Subcritical pattern languages for and/or trees

Abstract : Let $P_k(f)$ denote the density of and/or trees defining a boolean function $f$ within the set of and/or trees with fixed number of variables $k$. We prove that there exists constant $B_f$ such that $P_k(f) \sim B_f \cdot k^{-L(f)-1}$ when $k \to \infty$, where $L(f)$ denote the complexity of $f$ (i.e. the size of a minimal and/or tree defining $f$). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution $\pi$ defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for $\pi$.
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Roesler, Uwe. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science, pp.437-448, 2008, DMTCS Proceedings
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Jakub Kozik. Subcritical pattern languages for and/or trees. Roesler, Uwe. Fifth Colloquium on Mathematics and Computer Science, 2008, Kiel, Germany. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AI, Fifth Colloquium on Mathematics and Computer Science, pp.437-448, 2008, DMTCS Proceedings. 〈hal-01194664〉

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