考虑一个决策问题，其实例是一个函数。它的不可判定性程度，由算术（或Kleene-Mostowski）层次结构的相应类来衡量，可能取决于实例是部分递归函数还是原始递归函数。类似的情况也发生在像赖斯定理（对于原始递归函数是假的）这样的结果上。经典递归论主要研究部分递归函数的性质。本文研究了几个与原始递归函数有关的自然判定问题，并给出了它们的不可判定度。作为一个例子，我们证明了对于原始递归函数，内射性问题是Pi^0_1-完全的，而满射性问题是Pi_2-完全的.我们比较了几个原始的递归决策问题与经典递归理论的相应问题的不可判定性（由算术层次中的水平来衡量）的程度。例如，问题“一个函数的余域是否只有一个元素？“对于原始递归函数是Pi_1-完全的，对于部分递归函数属于类[Delta_2 -（Sigma_1 UNION Pi_1）]。一个重要的判定问题，“给定的原始递归函数是否至少有一个零？“进行了详细研究。
英文摘要：
Consider a decision problem whose instance is a function. Its degree of undecidability, measured by the corresponding class of the arithmetic (or Kleene-Mostowski) hierarchy hierarchy, may depend on whether the instance is a partial recursive or a primitive recursive function. A similar situation happens for results like Rice Theorem (which is false for primitive recursive functions). Classical Recursion Theory deals mainly with the properties of partial recursive functions. We study several natural decision problems related to primitive recursive functions and characterise their degree of undecidability. As an example, we show that, for primitive recursive functions, the injectivity problem is Pi^0_1-complete while the surjectivity problem is Pi_2-complete (omit superscripts from now on). We compare the degree of undecidability (measured by the level in the arithmetic hierarchy) of several primitive recursive decision problems with the corresponding problems of classical Recursion Theory. For instance, the problem "does the codomain of a function have exactly one element?" is Pi_1-complete for primitive recursive functions and belongs to the class [Delta_2 - (Sigma_1 UNION Pi_1)] for partial recursive functions. An important decision problem, "does a given primitive recursive function have at least one zero?" is studied in detail.
Axiomatic approach to the theory of algorithms and relativized computability
众所周知，递归论中的许多定理都可以“相对化”。这意味着，如果部分递归函数被相对于某个固定的Oracle集是部分递归的函数所替换，则它们仍然为真。乌斯彭斯基制定了三个“公理”，称为“计算记录公理”、“程序公理”和“算术性公理”。然后，使用这些公理（更确切地说，前两个），他证明了基本结果的递归论。这两个公理对于相对于某个固定的预言集是部分递归的函数类也是正确的。这个类在替换、原始递归式和最小化（$μ$ -运算符）下也是封闭的;这些（直观上显而易见的）封闭性也用于证明中。这一观察所作出的乌斯彭斯基解释了为什么许多定理的递归论可以相对化。事实证明，相反的陈述也是正确的：所有可相对化的结果都遵循前两个公理和封闭性。没错，就是\emph{every class of partial functions that is closed under substitution, primitive recursion and minimization that satisfies the first two axioms is the class of functions that are partial recursive relative to some oracle set $A$}。这是本文的主要结果。
Abstract Compilation as Abstraction of Operator Semantics, applied to Cost Analysis
Universidad Politécnica de Madrid（马德里理工大学）；IMDEA Software Institute（IMDEA软件研究所）；Spanish Council for Scientific Research (CSIC)（西班牙国家研究委员会）；Computer Science Laboratory of Sorbonne University (LIP6)（索邦大学计算机科学实验室）；STMicroelectronics（意法半导体）
