解读 HA^w 内部的 E-HA^w

Q4 Computer Science

Electronic Proceedings in Theoretical Computer Science Pub Date : 2023-11-17 DOI:10.4204/EPTCS.396.5

Félix Castro

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引用次数: 0

摘要

高级类型算术（HA$^w$）是一种一阶多排序理论。它是海因算术的保守扩展，通过将术语的语法扩展到系统 T 的所有部分而获得：这里感兴趣的对象是高级类型的函数。自然数之间的相等是由皮亚诺公理规定的，那么如何定义函数之间的相等呢？从这个问题出发，产生了不同版本的 HA$^w$，如扩展版本（E-HA$^w$）和意向版本（I-HA$^w$）。在这项工作中，我们将了解部分等价关系的研究如何引导我们设计一种从 E-HA$^w$ 到 HA$^w$ 的参数化转换。

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An Interpretation of E-HA^w inside HA^w

Higher Type Arithmetic (HA$^w$) is a first-order many-sorted theory. It is a conservative extension of Heyting Arithmetic obtained by extending the syntax of terms to all of System T: the objects of interest here are the functionals of higher types. While equality between natural numbers is specified by the axioms of Peano, how can equality between functionals be defined? From this question, different versions of HA$^w$ arise, such as an extensional version (E-HA$^w$) and an intentional version (I-HA$^w$). In this work, we will see how the study of partial equivalence relations leads us to design a translation by parametricity from E-HA$^w$ to HA$^w$.

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