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An Additively Optimal Interpreter for Approximating Kolmogorov Prefix Complexity.

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On the Empirical Agreement Between Compression and Program-Execution Approaches to Algorithmic Complexity: A

Zoe Leyva-Acosta1, Eduardo Acuña Yeomans1,2, Francisco Hernández-Quiroz3

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Practical estimators for algorithmic complexity, like compression-based and program-execution-based methods (Coding Theorem Method, CTM), show weak correlations. This suggests they capture different structural aspects, highlighting the need for explicit controls in complexity analysis.

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Block Decomposition MethodCoding Theorem Methodalgorithmic complexitycompresion-based estimates

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Area of Science:

  • Theoretical Computer Science
  • Information Theory
  • Algorithmic Complexity

Background:

  • Algorithmic complexity is crucial but incomputable, leading to practical estimators.
  • Existing estimators (compression-based, program-execution-based) lack understood correspondence.
  • The Coding Theorem Method (CTM) is a prominent program-execution-based estimator.

Purpose of the Study:

  • To systematically compare compression-based and program-execution-based algorithmic complexity estimators.
  • To investigate the correspondence between these two families of estimators across computational models.
  • To provide a framework for assessing cross-paradigm agreement in algorithmic complexity estimation.

Main Methods:

  • Developed a comparative framework using the Block Decomposition Method (BDM).
  • Extended CTM-based estimates to longer strings for direct comparison.
  • Introduced a control estimator (BDMId) to isolate block structure effects.

Main Results:

  • Cross-paradigm correlations were weak and decreased with model resolution.
  • Correlations vanished in long strings, with global correlations explained by length effects.
  • The control estimator (BDMId) outperformed BDM in low-resolution models, suggesting CTM information can hinder agreement.

Conclusions:

  • Compression-based and program-execution-based estimators capture fundamentally different structural aspects.
  • The study provides a methodology for assessing cross-paradigm correspondence in algorithmic complexity.
  • Emphasizes the importance of explicit controls in empirical comparisons of algorithmic complexity.