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Complexity and Probability of Some Boolean Formulas

Published online by Cambridge University Press:  01 December 1998

Institute of Computer Science, Academy of Sciences of Czech Republic, Prague, Czech Republic (e-mail:


For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, [oplus ] 1}. For every n and every k[les ]n/2, we describe a probabilistic distribution on formulas in the basis {∧, [oplus ] 1} in some given set of n variables and of size at most [lscr ](k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)[les ][lscr ](k)α, where α=log4(3/2), we have pn,k(f)>0. Moreover, for every function f, if pn,k(f)>0, then

formula here

where c>1 is an absolute constant. Although the upper and lower bounds are exponentially small in [lscr ](k), they are quasi-polynomially related whenever [lscr ](k)[ges ]lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.

Research Article
1998 Cambridge University Press

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