This paper presents a catalog of AND-EXOR expressions for representative function of four-variable NP-equivalence classes. Minimality is defined as minimizing first the number of product terms and then the total number of literal in the expression. Also, the propoerties of minimum expressions are discussed. Using this as a base, We compare minimum expressions of AND-OR type two-variable circuit with minimum expressions of AND-EXOR type two-variable circuit which used algorithm in this paper. As a results it was found that in the case of AND-OR type minimum expressions, number of product terms is under 8, and in the case of AND-EXOR type minimum expressions all functions are formed in which number of product terms is under 6, and generally number of product term is considerably small to realize four-vaiable function toward ADN-EXOR type minimum expression circuit. Algorithm suggested in this paper are realized on Sun 3/50, and through this, logic function under four-variable can get minimum immedately by a catalog suggested in this paper. As for five-variale function, we can do shanon-development a part of the function with suitable variable, and by applying four-variable minimum of this paper on this, it can be possible to get minimum in a short time and it can be said that it is possible to apply this method to functions over 6 variable.