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 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.
This research was partially supported by the A.N.R. project SADA and by the P.H.C. Amadeus project Probabilities and tree representations for boolean functions. The last author’s work has been supported by ÖAD, project Amadée, grant 3/2006, as well as by FWF (Austrian Science Foundation), National Research Area S9600, grant S9604.
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Fournier, H., Gardy, D., Genitrini, A., Gittenberger, B. (2008). Complexity and Limiting Ratio of Boolean Functions over Implication. In: Ochmański, E., Tyszkiewicz, J. (eds) Mathematical Foundations of Computer Science 2008. MFCS 2008. Lecture Notes in Computer Science, vol 5162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85238-4_28
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DOI: https://doi.org/10.1007/978-3-540-85238-4_28
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