CAS II: Symmetric Partitions as Kolmogorov Models

Problem

This work addresses a gap in algorithmic statistics concerning the explanation of strings by finite sets. It specifically focuses on the limitations of existing models in capturing the complexity of binary strings through finite representations. The paper is a preprint and has not undergone peer review.

Method

The core technical contribution is the introduction of symmetric partitions as hypotheses for binary strings, leveraging Kolmogorov's structure function, which records the smallest model at each complexity level. The method involves:

• Partition Type: Utilizing symmetric partitions to hypothesize about binary strings.
• Group Action: Employing orbit partitions of groups acting on strings to derive insights.
• Connection: Establishing a Galois connection between subgroups and partitions to enhance the understanding of string complexity.
• Structure Function: The symmetric regularity of strings is measured through the structure function, which is designed to capture the nuances of symmetry in the data.
• Symmetric Group: The model asserts that every partition is symmetric, allowing cells to recover all Kolmogorov models.
• GL(n,2): The cells correspond to linearly homogeneous sets, providing a structured approach to analyzing binary strings.
• Linear Symmetry Structure Function: This function is positioned between the sufficiency line and a trivial bound, indicating its potential for nuanced analysis.
• Coordinates: Each group is represented in a Burnside ring, detailing type and placement for enhanced clarity in modeling.
• Refinement: The method includes a refinement process where restriction moves refine partitions using the Mackey formula, allowing for more precise modeling of string complexity.

Results

The results indicate that:

• The cells of the symmetric group recover all Kolmogorov models, although no specific baselines are reported for comparison.
• Cells of cheap partitions are shown to recover exactly the strong models, again with no comparative baselines provided.
• The linear symmetry structure function achieves both edges between the sufficiency line and the trivial bound, with no reported baselines.
• The Maximal Gap Theorem suggests that a small enough search space may overlook simple structures, but no quantitative results are provided to substantiate this claim.

Limitations

The authors note several limitations:

• Stochastic normal strings may exhibit simple structures that are not detectable by the linear symmetry approach.
• Small search spaces may fail to capture simple structures, potentially leading to incomplete models. These limitations highlight the need for further exploration in more complex or larger search spaces to fully understand the implications of the proposed method.

Why it matters

This work has significant implications for downstream research in algorithmic statistics and complexity theory. By introducing symmetric partitions as a viable hypothesis for binary strings, it opens avenues for more robust models that can better explain the underlying structure of data. The connections made between group actions and string complexity could lead to advancements in both theoretical understanding and practical applications in areas such as data compression, information theory, and machine learning.