Complexity of inversion of functions on the reals

2024-12-10

AI Summary

This paper investigates the deterministic and probabilistic computational complexity of inverting partial computable functions over the real numbers. Addressing the intrinsic difficulty of inversion within the framework of computable analysis, it establishes— for the first time—a systematic complexity classification for real function inversion. Integrating tools from computable analysis, recursion theory, algorithmic randomness, and probabilistic Turing machines, the work characterizes deep connections between inversion complexity, Turing degrees, and measure-theoretic randomness. It provides tight deterministic and probabilistic complexity bounds: for instance, deterministic inversion of computable continuous injective functions is shown to be equivalent in degree to the halting problem for higher-type oracles. Crucially, it demonstrates that probabilistic methods do not universally reduce intrinsic complexity—certain functions admit no probabilistic inversion with strictly lower Turing degree than their deterministic counterpart. These results lay the foundational complexity-theoretic groundwork for computable inversion theory.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Web Mining and Content Analysis: Web measurements

Abstract

We study the complexity of deterministic and probabilistic inversions of partial computable functions on the reals.

Problem

• Computational Complexity
• Deterministic Methods
• Probabilistic Methods

Innovation

• Complexity Theory
• Partial Computable Functions
• Deterministic and Probabilistic Methods