On the asymptotic enumeration of accessible automata

Automata, Logic and Semantic

A large deviations principle for the Maki-Thompson rumour model

We consider the stochastic model for the propagation of a rumour within a population which was formulated by Maki and Thompson. Sudbury established that, as the population size tends to infinity, the proportion of the population never hearing the rumour converges in probability to $0.2032$. Watson later derived the asymptotic normality of a suitably scaled version of this proportion. We prove a corresponding large deviations principle, with an explicit formula for the rate function.Comment: 18 pages, 2 figure

Asymptotic enumeration of Minimal Automata

We determine the asymptotic proportion of minimal automata, within n-state accessible deterministic complete automata over a k-letter alphabet, with the uniform distribution over the possible transition structures, and a binomial distribution over terminal states, with arbitrary parameter b. It turns out that a fraction ~ 1-C(k,b) n^{-k+2} of automata is minimal, with C(k,b) a function, explicitly determined, involving the solution of a transcendental equation.Comment: 12+5 pages, 2 figures, submitted to STACS 201

The impatient collector

In the coupon collector problem with $n$ items, the collector needs a random number of tries $T_{n}\simeq n\ln n$ to complete the collection. Also, after $nt$ tries, the collector has secured approximately a fraction $\zeta_{\infty}(t)=1-e^{-t}$ of the complete collection, so we call $\zeta_{\infty}$ the (asymptotic) \emph{completion curve}. In this paper, for $\nu>0$, we address the asymptotic shape $\zeta (\nu,.)$ of the completion curve under the condition $T_{n}\leq \left( 1+\nu \right) n$, i.e. assuming that the collection is \emph{completed unlikely fast}. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Kor\v{s}unov

On the asymptotic enumeration of accessible automata

We simplify the known formula for the asymptotic estimate of the number of deterministic and accessible automata with n states over a k-letter alphabet. The proof relies on the theory of Lagrange inversion applied in the context of generalized binomial series.CNPq[311909/2009-4