Gödel’s functional interpretation and the concept of learning
Gödel’s functional interpretation and the concept of learning
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哥德尔的功能解释和学习的概念
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发表时间：
2016
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通讯作者：
Thomas Powell
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文献类型：
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作者：
Thomas Powell
本文从学习的角度对哥德尔的功能解释进行了研究。我们定义了学习算法的概念，并证明了根据这些算法可以给出归纳和各种理解图式的函数解释的直观实现。在算术理解的情况下，我们阐明了我们的学习实现器与传统的使用BAR递归获得的学习实现工具的比较，证明了BAR递归理解解释对应于“健忘”学习算法。这项工作的主要目的是更深入地了解使用函数解释提取的程序的语义。然而，通过这样做，我们的目的也是为了更好地理解它与经典逻辑的其他解释之间的关系，这些经典逻辑的概念是内在的，例如希尔伯特的epsilon微积分或最近对阿希耶里和贝拉尔迪的基于学习的可实现性解释。
In this article we study Gödel’s functional interpretation from the perspective of learning. We define the notion of a learning algorithm, and show that intuitive realizers of the functional interpretation of both induction and various comprehension schemas can be given in terms of these algorithms. In the case of arithmetical comprehension, we clarify how our learning realizers compare to those obtained traditionally using bar recursion, demonstrating that bar recursive interpretations of comprehension correspond to ‘forgetful’ learning algorithms. The main purpose of this work is to gain a deeper insight into the semantics of programs extracted using the functional interpretation. However, in doing so we also aim to better understand how it relates to other interpretations of classical logic for which the notion of learning is inbuilt, such as Hilbert’s epsilon calculus or the more recent learning-based realizability interpretations of Aschieri and Berardi.