The relation between tree size complexity and probability for Boolean functions generated by uniform random trees

We consider a probability distribution on the set of Boolean functions in n variables which is induced by random Boolean expressions. Such an expression is a random rooted plane tree where the internal vertices are labelled with connectives And and OR and the leaves are labelled with variables or negated variables. We study limiting distribution when the tree size tends to infinity and derive a relation between the tree size complexity and the probability of a function. This is done by first expressing trees representing a particular function as expansions of minimal trees representing this function and then computing the probabilities by means of combinatorial counting arguments relying on generating functions and singularity analysis

Strategic directions in constraint programming

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Eighth International Workshop "What can FCA do for Artificial Intelligence?" (FCA4AI at ECAI 2020)

International audienceProceedings of the 8th International Workshop "What can FCA do for Artificial Intelligence?" (FCA4AI 2020)co-located with 24th European Conference on Artificial Intelligence (ECAI 2020), Santiago de Compostela, Spain, August 29, 202

And/or trees:a local limit point of view

International audienceWe present here a new and universal approach for the study of random and/or trees,unifying in one framework many different models, including some novel models, not yet understood in the literature.An and/or tree is a Boolean expression represented in (one of) its tree shape.Fix an integer $k$, take a sequence of random (rooted) trees of increasing sizes, say$(t_n)_{n\ge 1}$, and label each of these random trees uniformly at random in order to get a random Boolean expression on $k$ variables.We prove that, under rather weak local conditions on the sequence of random trees $(t_n)_{n\ge 1}$, the distribution induced on Boolean functions by this procedure converges as $n\to\infty$. In particular, we characterise two different behaviours of this limit distribution depending on the shape of the local limit of $(t_n)_{n\ge 1}$: a degenerate case when the local limit has no leaves; and a non degenerate case, which we are able to describe in more details under stronger but reasonable conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples we cover include, in a unified way, trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton--Watson trees)