Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

Jason Bell, Stanley Burris, Karen Yeats
2012 Discrete Mathematics and Theoretical Computer Science DMTCS   unpublished
Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence ρ. If ρ ≥ 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 Pólya exponentiation operators En, E
more » ... on operators En, E (≥n). Let F be a monadic-second order class of finite forests, and let F(x) = n f (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.