Using games and universal trees to characterise the nondeterministic index of tree languages

2025-04-23

AI Summary

This paper addresses the long-standing open problem of parity index determination for regular tree languages: given a regular tree language $L$ and a priority set $J$, decide whether $L$ is recognized by a nondeterministic parity tree automaton with *exactly* the priority set $J$ (i.e., $J$-feasibility). Prior work resolved only isolated special cases. We introduce a counter-augmented variant of Lehtinen’s register games, yielding an exact characterization of $J$-feasibility. We parameterize attractor decomposition complexity via the $n$-Strahler number and establish equivalences among guided automata, universal trees, and $J$-feasibility. Our main contributions are: (1) a novel decidability characterization based on the enhanced register games; (2) a universal-tree representation parameterized by the $n$-Strahler number; and (3) a proof that Büchi feasibility is equivalent to the existence of a finite universal tree.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The parity index problem of tree automata asks, given a regular tree language $L$ and a set of priorities $J$, is $L$ $J$-feasible, that is, recognised by a nondeterministic parity automaton with priorities $J$? This is a long-standing open problem, of which only a few sub-cases and variations are known to be decidable. In a significant but technically difficult step, Colcombet and L""oding reduced the problem to the uniform universality of distance-parity automata. In this article, we revisit the index problem using tools from the parity game literature. We add some counters to Lehtinen's register game, originally used to solve parity games in quasipolynomial time, and use this novel game to characterise $J$-feasibility. This provides a alternative proof to Colcombet and L""oding's reduction. We then provide a second characterisation, based on the notion of attractor decompositions and the complexity of their structure, as measured by a parameterised version of their Strahler number, which we call $n$-Strahler number. Finally, we rephrase this result using the notion of universal tree extended to automata: a guidable automaton recognises a $[1,2j]$-feasible language if and only if it admits a universal tree with $n$-Strahler number $j$, for some $n$. In particular, a language recognised by a guidable automaton $A$ is B""uchi-feasible if and only if there is a uniform bound $nin mathbb{N}$ such that all trees in the language admit an accepting run with an attractor decomposition of width bounded by $n$, or, equivalently, if and only $A$ admits a extit{finite} universal tree. While we do not solve the decidability of the index problem, our work makes the state-of-the-art more accessible and brings to light the deep relationships between the $J$-feasibility of a language and attractor decompositions, universal trees and Lehtinen's register game.

Problem

• Characterizing J-feasibility of tree languages using games
• Relating J-feasibility to attractor decompositions and Strahler numbers
• Exploring connections between universal trees and parity automata

Innovation

• Enhanced Lehtinen's register game with counters
• Characterized feasibility via attractor decompositions complexity
• Linked guidable automata to finite universal trees