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- W4293497055 abstract "Let $cT$ be a monadic-second order class of finite trees, and let $bT(x)$ be its (ordinary) generating function, with radius of convergence $rho$. If $rho ge 1$ then $cT$ has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators $(n)$ and $(ge n)$. Using this, one has an explicit expression for $bT(x)$ in terms of the initial functions $x$ and $xcdot big(1-x^nbig)^{-1}$, the operations of addition and multiplication, and the P'olya exponentiation operators $sE_n, sE_{ge n}$. Let $cF$ be a monadic-second order class of finite forests, and let $bF(x)=sum_n f(n) x^n$ be its (ordinary) generating function. Suppose $cF$ is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class $cT$ of trees in $cF$, 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 $cF$ has a monadic second-order 0--1 law iff the radius of convergence of $bF(x)$ is 1 iff the radius of convergence of $bT(x)$ is $ge 1$." @default.
- W4293497055 created "2022-08-29" @default.
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- W4293497055 date "2010-04-07" @default.
- W4293497055 modified "2023-09-27" @default.
- W4293497055 title "Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law" @default.
- W4293497055 doi "https://doi.org/10.48550/arxiv.1004.1128" @default.
- W4293497055 hasPublicationYear "2010" @default.
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