Spectra and Systems of Equations

TL;DR

This paper extends a method for analyzing the periodicity of generating functions from single equations to systems of equations, with applications to monadic second-order classes of trees and related spectra.

Contribution

It generalizes the analysis of periodicity to systems of equations and simplifies proofs of periodicity and decidability results for monadic second-order classes.

Findings

Spectra of systems of equations can be analyzed for periodicity.

Monadic second-order classes of trees are eventually periodic.

Decidability of the monadic second-order theory of finite trees.

Abstract

In a previous work we introduced an elementary method to analyze the periodicity of a generating function defined by a single equation y=G(x,y). This was based on deriving a single set-equation Y = Gammma(Y) defining the spectrum of the generating function. This paper focuses on extending the analysis of periodicity to generating functions defined by a system of equations y = G(x,y). The final section looks at periodicity results for the spectra of monadic second-order classes whose spectrum is determined by an equational specification - an observation of Compton shows that monadic-second order classes of trees have this property. This section concludes with a substantial simplification of the proofs in the 2003 foundational paper on spectra by Gurevich and Shelah, namely new proofs are given of: (1) every monadic second-order class of $m$-colored functional digraphs is eventually periodic, and (2) the monadic second-order theory of finite trees is decidable.