Researching On Some Problems About Fuzzy Logic

Applications of fuzzy logic range over a wide field It can be used in the field of computer science, including automated theorem proving, approxinate reasoning, multi-agent systems, switching theory, and program verification, etc.. It can also be used in the field of pure mathematics, such as, independence or consistency proofs, generalized set theories, or the theory of particularly algebraic structures ( e.g., MV-algebras, BL-algebras, and Ro-algebras). And it can also be used in the field of humanities, e.g., linguistics and philosophy. Based on these facts, we research on fuzzy logic from four facets, namely, fuzzy operator, particularly algebraic theory, formal system and fuzzy reasoning.In the field of fuzzy operator, first, we recall the definitions of uninorm and the others related, and then relations among them are revealed. Structures of residuated (co)implicalors concerning uninonus comiuuous in ((0,1)2 and (-operators arc given after recalling the three kinds of residuated structures. In order to apply them, some simple qualities about residuated (co)implicators arc given. Moreover, for certain kinds of uninorms and (or) (-operators, modular condition equations and distributive equations are studied. Specifically, when nninorms continuous in (0. 1)2 and t-operators are only considered, the modular condition equations have no new non-trivial solutions. When idcmpotent uninorms and t-operators are considered, the above equations only have all new non- trivial solutions F =, U = R, and F = V, U = R* and the ones in Theorem 2.2.3.1. When the assumptions in the second ca.se hold, the new non-trivial solutions to the distributive equations are characterized as Theorem 2.3.1.1 and Theorem 2.3.2.1. Finally, we devote outself to generalizing nullnorms and introduce the left (right) uullnorms, and then characterize all kinds of the left ( right) nullnorms on [0,1] and on the finite chain C. In the field of particularly algebraic theory, first, the elementary properties are studies. And one to one correspondence between the set of ideals and the set of congruent relations are builded for any Ro-algebra. The sufficient and necessary conditions for any Ro-algebra becoming a Boolean-algebra are also given. Moreover, endowed with Stonean-topology on an Ro-algebra, we find out. the distributive lattices which are isomorphicalto Ro-algebras after stonean spaces are bnilded on the Ro--algebras. Furthermore. the suflicient and necessary conditions for an Ro-algehra with the irreducible representation are given, and then the free Ro-algebra.s are structured. Finally, the original definitions of the Ro-algebras are reduced.In the field of formal system, in order lo axioniatize the Zadeh s fuzzy logic, we pro-posted the extension system VTMTL of formal svslem MTL., and prove the completenesses of propositional logic system VTMTL and predicative logic system VTMTLV.In the field of fuzzy reasoning, the syntax interpretation of the triple I method is given, and then we analyze its essence and propose a new triple T method which is a good supplement, to the original method. Finally, we prove the triple I method is a universal approximator.