related arithmetics , 1961 .

[7]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[8]  Witold Pedrycz,et al.  The equivalence between fuzzy Mealy and fuzzy Moore machines , 2006, Soft Comput..

[9]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[10]  Qiu Daowen,et al.  Automata theory based on complete residuated lattice-valued logic , 2001 .

[11]  Lotfi A. Zadeh,et al.  Note on fuzzy languages , 1969, Inf. Sci..

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

[13]  J. Mordeson,et al.  Fuzzy Automata and Languages: Theory and Applications , 2002 .

[14]  Wolfgang Thomas,et al.  Languages, Automata, and Logic , 1997, Handbook of Formal Languages.

[15]  Olga Kosheleva,et al.  IEEE International Conference on Fuzzy Systems , 1996 .

[16]  Anil Nerode,et al.  Automata theory and its applications , 2001 .

[17]  Ping Li,et al.  Algebraic properties of LA-languages , 2006, Inf. Sci..

[18]  Daowen Qiu.  Automata theory based on complete residuated lattice-valued logic , 2007, Science in China Series : Information Sciences.