Differential information theory (2111.04335v2)

Abstract: This paper presents a new foundational approach to information theory based on the concept of the information efficiency of a recursive function, which is defined as the difference between the information in the input and the output. The theory allows us to study planar representations of various infinite domains. Dilation theory studies the information effects of recursive operations in terms of topological deformations of the plane. I show that the well-known class of finite sets of natural numbers behaves erratically under such transformations. It is subject to phase transitions that in some cases have a fractal nature. The class is \emph{semi-countable}: there is no intrinsic information theory for this class and there are no efficient methods for systematic search. There is a relation between the information efficiency of the function and the time needed to compute it: a deterministic computational process can destroy information in linear time, but it can only generate information at logarithmic speed. Checking functions for problems in $NP$ are information discarding. Consequently, when we try to solve a decision problem based on an efficiently computable checking function, we need exponential time to reconstruct the information destroyed by such a function. At the end of the paper I sketch a systematic taxonomy for problems in $NP$.

• The paper introduces a novel metric, information efficiency, that quantifies the differential loss and generation of information in recursive functions.
• It employs planar representations and dilation theory to uncover phase transitions and fractal-like behaviors in infinite data domains.
• The framework connects computational processes to complexity theory, suggesting that NP problems may require exponential time to recover lost information.

Differential Information Theory: A New Approach to Understanding Information and Computation

The paper by Pieter Adriaans introduces a novel foundational approach to information theory, termed Differential Information Theory (DIT). This framework is predicated on redefining our understanding of information in the context of recursive functions. The central thesis explores the differential between the information present in the input and output of these functions, facilitating an examination of information flow through computational processes.

Key Contributions

1. Information Efficiency: The concept of information efficiency is introduced, quantifying the difference in information content between inputs and outputs of recursive functions. One of the key insights is that in deterministic computations, information is destroyed linearly, whereas it can only be generated logarithmically.
2. Planar Representations and Dilation Theory: Adriaans extends traditional information theory using planar representations, arguing that infinite data domains, when transformed recursively, undergo phase transitions with potentially fractal characteristics. The paper of these transformations, coined as dilation theory, reveals the unpredictable behavior of finite natural number sets under these transformations.
3. Relation to Complexity Classes: A significant implication of this theory is its connection to complexity theory, specifically how deterministic functions relating to NP problems discard information. Notably, decision problems relying on efficiently computable checking functions necessitate exponential time to reconstruct the obliterated information.
4. Understanding NP: DIT provides a framework for understanding the complexities within the NP class by proposing systematic taxonomies based on data domain expressiveness, domain density, and the information efficiency of checking functions. This paradigm challenges traditional views, offering a fresh lens through which NP problems can be deconstructed.

Theoretical and Practical Implications

Theoretical Implications:

• Transfinite Information: The paper hints at the existence of "small infinities" or transfinite information measures that extend conventional information theoretic paradigms.
• Semi-Countable Sets: The paper introduces the notion of semi-countable sets, challenging the traditional binary distinction between countable and uncountable sets. This lack of intrinsic structure complicates attempts to standardize information measurements across these domains.

Practical Implications:

• Algorithm Design: Understanding the differential information efficiency offers a potent tool for developing algorithms, especially in contexts requiring optimization for information retention or minimization of loss.
• Data Compression and Encoding: The insights provided by DIT have potential applications in compressing data without significant loss of information, which could revolutionize fields like data storage and transmission.

Future Directions

The foundational nature of DIT opens several avenues for further exploration:

• Extension to Multi-Dimensional Domains: While the paper focuses on uni- and bi-dimensional data, extending these concepts to higher-dimensional spaces could yield further insights into complex data structures.
• Integration with Stochastic Models: Although DIT is fundamentally non-stochastic, its integration or comparison with stochastic models could provide a richer understanding of information dynamics in probabilistic environments.
• Real-World Applications: Further work is needed to apply these theoretical insights to practical problems, such as cryptography, AI model training, and complex systems analysis.

In conclusion, Pieter Adriaans’ Differential Information Theory provides a rich, nuanced understanding of information interactions in computational processes. By focusing on the differential nature of information flow, this framework allows for a deeper exploration of computational complexity and establishes a groundwork for future studies in both theoretical and applied domains of computer science.