5 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
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
Associative and commutative tree representations for Boolean functions
International audienceSince the 1990s, the probability distribution on Boolean functions, induced by some random formulas built upon the connectives And and Or, has been intensively studied. These formulas rely on plane binary trees. We extend all the results, in particular the relation between the probability and the complexity of a function, to more general formula structures: non-binary or non-plane trees. These formulas satisfy the natural properties of associativity and commutativity