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 $\pi$.
Highlights
/or trees are representations of boolean functions built from ∧, ∨ and positive and negative literals
It has been shown by Lefmann and Savicky in [8] that for every fixed number of variables k when n tends to infinity the induced distribution on boolean functions converge to some probability distribution which we denote by Pk
In the recent paper [5], Gardy and Woods focused on special functions. For such a function f they analysed behaviour of Pk(f ) when k tends to infinity
Summary
/or trees are representations of boolean functions built from ∧, ∨ and positive and negative literals. In the recent paper [5], Gardy and Woods focused on special functions (namely constant function and so called ”read-once” functions) For such a function f they analysed behaviour of Pk(f ) when k tends to infinity (in this approach f is treated as a boolean function of countably many variables, even though it depends only on the finite number of them). They stated the following conjecture: For a ”read-once” function f with complexity r, there exist constants bf and Bf such that πk(f ) ∼k bf k−r and Pk(f ) ∼k Bf k−r−1 as k → ∞. As a simple example of applications of this technique, we present (in the subsection 3.2) quite simple proof of the result announced by Woods in [9] that ”most of tautologies” are variations of formulae like x ∨ x ∨ φ
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