Complexity and Probability of Some Boolean Formulas

For any Boolean functionf letL(f) be its formulasizecomplexityin the basis f^ 1g. For every n and every k n=2, we describe a probabilistic distribution on formulas in the basis f^ 1g in some given set of n variables and of the size at most `(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) `(k) , where = log4(3=2), we have pn k(f) > 0. Moreover, for every function f, if pn k(f) > 0, then (4n);`(k) pn k(f) c;`(k)1=4 where c > 1 is an absolute constant. Although the upper and lower bounds are exponentially small in `(k), they are quasipolynomially related whenever `(k) ln (1) n. The construction is a step towards developping a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity. Keywords Complexity, probability, Boolean formulas 1This research was supported by GA CR, Grant No. 201/95/0976, and by Heinrich-Hertz-Stiftung w...

• Type: Journal
• Year: 1998
• Where: CPC
• Authors: Petr Savický
• Keywords: Basis F^, Boolean Functionf, CPC 1998, Probabilistic Distribution