Automata for Enriched Trees and Applications
Achim Blumensath
TL;DR
This work develops a uniform coalgebraic framework for enriched trees, providing automata–fixed-point logic correspondences that extend prior case-by-case results. By modeling enriched successor structures with polynomial functors and introducing L-automata, it establishes that μL-definable languages are exactly those recognised by L-automata, with robust translations in both directions. The authors extend these ideas to projection operations and a spectrum of MSO variants, obtaining automata characterisations for MSO, WMSO, CMSO, GSO, and CGSO, including well-founded and finite-branching cases. As a key application, they derive a simplified proof of the Theorem of Muchnik and variants for other logics via a Muchnik-iteration framework using a loop-detection extension μL, significantly broadening the decidability toolkit for enriched-tree structures.
Abstract
We study trees where each successor set is equipped with some additional structure. We introduce a family of automaton models for such trees and prove their equivalence to certain fixed-point logics. As a consequence we obtain characterisations of various variants of monadic second-order logic in terms of automata and fixed-point logics. Finally, we use our machinery to give a simplified proof of the Theorem of Muchnik and we derive several variants of this theorem for other logics.