History-Deterministic B\"uchi Automata are Succinct

2026-03-05

AI Summary

This work resolves a long-standing open problem concerning whether history-deterministic Büchi automata (HD-BA) can be strictly more succinct than equivalent deterministic Büchi automata (DBA). By combining formal theoretical analysis with computer-aided verification, the authors construct an HD-BA with only 65 states and rigorously prove that its language cannot be recognized by any DBA with 65 or fewer states. This result provides the first concrete evidence that HD-BAs can achieve strictly lower state complexity than DBAs for certain ω-regular languages, thereby establishing the theoretical advantage of history determinism in terms of representational succinctness. The finding conclusively settles a question that has remained open for over a decade in the field of automata theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

Abstract

We describe a history-deterministic B\"uchi automaton that has strictly less states than every language-equivalent deterministic B\"uchi automaton. This solves a problem that had been open since the introduction of history-determinism and actively investigated for over a decade. Our example automaton has 65 states, and proving its succinctness requires the combination of theoretical insights together with the aid of computers.

Problem

• history-determinism
• Büchi automata
• succinctness
• state complexity
• determinization

Innovation

• history-determinism
• Büchi automata
• succinctness
• state complexity
• automata theory