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What is a POLYNOMIAL-TIME Computable L2-Function?

Aras Bacho, Svetlana Selivanova, Martin Ziegler

TL;DR

The paper defines two natural notions of polynomial-time computability for $L^{2}$ functions and proves they are incomparble unless $FP_1$ contains $\#P_1$, contrasting them with existing polynomial-time notions for continuous functions. It develops three $L^{2}$-specific notions (Fourier-based, step-based, and mean-based) and analyzes their relationships, including $\#P$-based equivalences and separations via encoding discrete problems into continuous ones. The results show fundamental limits and separations among continuous-function and $L^{2}$-function computability notions, with a key application to the heat equation: polynomial-time computable initial data yield polynomial-time computable solutions for polynomially computable times. This work connects Parseval structure, gap- and counting- complexity classes, and average-case notions to the computability of solutions to PDEs in a quantitative, complexity-theoretic framework.

Abstract

We give two natural definitions of polynomial-time computability for L2 functions; and we show them incomparable (unless complexity class FP_1 includes #P_1).

What is a POLYNOMIAL-TIME Computable L2-Function?

TL;DR

The paper defines two natural notions of polynomial-time computability for functions and proves they are incomparble unless contains , contrasting them with existing polynomial-time notions for continuous functions. It develops three -specific notions (Fourier-based, step-based, and mean-based) and analyzes their relationships, including -based equivalences and separations via encoding discrete problems into continuous ones. The results show fundamental limits and separations among continuous-function and -function computability notions, with a key application to the heat equation: polynomial-time computable initial data yield polynomial-time computable solutions for polynomially computable times. This work connects Parseval structure, gap- and counting- complexity classes, and average-case notions to the computability of solutions to PDEs in a quantitative, complexity-theoretic framework.

Abstract

We give two natural definitions of polynomial-time computability for L2 functions; and we show them incomparable (unless complexity class FP_1 includes #P_1).
Paper Structure (14 sections, 9 theorems, 33 equations, 1 figure)

This paper contains 14 sections, 9 theorems, 33 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Computability and Complexity of Continuous Functions
  4. What is a Polynomial-Time Computable L2 Function?
  5. On Step-Computability in Mean
  6. Comparing Notions of Polynomial-Time Computability
  7. Proof of Theorem \ref{['t:SharpP']}: #P-equivalence
  8. Encoding Discrete Problems into Continuous Ones
  9. Proof of Theorem \ref{['t:FourierNotStep']}: Fourier but not Step polytime
  10. Proof of Theorem \ref{['t:StepNotFourier']}: Step but not Fourier polytime
  11. Proof of Theorem \ref{['t:Continuous1']}
  12. Proof Sketches of Theorems \ref{['t:Continuous2']}+\ref{['t:Continuous3']}+\ref{['t:Continuous4']}
  13. Applications
  14. Heat Equation

Key Result

lemma 1

Let $\mathcal{A}$ denote an algorithm computing on input $s\in\mathbb{N}$ an infinite sequence of Gaussian rational approximations $(h_{s,p})_{p=0}^{\infty}$ up to absolute error $2^{-p}$ to the $s$-th entry of some finite or infinite complex sequence $(h_{s})_{s\geq0}$. Let $\text{\rm time}\xspace_ In particular, for any probability distribution w.r.t. $s$, expected values $\mathbb{E}_s$ satisfy:

Figures (1)

  • Figure 1: Illustrating the construction of $g$ from the proof of Theorem \ref{['t:StepNotFourier']}

Theorems & Definitions (17)

  • definition 1
  • definition 2
  • lemma 1
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • theorem 6
  • theorem 7
  • ...and 7 more