度量结构与概率计算

摘要：

连续一阶逻辑用于将模型理论分析应用于分析结构（例如希尔伯特空间、巴拿赫空间、概率空间等）。经典可计算模型论用于研究可以用经典一阶逻辑描述的数学对象的算法结构。本文表明，概率计算（有时称为随机计算）可以发挥类似的作用，在连续一阶逻辑中描述的结构。本文的主要结果是一个有效的完备性定理，证明了每一个可判定的连续一阶理论都有一个概率可判定的模型。后面的章节给出了这个框架应用于各类结构和计算复杂性理论的一些问题的例子。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。