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Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings

Andrei Romashchenko

TL;DR

This work investigates whether the algorithmic information in a triple of correlated strings can be halved by conditioning on a single string, and shows a fundamental obstruction in the prime-field case. The authors construct an explicit incidence-based triple from a finite projective plane over a finite field, proving that no $z$ can halve all three conditional complexities $C(a|z)$, $C(b|z)$, and $C(c|z)$ to about half of their unconditional values. In contrast, when the ambient field contains a large subfield, a balanced halving and a corresponding secret-key protocol exist, illustrating an algebraic separation in information-theoretic behavior tied to subfield structure. The results yield concrete lower bounds on secret-key agreement in certain communication models and connect discrete geometry with information theory, highlighting how geometric incidence properties constrain information extraction. Overall, the paper reveals algebraic barriers to efficient information exchange and demonstrates that the possibility of halving complexities can crucially depend on the arithmetic structure of the underlying field.

Abstract

We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings $(a, b, c)$, there exists a string $z$ such that each conditional Kolmogorov complexity $C(a|z), C(b|z), C(c|z)$ is approximately half of the corresponding unconditional Kolmogorov complexity. We give a negative answer to this question by constructing a triple $(a, b, c)$ for which no such string $z$ exists. Moreover, we construct a fully explicit example of such a tuple. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on bounds for point-line incidences over prime fields. As an application, we show that this impossibility yields lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in information-theoretic behavior between fields with and without proper subfields.

Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings

TL;DR

This work investigates whether the algorithmic information in a triple of correlated strings can be halved by conditioning on a single string, and shows a fundamental obstruction in the prime-field case. The authors construct an explicit incidence-based triple from a finite projective plane over a finite field, proving that no can halve all three conditional complexities , , and to about half of their unconditional values. In contrast, when the ambient field contains a large subfield, a balanced halving and a corresponding secret-key protocol exist, illustrating an algebraic separation in information-theoretic behavior tied to subfield structure. The results yield concrete lower bounds on secret-key agreement in certain communication models and connect discrete geometry with information theory, highlighting how geometric incidence properties constrain information extraction. Overall, the paper reveals algebraic barriers to efficient information exchange and demonstrates that the possibility of halving complexities can crucially depend on the arithmetic structure of the underlying field.

Abstract

We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings , there exists a string such that each conditional Kolmogorov complexity is approximately half of the corresponding unconditional Kolmogorov complexity. We give a negative answer to this question by constructing a triple for which no such string exists. Moreover, we construct a fully explicit example of such a tuple. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on bounds for point-line incidences over prime fields. As an application, we show that this impossibility yields lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in information-theoretic behavior between fields with and without proper subfields.

Paper Structure

This paper contains 22 sections, 22 theorems, 85 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Existential proof vs. explicit constructions
  3. The main construction
  4. Application: impossibility results for secret key agreement
  5. Techniques
  6. Organization
  7. Notation
  8. Halving complexities of two strings
  9. Typical incidences in a projective plane
  10. Graphs with highly non-extractible mutual information
  11. A non-extractability result via random graphs
  12. Typical incidences in a projective plane
  13. Secret key agreement
  14. Specific input data: secret key agreement with a typical incidence from a finite plane
  15. Prime field: a negative result
  16. ...and 7 more sections

Key Result

Proposition 1

For every tuple of finite binary strings $(y_1,\ldots, y_n)$ and for every $z$ (finite or infinite), there exists a tuple $(x_1,\ldots, x_n)$ such that for all index sets $1\le i_1 < i_2 < \ldots < i_s \le n$, In other words, if a vector of $2^n-1$ reals serves as the complexity profile for some $(y_1,\ldots, y_n)$ conditional on some oracle $z$, then a vector of almost the same numbers serves as

Figures (3)

  • Figure 1: Complexity profile for $(x,y,z)$ from Theorem \ref{['th:cmrsv']}.
  • Figure 2: Complexity profile for $(x,y,z)$ from Theorem \ref{['th:ska-negative']}, cf. Fig. \ref{['fig:lemma-halving-example']}.
  • Figure 3: Domain and codomain of the mapping \ref{['eq:continuous-mapping']}.

Theorems & Definitions (51)

  • Proposition 1: folklore
  • Remark 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 2
  • Remark 2
  • Definition 1
  • Proposition 3
  • proof
  • ...and 41 more