The 2-Token Theorem: Recognising History-Deterministic Parity Automata Efficiently

2025-03-31

AI Summary

This paper addresses the decidability of history-determinism for parity automata—a long-standing open problem lacking efficient algorithms, with prior approaches requiring exponential time. We establish the first rigorous characterization: a parity automaton is history-deterministic if and only if it wins the 2-token game—thereby fully settling the property and confirming the Bagnol–Kuperberg conjecture. Leveraging this equivalence, we devise the first polynomial-time (PTIME) decision procedure for history-determinism, reducing the complexity from EXPTIME to PTIME for automata with fixed parity index. Our approach unifies techniques from game theory, parity game solving, and history-determinism analysis. The result provides a foundational algorithmic tool for synthesis, verification, and controller design of infinite-word automata.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Deterministic Planning
• 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-determinism is a restricted notion of nondeterminism in automata, where the nondeterminism can be successfully resolved based solely on the prefix read so far. History-deterministic automata still allow for exponential succinctness in automata over infinite words compared to deterministic automata (Kuperberg and Skrzypczak, 2015), allow for canonical forms unlike deterministic automata (Abu Radi and Kupferman, 2019 and 2020; Ehlers and Schewe, 2022), and retain some of the algorithmic properties of deterministic automata, for example for reactive synthesis (Henzinger and Piterman, 2006; Ehlers and Khalimov, 2024). Despite the topic of history-determinism having received a lot of attention over the last decade, the complexity of deciding whether a parity automaton is history-deterministic has, up till now, remained open. We show that history-determinism for a parity automaton with a fixed parity index can be checked in PTIME, thus improving upon 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-determinism, it suffices to find the winner of the 2-token game on it. They conjectured that this 2-token game based characterisation extends to parity automata. Boker, Kuperberg, Lehtinen, and Skrzypcak showed in 2020 that this conjecture holds for coB""uchi automata as well. We prove Bagnol and Kuperberg's conjecture that the winner of the 2-token game characterises history-determinism on parity automata.

Problem

• Efficiently check history-determinism in parity automata
• Resolve complexity of deciding parity automata history-determinism
• Prove 2-token game characterizes history-determinism for parity automata

Innovation

• Uses 2-token game for history-determinism check
• PTIME complexity for fixed parity index
• Extends conjecture to parity automata