Classification of Functions and Enumeration of Bases of Set Logic under Boolean Compositions

A small fraction of these functions are Boolean functions (including constants), that is functions that can be This paper discusses some classification and constructed from constants and variables, using union, The set p(r) is a enumeration problems in r-valued set logic, which is the intersection and complementation. logic of functions mapping n-tuples of subsets into subsets Boolean algebra (p (r), 0, r, U, A, ) when equipped with over r values. Boolean functions are convenient choice as building blocks in the design of set logic. B-maximal sets the set-theoretic operations U, n, and , which denote the are maximal sets containing d l Boolean functions, where union, the intersection and the complement operations respectively. The number of n-place r-valued Boolean Boolean functions are those obtained from U , n functions of set logic is (21721n [ 171. by composition (constants are not involved in the Let k be a fixed positive integer and let Lk = (0 , 1, ..., k compositions). We give the number of n-place functions in 1 }. The set of n-place k-valued logic functions f: Lk" -+ L, each B-maximal set and find some properties of intersection is denoted by Pk(n). The union of Pk(n) for n = 1 , 2, ..., is of B-maximal sets in r-valued set logic. These properties denoted by P The number of n-place k-valued logic k' functions is kk . functions according to the B-maximal sets to which they As mentioned in [ 151, every r-valued set logic function belong to. We prove that there are 8 and 200 such classes can be regarded as a k-valued logic function for k = 2'-, as of functions respectively in 2-valued and 3-valued set logic. follows. Without loss of generality we may use characteristic For each class of functions we give a one-place example function and its total number of one-place set logic binary vectors to represent the elements of ff (r) as binary functions. Finally, we study the B-Sheffer functions, i.e. numbers. A subset x E ff(r) is represented as binary functions which are complete under compositions with number XoXl ...Xr-2X,.l determined by xi = 1 if and only if ei E Boolean functions. Wefind the number of n-place B-Sheffer X, for i = 0, 1, ..., r 1. Next, we map X E P(r) into the functions of 2-val~ed set logic and give a lower bound and natural number x which is the binary number X , . ~ , X , ~ . . . X ~ X ~ , an upper bound on the " h r of n-place B-Sheffer then x = 2r-1x,l + 2r-2~,.2 + ... + 2x1 + x,. We also refer to functions of 3-valued set logic. Also we enumerate all the xi, i = r 1, ..., 0 as the i-th coordinate of x. classes of B-bases of 2-valued and 3-valued set logic. In general, x U y = U and x n y = v are determined by ui = max(xi, yi) and vi = min(xi, yi) for 0 5 i < I1, while X = k 1 x. We refer to these functions as union, intersection