login
Number of switching functions of n or fewer variables which cannot be realized as threshold gates.
5

%I #9 Sep 05 2017 03:11:37

%S 0,0,2,152,63654,4294872724,18446744073694523482,

%T 340282366920938463463374607423390140592,

%U 115792089237316195423570985008687907853269984665640564039457583990351590086990

%N Number of switching functions of n or fewer variables which cannot be realized as threshold gates.

%C The corresponding systems of linear inequalities are not solvable: linearly non-separable truth or switching functions. Truth functions which ar "non-neurons" and are realizable only as two levels threshold gate networks.

%H Wang Lan, <a href="/A064436/b064436.txt">Table of n, a(n) for n = 0..9</a>

%F a(n) = 2^(2^n) - A000609(n).

%e n=2: out of the 16 B^2 -> B^1 truth functions, 14 are linearly separable; the 2 exceptions are XOR and its negation: f(x,y) = !xz + x!y and !f(x,y) = xy + !x!y. So a(2)=2. With increasing n, the chance that a switching function belongs to this sequence tends to 1.

%Y Cf. A000609.

%K nonn

%O 0,3

%A _Labos Elemer_, Oct 01 2001