Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law
Authors: Jason P. Bell ^{1}; Stanley N. Burris ^{2}; Karen A. Yeats ^{1}
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Jason P. Bell;Stanley N. Burris;Karen A. Yeats
1 Department of Mathematics [Burnaby]
2 Department of Pure Mathematics [Waterloo]
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.