Weihrauch度、全知原理与弱可计算性
Weihrauch Degrees, Omniscience Principles and Weak Computability
摘要：
本文研究了表示空间上多值函数的Weihrauch约化。我们称相应的等价类为Weihrauch度，并证明了相应的偏序诱导出一个以多值函数的不交并为下确界运算的下半格.我们证明了平行化是这个半格的闭包算子，并且平行化的Weihrauch度甚至形成了一个以多值函数的乘积作为下确界运算的格.我们表明，梅德韦杰夫格，因此图灵度可以嵌入到一个自然的方式并行Weihrauch格。研究了全知LPO的有限原理和全知LLPO的小有限原理及其并行化。我们证明了并行LLPO等价于弱K“onig引理，从而等价于这种新的和非常强的意义下的Hahn-Banach定理。我们称一个多值函数弱可计算的，如果它是可约化的Weihrauch程度的并行LLPO，我们提出了一个新的证明，弱可计算的操作类是封闭的组合。这个证明是基于Kleene的三进制逻辑的计算版本。此外，我们的弱可计算操作的可计算度量空间的操作，承认上半可计算的紧值选择器，我们证明了任何单值弱可计算的操作已经在普通意义上的可计算。
英文摘要：
In this paper we study Weihrauch reducibility for multi-valued functions on represented spaces. We call the corresponding equivalence classes Weihrauch degrees and we show that the corresponding partial order induces a lower semi-lattice with the disjoint union of multi-valued functions as greatest lower bound operation. We prove that parallelization is a closure operator for this semi-lattice and the parallelized Weihrauch degrees even form a lattice with the product of multi-valued functions as greatest lower bound operation. We show that the Medvedev lattice and hence Turing degrees can be embedded into the parallelized Weihrauch lattice in a natural way. We study the limited principle of omniscience LPO, the lesser limited principle of omniscience LLPO and their parallelizations. We prove that parallelized LLPO is equivalent to Weak K"onig's Lemma and hence to the Hahn-Banach Theorem in this new and very strong sense. We call a multi-valued function weakly computable if it is reducible to the Weihrauch degree of parallelized LLPO and we present a new proof that the class of weakly computable operations is closed under composition. This proof is based on a computational version of Kleene's ternary logic. Moreover, we characterize weakly computable operations on computable metric spaces as operations that admit upper semi-computable compact-valued selectors and we prove that any single-valued weakly computable operation is already computable in the ordinary sense.
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