实例可归约性与Weihrauch度
Instance reducibility and Weihrauch degrees
AI总结：
本文提出实例可归约性概念，用于比较和分类逆向构造性数学中的各种原理，证明实例度构成完全格框架，并在相对可实现性拓扑斯中给出其显式表述，称为扩展Weihrauch度，改善了Weihrauch度的结构。
摘要：
我们确定了一个谓词之间的可归约性概念，称为实例可归约性，它常见于逆向构造性数学中。该概念可普遍用于比较和分类逆向构造性数学中研究的各种原理（形式化丘奇论题、布劳威尔连续性原理与扇定理、排中律、有限原理、函数选择、马尔可夫原理等）。我们证明实例度构成一个框架，即一个完全格，其中有限下确界分配于集合索引的上确界。它们被证明等价于真值上集框架，按逆Smyth偏序排序。我们研究该格的整体结构：子对象分类器以两种不同方式嵌入该格，一种单调，另一种反单调，且¬¬稠密度恰好与可归约于排中律度的那些度一致。\n我们在相对可实现性拓扑斯中给出实例度的显式表述，并称其为扩展Weihrauch度，因为在Kleene-Vesley可实现性中，¬¬稠密的适度实例度恰好对应于Weihrauch度。扩展度通过为Weihrauch度配备可计算的下确界和上确界、蕴含、控制参数访问和结果计算的能力，并普遍拓宽Weihrauch可归约性的范围，从而改善了Weihrauch度的结构。
英文摘要：
We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility.
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