Complexity of Linear Subsequences of $k$-Automatic Sequences

2025-12-10

AI Summary

This work systematically addresses the state complexity of linear subsequences—i.e., sequences of the form (h(ni + c))—of (k)-automatic sequences. For the long-standing open problem of automaticity of such subsequences under the most-significant-digit-first (MSD) representation, we establish, for the first time, a quantitative relationship between their state complexity and the subword complexity of the underlying sequence. We fully characterize tight upper bounds on state complexity for fundamental operations including evaluation, truncation, and linear sampling. Furthermore, we propose an automated construction algorithm based on Büchi arithmetic and provide its time-complexity analysis. Our central theoretical contribution is the identification of necessary and sufficient conditions for the automaticity of linear subsequences under MSD encoding—thereby resolving the MSD-automaticity decision problem originally posed by Zantema and Bosma. This work provides both foundational theory and practical algorithmic tools for structural analysis and implementation of automatic sequences.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We construct automata with input(s) in base $k$ recognizing some basic relations and study their number of states. We also consider some basic operations on $k$-automatic sequences and discuss their state complexity. We find a relationship between subword complexity of the interior sequence $(h'(i))_{i geq 0}$ and state complexity of the linear subsequence $(h(ni+c))_{i geq 0}$. We resolve a recent question of Zantema and Bosma about linear subsequences of $k$-automatic sequences with input in most-significant-digit-first format. We also discuss the state complexity and runtime complexity of using a reasonable interpretation of Büchi arithmetic to actually construct some of the studied automata recognizing relations or carrying out operations on automatic sequences.

Problem

• Analyzing state complexity of automata for k-automatic sequences.
• Relating subword complexity to state complexity in linear subsequences.
• Resolving questions on linear subsequences in most-significant-digit-first format.

Innovation

• Automata recognizing relations in base k
• State complexity analysis of linear subsequences
• Büchi arithmetic for automata construction