为什么可计算函数也称为递归函数？

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在可计算性理论中，可计算函数也称为递归函数。至少乍一看，它们与您在日常编程中所谓的“递归”（即调用自身的函数）没有任何共同之处。

在可计算性方面，递归的实际含义是什么？为什么将这些函数称为“递归”？

换句话说：“递归”的两种含义之间有什么联系？

computability terminology history

— 高洛·罗登（Golo Roden）
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μ递归函数

— Thomas Klimpel，

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它们作弊，因为它们包含μ运算符。这是一个最小化运算符，但是最小化与递归关系不大。因此，似乎有人（Kleene）认为“递归”听起来不错，因此他发明了使用该名称的借口。后来，罗伯特·索亚尔（Robert Soare）解释说，“可计算”听起来会好得多，而且“递归”只是早期的营销技巧，每个人都同意。

— Thomas Klimpel

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关于原始递归函数？从维基百科复制，它们定义为

和

h (0, x_{1}, \dots, x_{k}) = f (x_{1}, \dots, x_{k})

$h(0,x_{1},\ldots ,x_{k})=f(x_{1},\ldots ,x_{k})$

。那是一个自我调用的函数。

h (S (y), x_{1}, \dots, x_{k}) = g (y, h (y, x_{1}, \dots, x_{k}), x_{1}, \dots, x_{k})

$h(S(y),x_{1},\ldots ,x_{k})=g(y,h(y,x_{1},\ldots ,x_{k}),x_{1},\ldots ,x_{k})$

— Hendrik

3

@GoloRoden请注意，“可计算性”的标记描述（您在此问题中使用了它）说：“可计算性理论又称为递归理论”。Gödel将函数称为递归函数，但该术语演变为可计算的。可能是为了避免像您这样的困惑。深入研究可计算性理论的人们倾向于更多地使用术语递归理论来“尊重”它的根源。

— Auberon

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因为它们是递归定义的，即“ 根据先前定义的更简单的函数定义了更复杂的函数 ”

— Nikos M.

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定义一些基本功能：

零功能

$z e r o : N \to N : x \mapsto 0$ $zero: \mathbb{N} \rightarrow \mathbb{N} : x \mapsto 0$
后继功能

$s u c c : N \to N : x \mapsto x + 1$ $succ: \mathbb{N} \rightarrow \mathbb{N} : x \mapsto x + 1$
投影功能

p_{i}^{n} : N^{n} \to N : (x_{1}, x_{2}, \dots, x_{n}) \mapsto x_{i}

$p_i^n: \mathbb{N}^n \rightarrow \mathbb{N} : (x_1, x_2, \dots, x_n) \mapsto x_i$

从现在开始，我将使用来表示 $\bar{x_n}$ $(x_1, x_2, \dots, x_n)$

定义组成：

给定功能

分别带有签名 $g_1, g_2, \dots, g_m$ $\mathbb{N}^k \rightarrow \mathbb{N}$
$f : \mathbb{N}^m \rightarrow \mathbb{N}$

构造以下函数：

\begin{aligned} h : N^{k} \to N : \bar{x_{k}} \mapsto h (\bar{x_{k}}) = f ( & g_{1} (\bar{x_{k}}), g_{2} (\bar{x_{k}}), \dots, g_{m} (\bar{x_{k}})) \end{aligned}

$\begin{align} h: \mathbb{N}^k \rightarrow \mathbb{N}: \bar{x_k} \mapsto h(\bar{x_k}) = f ( & g_1(\bar{x_k}), g_2(\bar{x_k}), \dots, g_m(\bar{x_k}) ) \end{align}$

定义原始递归：

给定功能

$f: \mathbb{N}^k \rightarrow \mathbb{N}$
$g: \mathbb{N}^{k+2} \rightarrow \mathbb{N}$

构造以下（分段）函数：

h : N^{k + 1} \to N : (\bar{x_{k}}, y + 1) \mapsto {\begin{cases} f (\bar{x_{k}}), & y + 1 = 0 \\ g (\bar{x_{k}}, y, h (\bar{x_{k}}, y)), & y + 1 > 0 \end{cases}

$h : \mathbb{N}^{k+1} \rightarrow \mathbb{N} : \\ (\bar{x_k}, y + 1) \mapsto \begin{cases} f(\bar{x_k}), & y + 1 = 0 \\ g (\bar{x_k}, y, h(\bar{x_k}, y)), & y + 1 > 0 \end{cases}$

可以使用合成和基本功能上的原始递归实现的所有功能都称为原始递归。顾名思义就是这样称呼它。虽然存在带有可调用自身的函数的链接，但无需尝试将它们彼此链接。您可能会认为递归是谐音。

上面的定义和构造是由Gödel（也涉及其他一些人）构造的，试图捕获所有可计算的功能，即存在用于该功能的Turing Machine。请注意，图灵机的概念尚未描述，或者至少非常模糊。

不幸的是，一个叫Ackermann的人出现并定义了以下功能：

$Ack : \mathbb{N}^2 \rightarrow \mathbb{N}$
$Ack(0, y) = y+1$
$Ack(x+1, 0) = Ack(x, 1)$
$Ack(x+1, y+1) = Ack(x, Ack(x+1,y))$

This function is computable, but there's no way to construct it using only the constructions above! (i.e. $Ack$ is not primitive recursive) This means that Gödel and his posse failed to capture all computable functions in their construction!

