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Scheme-theoretic Approach to Computational Complexity I. The Separation of P and NP

Ali Çivril

TL;DR

This work introduces a scheme-theoretic viewpoint for computational complexity by parameterizing problem instances as moduli schemes via the Hilbert scheme. It defines an extended amplifying functor that links sub-problems to geometric objects, enabling connectivity-based lower bounds on computation and a structural route to separate $P$ from $NP$. For $3$-SAT, the authors construct a prime homogeneous simple sub-problem with exponentially many instances, and derive a concrete lower bound $1.296839^n$ for deterministic solving time, implying $P eq NP$ and related consequences such as $NP subseteq P/ ext{poly}$ and $BPP = P$ under the Exponential Time Hypothesis. The paper thus lays foundational groundwork bridging algebraic geometry and computational complexity, offering a novel method to obtain exponential lower bounds via geometric arguments, while acknowledging current limitations in extending to $2$-SAT and outlining future directions. The results, if validated, would significantly impact complexity theory by providing a new analytic framework for hardness via moduli spaces.

Abstract

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP. In particular, we show that no deterministic algorithm can solve \textsf{3-SAT} in time less than $1.296839^n$ in the worst case.

Scheme-theoretic Approach to Computational Complexity I. The Separation of P and NP

TL;DR

This work introduces a scheme-theoretic viewpoint for computational complexity by parameterizing problem instances as moduli schemes via the Hilbert scheme. It defines an extended amplifying functor that links sub-problems to geometric objects, enabling connectivity-based lower bounds on computation and a structural route to separate from . For -SAT, the authors construct a prime homogeneous simple sub-problem with exponentially many instances, and derive a concrete lower bound for deterministic solving time, implying and related consequences such as and under the Exponential Time Hypothesis. The paper thus lays foundational groundwork bridging algebraic geometry and computational complexity, offering a novel method to obtain exponential lower bounds via geometric arguments, while acknowledging current limitations in extending to -SAT and outlining future directions. The results, if validated, would significantly impact complexity theory by providing a new analytic framework for hardness via moduli spaces.

Abstract

We lay the foundations of a new theory for algorithms and computational complexity by parameterizing the instances of a computational problem as a moduli scheme. Considering the geometry of the scheme associated to 3-SAT, we separate P and NP. In particular, we show that no deterministic algorithm can solve \textsf{3-SAT} in time less than in the worst case.

Paper Structure

This paper contains 9 sections, 9 theorems, 28 equations, 11 tables.

Table of Contents

  1. Introduction
  2. Computational Problems and the Extended Amplifying Functor
  3. Computational Problems
  4. Representability of the Hilbert Functor
  5. Reductions and Prime Homogeneous Simple Sub-problems
  6. The Extended Amplifying Functor
  7. Lower Bounds via Prime Homogeneous Simple Sub-problems
  8. 3-SAT: The Separation of P and NP
  9. Final Remarks

Key Result

Theorem 2.1

Let $X$ be a projective scheme over $\overline{\mathbb{F}}_2$. Then for every polynomial $P \in \mathbb{Q}[x]$, there exists a projective scheme $\textrm{\normalfont{Hilb}}^P(X)$ over $\overline{\mathbb{F}}_2$, which represents the functor $\mathcal{H}_X^P$. Furthermore, the Hilbert functor $\mathca

Theorems & Definitions (27)

  • Theorem 2.1: Grothendieck62
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 17 more