Logic of Limit Computing and its Applications

该项目建立了极限可计算数学的基本理论，并取得了许多新的成果。其中的主要结果如下：1。改进LCM2的可实现性解释。sublearning hierarchy3。LCM game4。算术层次。LCM categories6。基于LCM的分析中的可计算性。不连续函数的可计算性理论。基于并行计算的Delta-0-2映射模型其中，子学习层次的发现是非常重要的。在LEM（排中律）层级之下，存在与构造数学的LLPO相对应的原理，它们与WKL（弱Koenig引理）有关。这一事实是一些研究人员凭直觉设想出来的。通过建立这一事实的计算和学习理论意义，我们最终探讨了WKL在LCM层次中的地位。我们发现该原理代表了有限过程非确定性可辩驳性计算的波普尔博弈的极限计算模型。充分探讨了包括子学习层次在内的LCM的完整算术层次。这是通过各种数学逻辑技术完成的，从数学的角度来看，这也是很重要的。根据这个层次的校准理论是由我们项目的一位研究人员开始的。我们还发现Coquand的游戏语义可以以一种非常自然的方式进行限制，以便与LCM相一致。

The fundamental theory of Limit Computable Mathematics (LCM) were founded and many new results have been obtained in the project. The followings are the main results among them :1. Improved realizability interpretations for LCM2. sublearning hierarchy3. LCM game4. the arithmetical hierarchy5. LCM categories6. Computability in analysis in the light of LCM. Computability theory of discontinuous functions7. Models of Delta-0-2 maps by means of concurrent computationAmong them, the discovery of sublearning hierarchy is quite important. Below the hierarchy of LEM (the laws of excluded middle), principles corresponding to LLPO of constructive mathematics exist and they are related to WKL (Weak Koenig Lemma). This fact has been conceived by some researchers intuitively. By establishing computational and learning theoretic meanings of this fact, we finally explored the position of WKL in the hierarchy of LCM. We found that the principle represents the limit computational model of Popperian game of non-deterministic computation of refutability by finite numbers of processes.The full arithmetical hierarchy of LCM including this sublearning hierarchy was fully explored.This was done through various techniques of mathematical logic, and it is important also from mathematical point of view. The calibration theory according to this hierarchy has been started by a researcher out of our project.We also found that Coquand's game semantics could be restricted in a very natural way so that it coincides with LCM.