Abstract: Given any set A whoso elements are functions,if there exists a finite set C,C≡A,such that each function in set A can be implemented by a combinatorial network of the functions in set C,it is said that C is a finite complete set of the functions in A.For example,the function set (x∧x,x∨x,x) is just a finite complete set of the two-valued logic functions.In this paper,a set consisting of 4 functions is given and proved to be a finite complete.set of the α-valued logic functions (α≥2).It is aiso proved that no finite complete set exists for the fuzzy logic functions.In addition to the above results,a formal inference system FUNC_c is created and it is proved that each combinatorial network of the functions in C is equivalent to a theorem of the FUNC_c.

Key words: combinatorial network of functions, complete set, multi-valued logic, fuzzy logic, formal inference

摘要： 对于某类函数的集合A,如果存在一个有限集C⊆A,使得A中的任意函数均可以由C的一个有限组合函数网络实现,则称C是A的一个有限完备集。如,函数集{x∧x,x∨x,x}是二值逻辑函数的有限完备集。本文给出一个4个函数组成的集合,并证明这个集合是一类多值逻辑函数的有限完备集。本文还证明,对于模糊逻辑函数,不存在有限完备集。此外,本文建立了形式演算系统FUNC_c,并证明C上的一个组合函数网络与FUNCT_c的一个定理等价。

关键词: 组合函数网络, 完备集, 多值逻辑, 模糊逻辑, 形式演算