寻址的数学演化：从尤先科的地址编程语言到数组数学——寻址如何演化为索引、数组计算与机器实现的一种数学理论

中文摘要 AI 辅助

本文认为，计算史上一个核心发展是寻址的数学演化。在卡捷琳娜·尤先科的地址编程语言（1958年）与莱诺尔·穆林的数组数学（1988年）相隔的三十年间，寻址从一种用于定位数据的机器级机制，演化为关于索引、归约和机器实现的形式理论，其间经历了肯尼斯·艾弗森的数组代数，以及菲利普·艾布拉姆斯将形状视为架构资源的认识。穆林的理论通过函数{\psi}将索引本身确立为原始操作，数组操作由之复合、归约为指称范式、转换为操作范式，并通过维度提升在硬件上实现。历史论断依据原始材料溯源，推断处予以标明；数学论断以恒等式表述；性能论断标注为已验证、理论确立或初步结果。该研究将编程语言、数组数学与计算机架构统一于一个叙述之中。

英文摘要

This article argues that a central development in the history of computing is the mathematical evolution of addressing. During the three decades separating Kateryna Yushchenko's Address Programming Language (1958) from Lenore Mullin's A Mathematics of Arrays (1988), addressing evolved from a machine-level mechanism for locating data into a formal theory of indexing, reduction, and machine realization, passing through Kenneth Iverson's algebra of arrays and Philip Abrams' recognition of shape as an architectural resource. Mullin's theory establishes indexing itself, via the function ψ, as the primitive from which array operations compose, reduce to a Denotational Normal Form, transform into an Operational Normal Form, and realize on hardware through dimension lifting. Historical claims are sourced against primary material, with inferences flagged as such; mathematical claims are stated as identities; performance claims are marked validated, theoretically established, or provisional. The result links programming languages, array mathematics, and computer architecture in one account.

发表机构

• University at Albany, State University of New York（纽约州立大学奥尔巴尼分校）
• Université Paris-Est Créteil (UPEC)（巴黎东克雷泰伊大学）

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