Metric Structures and Probabilistic Computation

TL;DR

This paper explores how probabilistic computation can be applied to continuous first-order logic structures, establishing an effective completeness theorem that links decidability with probabilistic decidability.

Contribution

It introduces a framework connecting probabilistic computation with continuous logic, demonstrating that decidable theories have probabilistically decidable models.

Findings

Every decidable continuous first-order theory has a probabilistically decidable model.

Application of the framework to various structures and complexity problems.

Extension of classical computability concepts to probabilistic settings.

Abstract

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role for structures described in continuous first-order logic. The main result of this paper is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. Later sections give examples of the application of this framework to various classes of structures, and to some problems of computational complexity theory.