Complexity and Limiting Ratio of Boolean Functions over Implication

Authors:
Hervé Fournier;Danièle Gardy;Antoine Genitrini;Bernhard Gittenberger
Affiliations:
Laboratoire PRiSM, CNRS UMR 8144 and Université de Versailles St-Quentin en Yvelines, Versailles, France 78035;Laboratoire PRiSM, CNRS UMR 8144 and Université de Versailles St-Quentin en Yvelines, Versailles, France 78035;Laboratoire PRiSM, CNRS UMR 8144 and Université de Versailles St-Quentin en Yvelines, Versailles, France 78035;Technische Universität Wien, Wien, Austria A-1040
Venue:
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Year:
2008

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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. We then show how to approximate the probability of a function fwhen 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.