Information Distance (1006.3520v1)

Abstract: While Kolmogorov complexity is the accepted absolute measure of information content in an individual finite object, a similarly absolute notion is needed for the information distance between two individual objects, for example, two pictures. We give several natural definitions of a universal information metric, based on length of shortest programs for either ordinary computations or reversible (dissipationless) computations. It turns out that these definitions are equivalent up to an additive logarithmic term. We show that the information distance is a universal cognitive similarity distance. We investigate the maximal correlation of the shortest programs involved, the maximal uncorrelation of programs (a generalization of the Slepian-Wolf theorem of classical information theory), and the density properties of the discrete metric spaces induced by the information distances. A related distance measures the amount of nonreversibility of a computation. Using the physical theory of reversible computation, we give an appropriate (universal, anti-symmetric, and transitive) measure of the thermodynamic work required to transform one object in another object by the most efficient process. Information distance between individual objects is needed in pattern recognition where one wants to express effective notions of "pattern similarity" or "cognitive similarity" between individual objects and in thermodynamics of computation where one wants to analyse the energy dissipation of a computation from a particular input to a particular output.

• The paper introduces algorithmic information distance as the minimum length program to transform one object into another.
• It demonstrates that maximal program overlap is achievable up to a logarithmic additive term, ensuring robustness in measuring similarity.
• The research extends the concept to reversible computation, linking informational measures with thermodynamic costs in computation.

Overview of Information Distance

The paper "Information Distance" by Bennett, Gács, Li, Vitànyi, and Zurek explores the concept of information distance within the framework of algorithmic information theory. While Kolmogorov complexity offers a measure of information content for individual objects, this work seeks to quantify the information distance between two arbitrary objects, exemplified by strings.

Key Contributions

1. Algorithmic Information Distance: The authors define the information distance between two strings x and y as the length of the shortest binary program that can transform x into y and vice versa. This metric serves as a universal cognitive distance, grounded in the notion that objects with smaller information distances are more similar.
2. Maximal Overlap and Minimal Correlation: The paper examines the conditions under which the information needed to convert between two strings can overlap maximally or exhibit minimal correlation. The results demonstrate that, up to an additive logarithmic term, this overlap can be maximally realized, providing a robust understanding of the informational relationship between strings.
3. Universal Cognitive Distance: By establishing an axiomatic foundation, the authors propose that the information distance is the most natural formalization of cognitive similarity. This measure sufficiently distinguishes between objects within algorithmically computable constraints, thereby revealing cognitive similarities.
4. Reversible Computation: The work extends the concept of information distance to reversible computations, exploring the implications of using reversible Turing machines where computations can be conducted without erasing intermediary information thereby minimizing thermodynamic costs.
5. Thermodynamic Implications: The paper highlights the thermodynamic cost of computation, connecting it to logical irreversibility. The authors align these notions with Landauer’s principle, positing that computation's thermodynamic cost is linked to the information difference between initial and final states of an object.
6. Density Properties: The discussion on density properties explores the distribution of objects within information metric spaces, pondering how many objects can exist within a given informational distance from any specific object. This perspective provides insights into the dimensional characteristics induced by the proposed metrics.

Strong Numerical Results

A pivotal result is the articulation of the relationship between various information distance metrics such as the optimal, reversible, and sum distances, and the rigorous demonstration of the bounds connecting these notions. The conversion theorem and its quantitative structure offer a nuanced understanding of informational overlaps between two objects.

Implications and Future Directions

The implications of this research are manifold, influencing domains such as pattern recognition, cognitive similarity assessment, and the thermodynamics of computation. On a practical note, the universal cognitive distance can inform the design of machine learning algorithms sensitive to intricate content similarities.

Future work could further explore resource-bounded versions of these distances or extend these ideas into other areas of artificial intelligence where the theoretical underpinnings may lead to more efficient algorithms or deeper insights into cognitive processing. The complex interplay between thermodynamic costs and computational processes remains a fertile ground for additional exploration, potentially influencing the development of energy-efficient computing technologies.

Conclusion

Overall, the paper provides a rigorous theoretical foundation for understanding information distance. By integrating concepts from algorithmic information theory and thermodynamics, the authors offer a comprehensive view of how information can be quantified and manipulated, laying the groundwork for advancements in both theoretical and applied computational disciplines.