History-Deterministic Parity Automata: Games, Complexity, and the 2-Token Theorem

2025-01-21

AI Summary

This paper addresses the decidability of history-determinism (HD) for parity automata. We introduce and prove the “2-token theorem,” which establishes for the first time a polynomial-time complete characterization of HD via the 2-token game—thereby resolving the Bagnol–Kuperberg conjecture and the open problem posed by Kuperberg and Skrzypczak (2015). For fixed parity index, we reduce the HD-decision complexity from EXPTIME to PTIME; for unbounded index, we prove NP-hardness. Furthermore, we present the first polynomial-time determinization algorithm for HD Büchi automata. Our approach integrates game-theoretic reasoning (via the 2-token game), algorithms for solving parity games, and fine-grained complexity analysis. This work unifies and advances both the theoretical foundations and algorithmic realizability of history-determinism in automata theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Behavioral Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

History-deterministic automata are a restricted class of nondeterministic automata where the nondeterminism while reading an input can be resolved successfully based on the prefix read so far. History-deterministic automata are exponentially more succinct than deterministic automata, while still retaining some of the algorithmic properties of deterministic automata, especially in the context of reactive synthesis. This thesis focuses on the problem of checking history-determinism for parity automata. Our main result is the 2-token theorem, due to which we obtain that checking history-determinism for parity automata with a fixed parity index can be checked in PTIME. This improves the naive EXPTIME upper bound of Henzinger and Piterman that has stood since 2006. More precisely, we show that the so-called 2-token game, which can be solved in PTIME for parity automata with a fixed parity index, characterises history-determinism for parity automata. This game was introduced by Bagnol and Kuperberg in 2018, who showed that to decide if a B""uchi automaton is history-deterministic, it suffices to find the winner of the 2-token game on it. They conjectured that this 2-token game based characterisation of history-determinism extends to parity automata. We prove Bagnol and Kuperberg's conjecture that the winner of the 2-token game characterises history-determinism on parity automata. We also give a polynomial time determinisation procedure for history-deterministic B""uchi automata, thus solving an open problem of Kuperberg and Skrzypczak from 2015. This result is a consequence of our proof of the 2-token theorem. Finally, we also show NP-hardness for the problem of checking history-determinism for parity automata when the parity index is not fixed. This is an improvement from the lower bound of solving parity games shown by Kuperberg and Skrzypczak in 2015.

Problem

• Parity Automata
• Historical Determinism
• Algorithmic Complexity

Innovation

• 2-marked theorem
• polynomial-time determinacy
• history-deterministic parity automata