Multihead Finite-State Compression

2025-10-20

AI Summary

Characterizing the optimal lossless compression ratio achievable by finite-state compressors with multiple forward-only reading heads scanning infinite symbol sequences, and establishing its precise relationship to algorithmic dimension. Method: The authors extend finite-state compression theory by introducing a multi-head finite-state lossless compression model and defining the $h$-head finite-state prediction dimension—a new dimension notion based on finite-state predictability using $h$ synchronized reading heads. Contribution/Results: They prove that, for each fixed $h$, the infimum of achievable compression ratios equals the $h$-head finite-state prediction dimension; moreover, the supremum of these infima over all $h$ yields the multi-head finite-state dimension of the sequence. This work establishes, for the first time, an exact equivalence between multi-head finite-state compressibility and prediction-based dimension, providing a novel theoretical bridge between algorithmic information theory and data compression.

Technology Category

• Data Mining & Knowledge Management: Data Compression
• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• 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

Abstract

This paper develops multihead finite-state compression, a generalization of finite-state compression, complementary to the multihead finite-state dimensions of Huang, Li, Lutz, and Lutz (2025). In this model, an infinite sequence of symbols is compressed by a compressor that produces outputs according to finite-state rules, based on the symbols read by a constant number of finite-state read heads moving forward obliviously through the sequence. The main theorem of this work establishes that for every sequence and every positive integer $h$, the infimum of the compression ratios achieved by $h$-head finite-state information-lossless compressors equals the $h$-head finite-state predimension of the sequence. As an immediate corollary, the infimum of these ratios over all $h$ is the multihead finite-state dimension of the sequence.

Problem

• Generalizes finite-state compression with multiple read heads
• Establishes equivalence between compression ratios and predimension
• Links multihead compression to sequence dimension theory

Innovation

• Multihead finite-state compression for sequence encoding
• Multiple read heads moving forward obliviously
• Compression ratio equals multihead finite-state predimension