In article <3922C5E4.459C@cs.bham.ac.uk>, Tim Kovacs <T.Kovacs@cs.bham.ac.uk> writes: > I'm interested in the distribution of N input boolean functions > according to the length of their minimal representation. > > I suspect that for a given N and a given representation, > some functions can be represented very compactly, some can > only be represented in a very uncompact way, and most are in > between. But what's the distribution? > > Does anyone have any references to this sort of thing?
My intuition disagrees with yours. Whilst I agree that some functions can be represented very compactly, I would not expect much of a tail at the other end of the distribution. I'd expect that an "averagely difficult to represent" function will be not much more compactly representable than the most difficult functions. (I'm assuming N is not very small.)
Using conjunctive or disjunctive normal form you can obtain an upper bound on the difficulty of representing the most troublesome functions. Using the fact that there are 2^(2^N) possible functions, and an estimate for how many syntactically correct formulae there are of length M, you can find a lower bound for the length of the shortest representation of average functions.
Others will probably have more specific answers.
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