The Decision On The Minimal Covering Of Function Sets Preserving 3-ary Full Symmetric Relation In Partial Multiple-Valued Logic

The decision on Sheffer function is an important problem in multiple-valued logic completeness theory, which depends on deciding the minimal covering of the precomplete classes. For the complete multiple-valued logic functions, it was solved by Schofield and Kudrjavcev etc. For the partial multiple-valued logic functions, it has not been solved thoroughly. The main work of this thesis is decision on minimal covering of precomplete classes in partial multiple-valued logic. We focus on function sets preserving 3-ary full symmetric relation and prove that one class of function sets preserving 3-ary full symmetric relation must be the component part of the minimal covering of precomplete classes in P_k~* by means of the concept of similar relationship among precomplete sets. Moreover, some properties of simply separable function set and the structure of simply separable relation are discussed. The thesis is divided into four chapters. In chapter one, it summarizes the structure theory of multiple-valued logic functions. First, it introduces the basic concept of the structure theory of the complete multiple-valued logic functions, and then it introduces the precomplete class in the complete k-valued logic functions and in the partial k-valued logic functions. In the second chapter, it introduces the minimal covering of precomplete class in partial k-valued logic. First, it introduces the concept of minimal covering and the relation between the concept and decision problem of Sheffer function. Then, as a special example, the minimal covering of precomplete class in the partial 2-valued logic functions is introduced. At last, it introduces the concept of similar relation among the precomlete sets, summarizes the research status and main production of minimal covering of precomplete classes in P_k~*. In the third chapter, it studies the function set preserving 3-ary full symmetric relation in partial multiple-valued logic, and proves that the one class of function set preserving 3-ary full symmetric relation must be the component part of the minimal covering. In the fourth chapter, some properties of simply separable function set which is another kind of precomplete class in partial multiple-valued logic and the structure of simply separable relation are discussed. We get some conclusions and it will lay the foundation for more in-depth understanding of the structure of simply separable function set.