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

## Authors

• ### Hervé Fournier,

1. Laboratoire PRiSM, CNRS UMR 8144, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, 78035 Versailles, France
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• ### Danièle Gardy,

1. Laboratoire PRiSM, CNRS UMR 8144, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, 78035 Versailles, France
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• ### Antoine Genitrini,

1. Laboratoire LIP6, CNRS UMR 7606, Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris cedex 05, France
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• ### Bernhard Gittenberger

Corresponding author
1. Technische Universität Wien, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria
• Technische Universität Wien, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria
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• 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.

## Abstract

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