12 research outputs found
Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability
This article is motivated by the following satisfiability question: pick
uniformly at random an and/or Boolean expression of length n, built on a set of
k_n Boolean variables. What is the probability that this expression is
satisfiable? asymptotically when n tends to infinity?
The model of random Boolean expressions developed in the present paper is the
model of Boolean Catalan trees, already extensively studied in the literature
for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this
paper is to generalise the previous results to any (reasonable) sequence of
integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above
satisfiability question.
We also analyse the effect of introducing a natural equivalence relation on
the set of Boolean expressions. This new "quotient" model happens to exhibit a
very interesting threshold (or saturation) phenomenon at k_n = n/ln n.Comment: Long version of arXiv:1304.561
On reliable computation by noisy random Boolean formulas
We study noisy computation in randomly generated k-ary Boolean formulas. We
establish bounds on the noise level above which the results of computation by
random formulas are not reliable. This bound is saturated by formulas
constructed from a single majority-like gates. We show that these gates can be
used to compute any Boolean function reliably below the noise bound.Comment: A new version with improved presentation accepted for publication in
IEEE TRANSACTIONS ON INFORMATION THEOR
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
The distribution of height and diameter in random non-plane binary trees
This study is dedicated to precise distributional analyses of the height of
non-plane unlabelled binary trees ("Otter trees"), when trees of a given size
are taken with equal likelihood. The height of a rooted tree of size is
proved to admit a limiting theta distribution, both in a central and local
sense, as well as obey moderate as well as large deviations estimates. The
approximations obtained for height also yield the limiting distribution of the
diameter of unrooted trees. The proofs rely on a precise analysis, in the
complex plane and near singularities, of generating functions associated with
trees of bounded height
On density of truth of the intuitionistic logic in one variable
In this paper we focus on the intuitionistic propositional logic with one propositional variable. More precisely we consider the standard fragment of this logic and compute the proportion of tautologies among all formulas. It turns out that this proportion is different from the analog one in the classical logic case
Subcritical pattern languages for and/or trees
Let denote the density of and/or trees defining a boolean function within the set of and/or trees with fixed number of variables . We prove that there exists constant such that when , where denote the complexity of (i.e. the size of a minimal and/or tree defining ). This theorem has been conjectured by Danièle Gardy and Alan Woods together with its counterpart for distribution defined by some critical Galton-Watson process. Methods presented in this paper can be also applied to prove the analogous property for
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Probability, Trees and Algorithms
The subject of this workshop were probabilistic aspects of algorithms for fundamental problems such as sorting, searching, selecting of and within data, random permutations, algorithms based on combinatorial trees or search trees, continuous limits of random trees and random graphs as well as random geometric graphs. The deeper understanding of the complexity of such algorithms and of shape characteristics of large discrete structures require probabilistic models and an asymptotic analysis of random discrete structures. The talks of this workshop focused on probabilistic, combinatorial and analytic techniques to study asymptotic properties of large random combinatorial structures
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 , take a sequence of random (rooted) trees of increasing sizes, say, and label each of these random trees uniformly at random in order to get a random Boolean expression on variables.We prove that, under rather weak local conditions on the sequence of random trees , the distribution induced on Boolean functions by this procedure converges as . In particular, we characterise two different behaviours of this limit distribution depending on the shape of the local limit of : 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)