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Thermodynamic Perspectives on Computational Complexity: Exploring the P vs. NP Problem

Florian Neukart

TL;DR

This research presents a structured framework for establishing entropy profiles within computational tasks, enabling a clear distinction between P and NP-classified problems, and introduces Entropy-Driven Annealing (EDA) as a new method to decipher the energy landscapes of computational problems.

Abstract

The resolution of the P vs. NP problem, a cornerstone in computational theory, remains elusive despite extensive exploration through mathematical logic and algorithmic theory. This paper takes a novel approach by integrating information theory, thermodynamics, and computational complexity, offering a comprehensive landscape of interdisciplinary study. We focus on entropy, a concept traditionally linked with uncertainty and disorder, and reinterpret it to assess the complexity of computational problems. Our research presents a structured framework for establishing entropy profiles within computational tasks, enabling a clear distinction between P and NP-classified problems. This framework quantifies the 'information cost' associated with these problem categories, highlighting their intrinsic computational complexity. We introduce Entropy-Driven Annealing (EDA) as a new method to decipher the energy landscapes of computational problems, focusing on the unique characteristics of NP problems. This method proposes a differential thermodynamic profile for NP problems in contrast to P problems and explores potential thermodynamic routes for finding polynomial-time solutions to NP challenges. Our introduction of EDA and its application to complex computational problems like the Boolean satisfiability problem (SAT) and protein-DNA complexes suggests a potential pathway toward unraveling the intricacies of the P vs. NP problem.

Thermodynamic Perspectives on Computational Complexity: Exploring the P vs. NP Problem

TL;DR

This research presents a structured framework for establishing entropy profiles within computational tasks, enabling a clear distinction between P and NP-classified problems, and introduces Entropy-Driven Annealing (EDA) as a new method to decipher the energy landscapes of computational problems.

Abstract

The resolution of the P vs. NP problem, a cornerstone in computational theory, remains elusive despite extensive exploration through mathematical logic and algorithmic theory. This paper takes a novel approach by integrating information theory, thermodynamics, and computational complexity, offering a comprehensive landscape of interdisciplinary study. We focus on entropy, a concept traditionally linked with uncertainty and disorder, and reinterpret it to assess the complexity of computational problems. Our research presents a structured framework for establishing entropy profiles within computational tasks, enabling a clear distinction between P and NP-classified problems. This framework quantifies the 'information cost' associated with these problem categories, highlighting their intrinsic computational complexity. We introduce Entropy-Driven Annealing (EDA) as a new method to decipher the energy landscapes of computational problems, focusing on the unique characteristics of NP problems. This method proposes a differential thermodynamic profile for NP problems in contrast to P problems and explores potential thermodynamic routes for finding polynomial-time solutions to NP challenges. Our introduction of EDA and its application to complex computational problems like the Boolean satisfiability problem (SAT) and protein-DNA complexes suggests a potential pathway toward unraveling the intricacies of the P vs. NP problem.
Paper Structure (256 sections, 2 theorems, 80 equations, 2 algorithms)

This paper contains 256 sections, 2 theorems, 80 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background on the P vs. NP Problem
  3. Importance of the Problem in Computational Theory
  4. Overview of Information Theory and Thermodynamics
  5. Rationale for Merging These Domains to Approach the P vs. NP Conundrum
  6. Background
  7. Classical Computational Complexity
  8. Basics of P
  9. Basics of NP
  10. NP-complete Problems
  11. Fundamentals of Information Theory
  12. Entropy
  13. Mutual Information
  14. Shannon's Theorems
  15. Negentropy and the Landauer Limit
  16. ...and 241 more sections

Key Result

theorem 1

For any computational problem $C$, the entropy profiles $H_P(C)$ and $H_{NP}(C)$ corresponding to its classification as a problem in P and NP, respectively, exhibit distinct characteristics that are quantifiable and fundamentally impact the computational approach to solving $C$.

Theorems & Definitions (2)

  • theorem 1
  • theorem 2