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The aim is to find a formula for the question.

For $n=2$ i get $2^{2^n}=16$ possible functions.

This is the solution for a boolean function with 2 attributes:

01) a     ->  separable
02) b     ->  separable
03) not a    ->  separable
04) not b    ->  separable
05) a and b   ->  separable
06) a or b    ->  separable
07) a xor b   -> not separable
08) a nand b   ->  separable
09) a nor b   ->  separable
10) a xnor b   -> not separable
11) (not a) and b  ->  separable
12) a and (not b)  ->  separable
13) (not a) or b  ->  separable
14) a or (not b) ->  separable
15) (not a) xor b  -> not separable
16) a xor (not b) -> not separable

But what is the formular for n?

Chain Markov
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  • This looks similar to this question: http://math.stackexchange.com/questions/17852/how-many-different-sets-can-be-formed-by-two-nonempty-sets/17853#17853 – NebulousReveal Jan 19 '11 at 01:58
  • @Brainlag: Something is wrong with your list of functions; because you lack the two trivial functions in your list: $f(a,b)=0$ and $f(a,b)=1$

    But you are still correct that there are $2^{2^n}$ boolean functions; your (10), (15), and (16) all represent the same function.

     – Chas Brown Jan 19 '11 at 05:33
  • @PEV : Yes but i know how many functions are possible. The Question is how many functions are separable for n (not only n=2) –  Jan 19 '11 at 12:24
  • @Chas Brown I don't get it. I don't see they are the same. –  Jan 19 '11 at 12:24
  • all three have the truth table: – Chas Brown Jan 19 '11 at 17:17
  • @Brainlag: (Sorry, previous comment timed out) Try writing out the truth table for the three functions. Alternatively, think of a xor b as not (a == b); then a xor (not b) <-> not (a == not b) <-> not (a != b) <-> (a == b) <-> a xnor b; etc. – Chas Brown Jan 19 '11 at 17:23

1 Answers1

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It may be easier to think in terms of linearly separable subsets of $\{0,1\}^n$ (that is, those that are cut off from the cube by a hyperplane). For $n=2$ we are looking at subsets of the vertex set of a square: the only two that are not linearly separable are the diagonals. This makes $2^{2^2}-2=14$ separable functions: all except XOR and NOT XOR.

There is no formula for general $n$, in fact only a handful of counts are known. This is the OEIS sequence A000609, labeled "hard". By the way, the Wikipedia page on linear separability references OEIS.

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