比较计算能力
Comparing Computational Power
摘要：
比较不同计算模型的计算能力是常见的做法。例如，递归函数严格来说比原始递归函数更强大，因为后者是前者的真子集（包括阿克曼函数）。与这种测量功率的“遏制”方法并行，使用基于“模拟”的方法是标准的。例如，有人说（无类型）λ演算在计算上与部分递归函数一样强大，因为λ演算可以通过将自然数编码为Church数来模拟所有部分递归函数。问题是，不受约束地使用这两种比较能力的方法可以表明，某些计算模型严格地比它们自己更强！我们认为，一个更好的定义是，模型A严格强于B，如果A可以模拟B通过某种编码，而B不能模拟A下的任何编码。然后，我们表明，递归函数是严格强在这个意义上比原始递归。我们还证明了递归函数，部分递归函数，和图灵机是“完整的”，在这个意义上，没有内射编码可以使它们等价于任何“超计算”模型。
英文摘要：
It is common practice to compare the computational power of different models of computation. For example, the recursive functions are strictly more powerful than the primitive recursive functions, because the latter are a proper subset of the former (which includes Ackermann's function). Side-by-side with this "containment" method of measuring power, it is standard to use an approach based on "simulation". For example, one says that the (untyped) lambda calculus is as powerful--computationally speaking--as the partial recursive functions, because the lambda calculus can simulate all partial recursive functions by encoding the natural numbers as Church numerals. The problem is that unbridled use of these two ways of comparing power allows one to show that some computational models are strictly stronger than themselves! We argue that a better definition is that model A is strictly stronger than B if A can simulate B via some encoding, whereas B cannot simulate A under any encoding. We then show that the recursive functions are strictly stronger in this sense than the primitive recursive. We also prove that the recursive functions, partial recursive functions, and Turing machines are "complete", in the sense that no injective encoding can make them equivalent to any "hypercomputational" model.
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