Multihead Finite-State Compression

TL;DR

This paper introduces multihead finite-state compression, a novel model where multiple read heads compress sequences based on finite-state rules, linking compression ratios to sequence dimensions.

Contribution

It generalizes finite-state compression to multiple heads and establishes a fundamental theorem connecting compression ratios with sequence predimensions.

Findings

The infimum of compression ratios equals the sequence's $h$-head finite-state predimension.

The overall infimum over all $h$ equals the sequence's multihead finite-state dimension.

The model extends the theoretical understanding of sequence compressibility with multiple read heads.

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.