IMTL System And IMTL-algebra

Fuzzy logic, as an important branch of non-classical mathematical logics, is the basis of inference mechanism of many fields, such as artificial intelligence, information science, and etc.. L.A.Zadeh distinguished fuzzy logic into broad and narrow senses. In narrow sense, fuzzy logic is the extension of many-valued logic, and in broad sense ,fuzzy logic is the theory of fuzzy sets. In engineering fields, fuzzy logic means broad one, but in theoretical fields, narrow fuzzy logic is studied generally. Accordingly, this paper studies narrow fuzzy logic.In fact, fuzzy logic has been studied by many scholars for many years, and has been developed deeply arid widely .In the course of studying , some new methods have been used to improve fuzzy logic. The one is algebraic method, that is, the fuzzy logic and it's related algebra structure will be studied together. We can use the results in algebraic fields to solve the problems in fuzzy logic. As a matter of fact, there are two successful examples, MV-algebra and R0-algebra,which function well in proving the completeness of the related fuzzy logic respectively. On the other hand, in recent study, t-norm has been introduced in fuzzy logic, forming a kind of logic systems which base on i-norm. In fact, BL introduced by P.Hajek is a continuous t-norm based logic. What's more, recently a left-continuous t-norm based logic, MTL, has been introduced by F.Esteva and L.Godo. IMTL system is an extension of MTL.This paper aims at studying IMTL system, IMTL-algebra, it's related algebra structure, and some questions in it's two complete systems, Lukasiewicz and L* system. It is divided into three chapters.In the first chapter, IMTL system and IMTL-algebra are studied, an equivalent IMTL axiom system is given, and the equivalence of IMTL- algebra, weak R0-algebra and weak MV-algebra is proved. Then, a new left-continuous t-norm is given and it is proved that ([0,1], A, V, , →p, 0,1) is an IMTL-algebra. According to the above, Rp-algebra is introduced and related Lp system is constructed.In the second chapter, the accessible generalized tautologies in Lukasiewicz logic system are studied by means of McNaughton function. The main result is that when a is an irrational number, the set of accessible ct-tautology is empty. Consequently, akind of partition on F(S) is given. In addition, the theorem that a tautology can be got by using upgrade algorithm to non-tautologies within finite times in Lukasiewicz logic system is also proved.In the third chapter, it is proved that in L* system, MP filters of ?-Lindenbaum algebra are all in the form of [D(T)], . Also, there is a proof that a maximal filter in the sense of lattice theory is a maximal MP filter. What's more, a sufficient condition about maximal niters is given.