Some Typical Properties of Large AND/OR Boolean Formulas

In this paper typical properties of large random Boolean AND/OR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees with m leaves, where n is fixed and m tends to infinity. The leaves are labeled by literals and the inner nodes by the connectives AND/OR, both uniformly at random. In extending the investigation to infinite trees, we obtain a close relation between the formula size complexity of an arbitrary Boolean function f and the probability of its occurrence under this distribution, i.e., the negative logarithm of this probability differs from the formula size complexity of f only by a polynomial factor.

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