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Recursive definition

软件工程师,对于definition(定义)这个词肯定不会陌生,因为我们每天都在“定义一个函数”、“定义一个类”。wikipedia definition内容比较深奥,本文将简单地来说definition,定义就是在前文中所提及的“描述”(description),不过定义有着更多限制。本文重点简述的是Recursive definition,在计算机科学中,recursion无处不在,软件工程师应该对它保持敏感,对于所有具备recursion特性的(包括类型、过程等),都能够给出其Recursive definition

wikipedia Recursive definition

In mathematics and computer science, a recursive definition, or inductive definition(归纳定义), is used to define the elements in a set in terms of other elements in the set (Aczel 1977:740ff). Some examples of recursively-definable objects include factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set.

NOTE: 上述对recursive definition的描述使用的是数学中的set的概念,数学中的set表示所有具备某一特性的object的集合,比如我们可以将所有具备相同type的object放到一个set中。下面我将它转换为software engineer更加熟悉的type(类型)的概念:

规定属于同一个set的所有元素都是具有相同类型的元素,则recursively-definable object是由相同类型的其他object(即这个集合中的其他元素)来进行定义的(构成)。

可以看到,当我们将set理解为type后,原来的描述就变成了software engineer非常容易理解的了。

上述定义是非常严谨的,使用了数学中的 set 的概念,可以认为它使用的是一种数学语言。在其他文章中都可以看到完全使用自然语言描述的recursive definition,比如:

在wikipedia Recursion中:

Recursion (adjective: recursive) occurs when a thing is defined in terms of itself or of its type.

在wikipedia Recursive acronym中:

A recursive acronym is an acronym that refers to itself.

显然使用自然语言的描述是更加容易理解的。

如何理解recursively defined function?

显然,一个recursively defined function就是“ defined in terms of itself or of its type ”

A recursive definition of a function defines values of the function for some inputs in terms of the values of the same function for other (usually smaller) inputs. For example, the factorial function n! is defined by the rules

0! = 1.

(n + 1)! = (n + 1)·n!.

NOTE: 原文中给出了两种**recursive definition**:recursively defined functions and recursive defined objects

Form of recursive definitions

Most recursive definitions have two foundations: a base case (basis) and an inductive clause.

NOTE: “inductive clause”的含义是“归纳子句”,关于“inductive ”,参见"Induction"。

That recursive definitions are valid – meaning that a recursive definition identifies a unique function – is a theorem of set theory known as the recursion theorem, the proof of which is non-trivial. Where the domain of the function is the natural numbers, sufficient conditions for the definition to be valid are that the value of f(0) (i.e., base case) is given, and that for n > 0, an algorithm is given for determining f(n) in terms of f(0),f(1), ... ,f(n-1) (i.e., inductive clause).

More generally, recursive definitions of functions can be made whenever the domain is a well-ordered set, using the principle of transfinite recursion. The formal criteria for what constitutes a valid recursive definition are more complex for the general case.

NOTE: 上面这段话并没有理解。

总结

按照原文第一段中对recursive definition的描述,发现其实它非常类似于induction(归纳法),我们知道,induction是bottom-up的,尤其是natural numbers的例子。而我对recursion的惯常印象是它是top-down。这两者不是矛盾吗?

“recursive definition”是一种描述方式,它不涉及实现的问题,它的描述可以是类似induction的自底向上,也可以是自顶向下,正如原文第一段所述的:recursive definition也可以叫做**inductive definition**。“recursive definition”本质在于“ defined in terms of itself or of its type ”,正如本文第一段所描述的:

In mathematics and computer science, a recursive definition, or inductive definition(归纳定义), is used to define the elements in a set in terms of other elements in the set

关于“recursive definition”的一个例子就是Recursive grammar,比如在维基百科Recursion (computer science)Recursive data types段中给出了这样的一个例子:

 <expr> ::= <number>
          | (<expr> * <expr>)
          | (<expr> + <expr>)

可以看到,<expr>的定义中就包含了"itself"。

“我对recursion的惯常印象是它是top-down”是源于我是从recursion function的角度来看待的。显然,这是描述与实现的差异。在原文的Form of recursive definitions中给出了recursive definition所对应的function,显然对于这个function的实现,我们可以采用的实现方式有两种:

  • 自底向上
  • 自顶向下

Examples of recursive definitions

Recursive grammar

参见:

1、维基百科Recursive grammar

2、龙书2.2.1 Definition of Grammars

Recursive definition in computer science

有太多太多的算法、结构都是可以使用recursive definition的,能够recursive definition的,称它具备递归性。

Sub structure

一般带有“sub”的都是可以进行recursive definition的,它是一种典型的**containing关系**,它是可以使用CFG来进行描述的。下面是典型的sub structure:

1、subtree,sublist,在维基百科Structural induction中所提及的:

recursively defined structure, such as formulas, lists, or trees

下面是一些例子:

2、Subobjects(参见cppreference Object#Subobjects

3、维基百科 Optimal substructure

Pointer of pointer

无论是在computer science还是在实际生活中,都存在着大量的类似于pointer of pointer的例子:

1、pointer of pointer

2、array of array(其实是一种sub structure,一个array可以看做是由多个sub array 组成)

3、mother of mother (家族hierarchy)

它们都是都是非常典型的recursion definition。下面对array of array即multiple-dimensional array进行说明。

multiple-dimensional array

在阅读 https://stackoverflow.com/a/4810676 时,其中有这样的描述:

Multidimensional arrays are often referred to as "arrays of arrays"

显然,这是典型的recursive definition。

与此类似的还有:multiple-level pointer。

multiple-dimensional array和multiple-level pointer可以看做是具备nesting关系(参见工程data-structureGraph\Tree\Tree-structure.md章节)。

multiple-dimensional array和multiple-level pointer可以递归的进行处理,在文章 https://stackoverflow.com/a/5580952 中给出了示例程序。

Recursive definition and closure

按照recursively defined sets的概念,这个set在构造新元素的operation下是closed的(参见 Closure (mathematics) )。

Recursive definition and structure

Recursive data types中专门描述可以使用recursive definition的structure,这在"Data-structure"章节中会进行讨论。

可以使用recursive definition进行定义的structure,都具备递归特性。