Multihead Finite-State Compression
Neil Lutz
TL;DR
This work extends finite-state compression to multihead finite-state compressors (h-FSCs) and proves an exact equivalence between the infimum compression ratios of $h$-head, information-lossless compressors and the $h$-head predimensions, thereby characterizing the multihead FS dimension. It develops two constructive directions: (i) from compressors to gamblers, via head-synchronizing block constructions that bound compressor output by gambler martingales, and (ii) from gamblers to compressors, via non-vanishing gambler→information-lossless compressors that mirror betting behavior through Shannon-Fano– Elias coding. Consequently, for each fixed $h$, $\dim_{\textup{FS}}^{(h)}(S)=\rho_{\textup{FS}}^{(h)}(S)$ and $\mathop{\mathrm{Dim}}_{\textup{FS}}^{(h)}(S)=R_{\textup{FS}}^{(h)}(S)$, and taking infimum over $h$ yields the multihead FS dimension equal to its predictive analogue. The results generalize the FSD/AHLM equivalences to the multihead setting and establish a strict MFSD hierarchy where more heads improve predictive power for suitable sequences.
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.