Dwrete Applied Mathematics 4 (1982) 153-156 North-Holland Publishing Company 153 NOTE ON EQUIVALENT SETS OF FUNCTIONS Jaak HENNO Department of Information Processing, Tallinn Polytechnic Institute, 20026 Tallinn, USSR Received 13 August 1981 Two complete sets of functions are called equivalent [l] if depths of any function over these sets differ at most in an additive constant. Based on symmetry properties of functions, conditions for equivalence of sets of functions of many-valued logic are found. Using these conditions, series of complete sets of binary Boolean functions are described, equivalent to the set & of all binary Boolean functions or to the set U, of all binary unate functions (in [I] they were investigated by direct methods). Let F be the set of all total functions on the set M= zyxwvutsrqponmlkjihgfedcbaZYXW (0, 1, . . . , m - 11, m L 2. For TC F denote by [T] the set of functions which can be obtained from functions in T using superposition. Every f E [T] can be represented by a formula composed of variables xl, x2, . . . and functions from T. The depth of a formula is the maximum number of functions from T on any path in the tree, representing this formula (variables and constants are not counted) and the depth dT(f) of a function zyxwvutsrqponmlkj f over T is the depth of a formula of minimal depth for zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP f. Call sets T, T'cF equivalent if [T] = [T'] and there exists a constant c such that idr(f)-Mf)I- f -= c or every f E [T] . A set T is called complete if [T] = F. Let G be the full substitution group on the set M, and H the full substitution group on the set { 1,2, . ..}. Call functions f, f J?-equivalent, if there exist g,, . . . ,gne G, h EH such that f(x It *-* ,x,1 =fYsl(X/l(l))r * . . ,&(X/I(n))); Y-equivalent, if there exist g E G, h E H such that zyxwvutsrqponmlkjihgfedcbaZYXWVU f(x I,...,Xn)=g(f’(Xh(l),...,Xh(n))); Y-equivalent, if there exists f” E F such that f 9 f” and f” k”f I. Clearly $I= :$iu= SJ? =_!?uw. For an equivalence E on a set K and for a subset L C K define the E-closure E(L) ofLby E(L)={~J~EK,ZIL kEl}. 0166-218X/82/0000-0000/$02.75 0 1982 North-Holland