Principle of Conservation of Computational Complexity

TL;DR

This paper introduces the principle of conservation of computational complexity, linking problem solution space and decision transfer, and discusses implications for P vs NP and the halting problem.

Contribution

It formulates a new principle relating computational complexity to decision transfer, offering insights into P vs NP and undecidability.

Findings

Demonstrates the principle using SAT problem

Provides an alternative explanation for halting problem undecidability

Suggests P ≠ NP based on complexity conservation

Abstract

In this manuscript, we derive the principle of conservation of computational complexity. We measure computational complexity as the number of binary computations (decisions) required to solve a problem. Every problem then defines a unique solution space measurable in bits. For an exact result, decisions in the solution space can neither be predicted nor discarded, only transferred between input and algorithm. We demonstrate and explain this principle using the example of the propositional logic satisfiability problem ($SAT$). It inevitably follows that $SAT \not\in P \Rightarrow P\neq NP$. We also provide an alternative explanation for the undecidability of the halting problem based on the principle.