Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

Source : oai:HAL:hal-00990562v1

Volume: Vol. 14 no. 1

Section: Automata, Logic and Semantics

Published on: May 8, 2012

Submitted on: April 1, 2010

Keywords: Trees,forests,monadic second-order class,ordinary generating function,radius of convergence,Compton equations,0-1 law,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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