The Shortest Disjunctive Normal Form of a Random Boolean Function

This paper gives a new upper bound for the average length ℓ(n) of the shortest disjunctive normal form for a random Boolean function of n arguments, as well as new proofs of two old results related to this quantity. We consider a random Boolean function of n arguments to be uniformly distributed over all 2^(2^n) such functions. (This is equivalent to considering each entry in the truth-table to be 0 or 1 independently and with equal probabilities.) We measure the length of a disjunctive normal form by the number of terms. (Measuring it by the number of literals would simply introduce a factor of n into all our asymptotic results.) We give a short proof using martingales of Nigmatullin\u27s result that almost all Boolean functions have the length of their shortest disjunctive normal form asymptotic to the average length ℓ(n). We also give a short information-theoretic proof of Kuznetsov\u27s lower bound ℓ(n) ≥ (1 + o(1)) 2n/log n log log n. (Here log denotes the logarithm to base 2.) Our main result is a new upper bound ℓ(n) ≤ (1 + o(1)) H(n) 2n/log n log log n, where H(n) is a function that oscillates between 1.38826 … and 1.54169 … . The best previous upper bound, due to Korshunov, had a similar form, but with a function oscillating between 1.581411 … and 2.621132 … . The main ideas in our new bound are (1) the use of Rödl\u27s “nibble” technique for solving packing and covering problems, (2) the use of correlation inequalities due to Harris and Janson to bound the effects of weakly dependent random variables, and (3) the solution of an optimization problem that determines the sizes of “nibbles” and larger “bites” to be taken at various stages of the construction