Coloring rules for finite trees, and probabilities of monadic second order sentences

Abstract

ABSTRACT:A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex trees with a given root color is investigated. As a primary application, if is any monadic second-order sentence, then the fractions Ž . Ž .

Formulae and Asymptotics for Coefficients of Algebraic Functions

“…For a large class of strongly connected positive systems (with a Jacobian condition), it leads to the Airy function, and it is expected that it will also be the case for a class of functional equations allowing negative coefficients. theorem from [53] by calling it the Drmota-Lalley-Woods theorem, since similar results were obtained independently by Lalley [84] and Woods [121].…”

Counting finite models

“…(Additionally, they show that there is no algorithm for deciding, given ip, whether or not /i(<p) -0.) But for monadic second order sentences, the work reported here and in [45] (see Theorem 5.6 below) establishes convergence, without zero-one laws holding. §2.…”

Recursion and Growth Estimates in Renormalizable Quantum Field Theory

“…Nonnegative polynomial recursive systems give the universal law under some reasonable conditions, as was shown independently by Drmota [6], Lalley [16], and Woods [25]. For our purposes the full generality of the above are not necessary and we'll follow the presentation of Flajolet and Sedgewick [7,VII.6.3].…”

“…Then multiplying the system by v and expanding around ρ gives the desired asymptotics. Woods [25] also uses the Jacobian and the largest eigenvalue 1. He continues the analysis on block upper triangular matrices in order to deal with certain nonirreducible cases.…”