What is a POLYNOMIAL-TIME Computable L2-Function?

2026-01-23

AI Summary

This work investigates how to appropriately define polynomial-time computability for functions in the L² space. The authors propose two natural notions of polynomial-time computability for L² functions and, for the first time, systematically compare their computational complexity relationships. By integrating models of real-number computation, functional analysis in L² spaces, and classical complexity theory, they demonstrate that—under the standard complexity-theoretic assumption that FP₁ does not contain #P₁—the two definitions are incomparable. This result highlights the intrinsic diversity of computability concepts in continuous computation and provides a foundational contribution to the theory of computational complexity over continuous domains.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Learning Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Computational social science
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation

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).

Problem

• polynomial-time computability
• L2 functions
• computational complexity
• FP_1
• #P_1

Innovation

• polynomial-time computability
• L2 functions
• computational complexity
• FP_1
• #P_1