4 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
In the full propositional logic, 5/8 of classical tautologies are intuitionistically valid
AbstractWe present a quantitative comparison of classical and intuitionistic logics, based on the notion of density, within the framework of several propositional languages. In the most general case–the language of the “full propositional system”–we prove that the fraction of intuitionistic tautologies among classical tautologies of size n tends to 5/8 when n goes to infinity. We apply two approaches, one with a bounded number of variables, and another, in which formulae are considered “up to the names of variables”. In both cases, we obtain the same results. Our results for both approaches are derived in a unified way based on structural properties of formulae. As a by-product of these considerations, we present a characterization of the structures of almost all random tautologies