避免线性秩序中的梅德韦杰夫减少

Pub Date : 2023-07-24 DOI:10.1002/malq.202200059
Noah Schweber
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引用次数: 0

摘要

虽然线性阶J中的每个端点区间I都被认为是线性阶,但Muchnik平凡地可约为J本身,这对于Medvedev约简是失败的。我们构造了一个极端的例子:一个线性阶,其中没有端点区间是Medvedev可约为任何其他区间的,甚至允许参数,除非两个区间有有限差。我们还构造了一个离散线性阶,它有许多自己无法比拟的端点区间Medvedev;这种线性阶的唯一其他已知构造产生了一个复杂度极高的序数,而这种构造产生了低级别的算术示例。此外,这里的结构是“粗糙的”,因为它们提升到了其他一致可约性概念,而不是梅德韦杰夫可约性本身。
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Avoiding Medvedev reductions inside a linear order

While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik-reducible to J itself, this fails for Medvedev-reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev-reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev-incomparable to itself; the only other known construction of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low-level-arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself.

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