Gödel had to expand his class of functions so $Ack$ could be constructed. He did this by defining the following:

Unbounded minimisation

$g : \mathbb{N}^k \rightarrow \mathbb{N}$
IF $\left[f(\bar{x_k}, y) = 0 \text{ AND } \\f(\bar{x_k}, z) \text{ is defined } \forall z < y \text{ AND } \\f(\bar{x_k}, z)\neq 0\right]$
THEN
$g(\bar{x_k}) = y$
ELSE
$g(\bar{x_k})$ is not defined.

This last one may be hard to grasp, but it basically means that $g((x_1, x_2, \dots, x_k))$ is the smallest root of $f$ (if a root exists).

All functions that can be constructed with all the constructions defined above are called recursive. Again, the name recursive is just by definition, and it doesn't necessarily have correlation with functions that call themselves. Truly, consider it a homonym.

Recursive functions can be either partial recursive functions or total recursive functions. All partial recursive functions are total recursive functions. All primitive recursive functions are total. As an example of a partial recursive function that is not total, consider the minimisation of the successor function. The successor function doesn't have roots, so its minimisation is not defined. An example of a total recursive function (which uses minimisation) is $Ack$ .

Now Gödel was able to construct the $Ack$ function as well with his expanded class of functions. As a matter of fact, every function that can be computed by a Turing machine, can be represented by using the constructions above and vice versa, every construction can be represented by a Turing machine.

If you're intrigued, you could try to make Gödel's class bigger. You can try to define the 'opposite' of unbounded minimisation. That is, unbounded maximisation i.e. the function that finds the biggest root. However, you may find that computing that function is hard (impossible). You can read into the Busy Beaver Problem, which tries to apply unbounded maximisation.

— Auberon
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I know realise the given definitions don't really answer the question, but my answer describes the evolution of recursion/computability theory, kind of. Might be worth a read.

— Auberon

I like it, thanks for your efforts :-)

— Golo Roden

In "if

h ((x_{1}, x_{2}, . . ., x_{k}), 0) = f ((x_{1}, x_{2}, . . ., x_{k}))

$h((x_1, x_2, ..., x_k), 0) = f((x_1, x_2, ..., x_k))$ ", I think you mean

h ((x_{1}, x_{2}, . . ., x_{k}, 0))

$h((x_1, x_2, ..., x_k, 0))$ . Also, there is no then clause prior to the next bullet point's else clause.

— Eric Towers

2

Afaik, this is subtly wrong. The set if

μ

$\mu$ -recursive functions is called the set of partially recursive functions whereas recursive functions are always total. That's why the set of all total functions (resp. languages that can be decided) is called R.

— Raphael

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There are quite a few incorrect statements in your answer. You should not make up history for an answer.

— 卡夫

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The founders of computability theory were mathematicians. They founded what is now called computability theory before there was any computers. What was the way mathematicians defined functions that could be computed? By recursive definitions!

So there were recursive function before there were any other model of computation like Turing machines or lambda calculus or register machines. So people referred to these function as recursive functions. The fact that they turned out to be exactly what Turing machines and other models can compute is a later event (mostly proven by Kleene).

We have the simple definition of a recursive function which is now called primitive recursive function. There were not general enough (e.g. Ackermann's function) so people developed more general notions like $\mu$ -recursive functions and Herbrand-Gödel general recursive functions that did capture all computable functions (assuming the Church's thesis). Church claimed that his model of lambda calculus captured all computable functions. Many people, and in particular Gödel, were not convinced that these capture all functions that can be computed. Until Turing's analysis of computation and introduction of his machine model.

The name of the field used to recursion theory. However there has been a successful push in recent decades to change the name to something more appealing from recursion theory to something more computer sciency (vs. mathy). As a result the field is now called computability theory. However if you look at books, papers, conferences, etc. in the early decades they are called recursion theory and not computability theory. Even the title of Soare's own 1987 book (who was the main person behind the push to change the name to computability theory) is "Recursively Enumerable Sets and Degrees".

If you want to know more about the history a fun and good place to read about it is the first chapter of Classical Recursion Theory by Odifreddi.

— Kaveh
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Robert Soare wrote an essay about this issue. According to him, the term (general) recursive functions was coined by Gödel, who defined them using some sort of mutual recursion. The name stuck, though later on other equivalent definitions were found.

For more information, I recommend Soare's essay.

— Yuval Filmus
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instead of putting a long comment decided to add an answer:

Because they are defined recursively, i.e "more complex functions are defined in terms of previously defined, simpler functions"

This kind of iterative or incremental procedure creates well-defined functions (in the mathematical sense)

This is the meaning of recursiveness in mathematical parlance. See below how this relates to recursion in programming parlance.

Compare this procedure with techniques and methods like (mathematical) induction which is also an example of recursiveness in mathematics.

Programming has a mathematical vein as well as an engineering one.

This (usualy constructive) procedure is also refered as "bootstrapping" in Operating Systems parlance.

However a runtime recursion of the same function (i.e caling itself during its runtime), since it must (hmm, should) happen on already computed values (or arguments), or in other words, in the part of the result set already computed, is also recursive in the above sense, i.e "defined w.r.t previously defined functions (and their values)"

Else is not well-defined, and leads to such things like Stack Overflow :))))

To give a further example from Operating Systems, a runtime recursion (calling itself) can be taken as the analog of an operating system rebooting after a certain update (e.g core update). Many OSes do the following procedure:

an initial boot to load low-level routines (e.g I/O)
do the necessary updates (using the low-level routines)
re-boot (effectively, re-calling itself), but this time loading the more complex routines (or even the whole system)

Auberon's beautiful answer demonstrates a procedure of this kind in more detail.

— Nikos M.
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