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On SAT information content, its polynomial-time solvability and fixed code algorithms

Maciej Drozdowski

TL;DR

The paper investigates how information-theoretic limits relate to SAT solvability by formalizing a framework with fixed-code algorithms, information conservation, and string-relations representations. It shows that SAT information content grows exponentially with instance size $|I|$, implying polynomial-time solvability would require exponential information access, and that fixed-code approaches cannot achieve this under bitrate constraints. By contrasting SAT with polynomial problems (which can exhibit polynomial information content) and by analyzing non-fixed-code methods, the work highlights fundamental barriers to solving SAT and informs implications for learning-based solvers and metaheuristics. Overall, the results bridge information theory and computational complexity to clarify why SAT remains intractable under common information-restricted models and what this means for future solver design.

Abstract

The amount of information in satisfiability problem (SAT) is considered. SAT can be polynomial-time solvable when the solving algorithm holds an exponential amount of information. It is also established that SAT Kolmogorov complexity is constant. It is argued that the amount of information in SAT grows at least exponentially with the size of the input instance. The amount of information in SAT is compared with the amount of information in the fixed code algorithms and generated over runtime.

On SAT information content, its polynomial-time solvability and fixed code algorithms

TL;DR

The paper investigates how information-theoretic limits relate to SAT solvability by formalizing a framework with fixed-code algorithms, information conservation, and string-relations representations. It shows that SAT information content grows exponentially with instance size , implying polynomial-time solvability would require exponential information access, and that fixed-code approaches cannot achieve this under bitrate constraints. By contrasting SAT with polynomial problems (which can exhibit polynomial information content) and by analyzing non-fixed-code methods, the work highlights fundamental barriers to solving SAT and informs implications for learning-based solvers and metaheuristics. Overall, the results bridge information theory and computational complexity to clarify why SAT remains intractable under common information-restricted models and what this means for future solver design.

Abstract

The amount of information in satisfiability problem (SAT) is considered. SAT can be polynomial-time solvable when the solving algorithm holds an exponential amount of information. It is also established that SAT Kolmogorov complexity is constant. It is argued that the amount of information in SAT grows at least exponentially with the size of the input instance. The amount of information in SAT is compared with the amount of information in the fixed code algorithms and generated over runtime.
Paper Structure (9 sections, 4 theorems, 1 equation, 1 table)

This paper contains 9 sections, 4 theorems, 1 equation, 1 table.

Table of Contents

  1. Introduction
  2. Notions and Postulates
  3. SAT Polynomial-time Solvability
  4. Kolmogorov Complexity of SAT
  5. Amount of Information in SAT
  6. On the Consequences
  7. Polynomial-time Solvability and Information Size
  8. On Polynomial Problems Information Content
  9. On Non-fixed Code Algorithms

Key Result

Theorem 8

The amount of information in SAT grows at least exponentially with instance size.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 4
  • Theorem 8
  • Corollary 9
  • Proposition 10
  • Proposition 11