Paper: Randomness Extraction Fails for Finite-State Dimension (at LICS 2026)

Authors: Subin Pulari Akhil S

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.78

Abstract

Finite-state dimension, introduced as a finite-state analogue of Hausdorff dimension, quantifies the lower asymptotic density of information in an infinite sequence as perceived by finite-state automata. It admits several equivalent formulations; two particularly useful are via finite-state gambling strategies and via the optimal asymptotic compression ratio achieved by information-lossless finite-state compressors. Normal sequences represent the highest level of algorithmic randomness visible to finite automata, and are exactly those sequences having finite-state dimension equal to 1. This motivates a bounded-memory notion of randomness extraction: can a finite-state transducer, reading a single sequence streamingly, extract a normal output from a single input source? More modestly, can it always transform the input into an output of strictly higher finite-state dimension? Finite-state transducers can perform surprisingly effective one-pass transformations: even with constant memory they can implement variable-length coding schemes including Shannon-Fano coding, remove local redundancy, and increase the apparent randomness rate on many structured or stochastic inputs. We show randomness extraction using transducers is impossible in a strong, explicit form. For every rational s ∈ (0,1), we construct a near linear-time computable binary sequence X with dim_FS(X) = s such that for every finite-state transducer T, the output satisfies dim_FS(T(X)) ≤ s. Thus, for these sequences, finite-state transduction cannot extract normality - indeed it cannot even improve finite-state dimension. Our proof proceeds by a structural analysis of finite-state transducers together with a dimension-preserving diagonal construction that, for each target s, builds a sequence whose organization defeats every such transducer’s attempt to concentrate randomness. The result is a finite-state analogue of Miller’s non-extractability phenomenon for effective dimension, but its proof relies on substantially different techniques, tailored to the finite-state setting. Furthermore, we show that the impossibility persists even with multiple independent input streams. We treat two notions of independence: (i) Kolmogorov-complexity–based independence (via joint prefix complexity), and (ii) a finite-state notion of relative independence, formulated via relative finite-state dimension. By sharp contrast with the effective-dimension setting - where two independent sources suffice for a uniform effective procedure that boosts randomness rate arbitrarily close to 1 - we show that finite-state dimension exhibits no comparable multi-source extraction phenomenon. Specifically, for every rational s ∈ (0,1) and every fixed k ≥ 2, there exist k independent sources, each of finite-state dimension s, such that for every k-input finite-state transducer T, the output satisfies dim_FS(T(X_1,… ,X_k)) ≤ s. Thus, even independent streams do not allow bounded-memory transduction to output a normal sequence or to increase finite-state dimension.

BibTeX

  @InProceedings{PulariS-RandomnessExtractio,
    author = 	 {Subin Pulari and Akhil S},
    title = 	 {Randomness Extraction Fails for Finite-State Dimension},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {78:1--78:26},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.78}
  }