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.
We consider random balanced Boolean formulas, built on the two connectives and and or, and a fixed number of variables. The probability distribution induced on Boolean functions is shown to have a limit when letting the depth of these formulas grow to infinity. By investigating how this limiting distribution depends on the two underlying probability distributions, over the connectives and over the Boolean variables, we prove that its support is made of linear threshold functions, and give the speed of convergence towards this limiting distribution.
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