A categorical approach to lattice-valued fuzzy automata

Some uniform categorical theoretical treatment of automata and lattice-valued fuzzy automata using quantale theory is studied in this paper. First, L-relational sheaves on a monoid M and Q-enriched categories are introduced for quantales L and Q, the equivalence of the corresponding categories are proved next. Then lattice-valued (fuzzy) automata are described by Q-enriched categories. In fact, lattice-valued (fuzzy) automata are characterized by the category of generalized lattice-valued automata using the notions of Q-bimodules. Finally, some of the algebraic properties of behaviors of generalized lattice-valued automata are studied by using the technique of gluing of Q-bimodules.

[1]  K. I. Rosenthal.  The Theory of Quantaloids , 1996 .

[2]  K. I. Rosenthal.  Quantales and their applications , 1990 .

[3]  Steven J. Vickers,et al.  Quantales, observational logic and process semantics , 1993, Mathematical Structures in Computer Science.

[4]  G. Grätzer.  General Lattice Theory , 1978 .

[5]  Eugene S. Santos,et al.  Maximin Automata , 1968, Inf. Control..

[6]  M. W. Shields.  An Introduction to Automata Theory , 1988 .

[7]  José Meseguer,et al.  From Petri Nets to Linear Logic , 1989, Category Theory and Computer Science.

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

[9]  Samuel Eilenberg,et al.  Automata, languages, and machines. A , 1974, Pure and applied mathematics.

[10]  Peter R. J. Asveld,et al.  Algebraic aspects of families of fuzzy languages , 2001, Theor. Comput. Sci..

[11]  Stefano Kasangian,et al.  Tree automata and enriched category theor , 1985 .

[12]  L. Peeva,et al.  Equivalence, Reduction and Minimization of Finite Automata over Semirings , 1991, Theor. Comput. Sci..

[13]  Jiri Mockor.  Semigroup homomorphisms and fuzzy automata , 2002, Soft Comput..

[14]  Stefano Kasangian,et al.  Glueing enriched modules and composition of automata , 1990 .

[15]  Li Yongming,et al.  Projective and injective objects in the category of quantales , 2002 .

[16]  A. Joyal,et al.  An extension of the Galois theory of Grothendieck , 1984 .

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

[18]  H. Priestley,et al.  Distributive Lattices , 2004 .

[19]  W. Wee.  On generalizations of adaptive algorithms and application of the fuzzy sets concept to pattern classification , 1967 .

[20]  LI Yong-ming.  Noncommutative linear logic and its Quantale semantics , 2001 .

[21]  K. Peeva.  Fuzzy automata and languages: theory and applications , 2004 .

[22]  Stefano Kasangian,et al.  Decomposition of automata and enriched category theory , 1986 .

[23]  Fabio Rossi,et al.  Cofibrations and the realization of non-deterministic automata , 1983 .

[24]  Eugene S. Santos.  Fuzzy automata and languages , 1976, Information Sciences.