步进递归：精确深度不决定代数表达能力
Step Recursion: Exact Depth Does Not Determine Algebraic Expressiveness
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中文总结 AI 辅助
本文证明步进递归的精确深度不能决定代数表达能力：即使深度映射相同，不同的前驱选择仍可产生不同代数，深度并非完备不变量。
中文摘要 AI 辅助
步进递归是有界递归的一种形式，其中每次递归调用从输入y移动到指定的前驱ρ_g(y)。其深度D_g(y)是达到零所需的此类移动的精确次数。一个自然的问题是，知道每个输入的该深度是否决定了所得到的函数代数的表达能力。我们证明它不能。我们首先构造两个简单的生成器，它们具有完全相同的深度映射但不同的步进递归代数。一个给出普通的二进制减半b(x)=2x+1；另一个是p(0)=1，对于x>0，p(x)=x+2^{λ(x)}，其中λ是二进制长度。尽管D_p=D_b逐点成立，但前驱ρ_p不能从任何固定步长的二进制减半下降在基零处定义。因此，两个递归方案可能在每个输入上采取完全相同的步数，但仍然具有不同的表达能力。该现象远比此例广泛。每当无限多个深度级别允许多于一种前驱安排时，单个精确深度轮廓支持在包含零和投影的每个可数基上存在2^{ℵ₀}个不同的步进递归代数。在可计算设置中，相应的有效族恰好有ℵ₀个不同的代数。因此，精确递归深度是一种信息丰富的资源度量，但它不是一个完整的完备不变量：每个深度级别内前驱选择的几何结构携带额外的代数信息。
英文摘要
Step recursion is a form of bounded recursion in which each recursive call moves from an input y to a prescribed predecessor $ρ_g(y)$. Its depth $D_g(y)$ is the exact number of such moves needed to reach zero. A natural question is whether knowing this depth for every input determines the expressive power of the resulting function algebra. We prove that it does not. We first construct two simple generators with exactly the same depth map but different step-recursion algebras. One gives ordinary binary halving, $b(x)=2x+1$; the other is $p(0)=1$, $p(x)=x+2^{λ(x)}$ for $x>0$, where $λ$ is binary length. Although $D_p=D_b$ pointwise, the predecessor $ρ_p$ cannot be defined from any fixed-stride binary-halving descent at basis zero. Thus two recursion schemes may take exactly the same number of steps on every input and still have different expressive power. The phenomenon is much larger than this example. Whenever infinitely many depth levels allow more than one predecessor arrangement, a single exact depth profile supports $2^{\aleph_0}$ distinct step-recursion algebras over every countable basis containing zero and the projections. In the computable setting the corresponding effective family has exactly $\aleph_0$ distinct algebras. Hence exact recursion depth is an informative resource measure, but it is not a complete invariant: the geometry of predecessor choices inside each depth level carries additional algebraic information.
