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The fraction of large random trees representing a given Boolean function in implicational logic


  • A preliminary version of this work appeared in MFCS'08. Supported by FWF (Austrian Science Foundation), National Research Area S9600 (S9604), ÖAD (F03/2010); A.N.R. projects SADA, BOOLE, P.H.C. Amadeus project Probabilities and tree representations for Boolean functions.


We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all read-once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011