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Dimensionality and randomness

George Barmpalias, Xiaoyan Zhang

TL;DR

This work quantifies how distributing the bits of a random real into arrays and trees induces a loss of randomness measured by deficiency in Kolmogorov complexity. It develops probabilistic constructions showing the existence of $n/\! (\log n)^2$-fat incompressible structures and connects width growth to the required deficiency, establishing both upper and lower bounds. The authors demonstrate that random reals can compute perfect weakly-incompressible trees, while certain strong incompressible structures (e.g., proper pruned incompressible trees) would compute $\emptyset'$ and hence are not obtainable from all random reals. They further extend negligibility and depth phenomena beyond $\Pi^0_1$ classes to $\Pi^0_2$ classes, providing topological and computational separations, including the existence of positive incompressible trees not computing any random real. The results raise open questions about defining depth for higher arithmetical levels and about precise deficiency-width trade-offs in computable structures derived from randomness.

Abstract

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.

Dimensionality and randomness

TL;DR

This work quantifies how distributing the bits of a random real into arrays and trees induces a loss of randomness measured by deficiency in Kolmogorov complexity. It develops probabilistic constructions showing the existence of -fat incompressible structures and connects width growth to the required deficiency, establishing both upper and lower bounds. The authors demonstrate that random reals can compute perfect weakly-incompressible trees, while certain strong incompressible structures (e.g., proper pruned incompressible trees) would compute and hence are not obtainable from all random reals. They further extend negligibility and depth phenomena beyond classes to classes, providing topological and computational separations, including the existence of positive incompressible trees not computing any random real. The results raise open questions about defining depth for higher arithmetical levels and about precise deficiency-width trade-offs in computable structures derived from randomness.

Abstract

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.
Paper Structure (11 sections, 32 theorems, 79 equations, 1 figure)

This paper contains 11 sections, 32 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Relevant literature
  4. Incompressible arrays
  5. Incompressible and weakly-incompressible trees
  6. Nearly incompressible trees
  7. Negligibility and depth of classes of reals
  8. Negligibility beyond effectively closed classes
  9. Depth beyond effectively closed classes
  10. Positive incompressible trees
  11. Conclusion

Key Result

Theorem 1.1

Every random real computes an $n/(\log n)^2$-fat set of incompressible strings.

Figures (1)

  • Figure 1: Loss of complexity when arranging a real into columns: black and gray columns indicate segments of high and low complexity, respectively.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10: Essentially luliumajma
  • ...and 44 more