Discrete Math. AppL, Vol. 8, No. 4, pp. 331-355 (1998) © VSP 1998. On the asymptotics of the logarithm of the number of threshold functions in K- valued logic A. A. IRMATOV and Z. D. KOVIJANIC Abstract — For the number P(K,n) of threshold n-ary functions of A"- valued logic, we obtain the lower bound The key argument in our investigation is the generalization of a result obtained by Odlyzko on the subspaces spanned by ñ randomly chosen (±1)- vectors. Namely, we prove that, as ç ->· oo, for any Ñ < ç - (3 + Iog 2 36)n/log 2 n if Ê — 2(2, respectively, for any Ñ < ç - (3 + log 20 +! 36)n/log 2 + 1 n if K = 2g + 1, the probability that the linear span of p randomly chosen vectors vi, v 2 ,... ,í,, € (E' K ) n = {±1,±3,... ,±(2â- 1)} ç , respectively, from E% = {0, ±1 , . . . , ±â} ç » contains at least one vector from respectively, from é=1 equals, for even K = 2g, Q ^ 1, and for â = 1, and, respectively, for odd K = IQ + 1, â ö 1 , (ú - 4(2 W)") ' *UDC 519.7. Originally published in Diskretnaya Matematika (1998) 10, No. 3 (in Russian). Received June 22, 1998. Translated by the authors. Brought to you by | University of Arizona Authenticated Download Date | 6/8/15 7:08 PM