《how to design programs》15章 相互引用的数据定义

由结构体组成的表与结构体中的表。

在用追溯形式建立家家谱树时，我们通常从某个后代除法，依次处理它的父母，组父母等。而构建树时，我们会不断添加谁是谁的孩子，而不是写出谁是谁的父母，从而建立一颗后代家谱树。

绘制后代数与绘制祖先树一样，只是将所有箭头的方向都反了过来：

(define-struct parent (children name date eyes))
children数量不固定，怎么办？
自然的选择是另children代表由parent结构体组成的表，这个表代表孩子，
parent是结构体:
(make-parent loc n d e) loc是孩子的表。
不幸的是，这个数据定义违反了我们关于定义的标准，具体说，他提到了一个还没定义的集合：孩子的表。
既然无法在不知道孩子的表示什么的情况下定义父母类型，我们就先定义孩子的表:
list of childrens 是下列之一
1 empty
2 (cons p loc)；其中p是parent，loc是孩子的表。
第二个定义时标准的，但是他遇到和parent一样的问题。
结论是，这2个定义在相互引用对方，他们只在同时定义的情况下才有意义：
hen two (or more) data definitions refer to each other, they are said to be MUTUALLY RECURSIVE or MUTUALLY REFERENTIAL.

如果2个或更多的数据定义相互引用，我们就称他们为相互引用的。
现在，我们可以把上图中的家谱树转成scheme表达式了，当然，在建立parent结构体之前，必须先定义所偶有表示其孩子的阶段，最好的方法是，在使用某个parent结构体之前先给他命名，如： 

(define Gustav (make-parent empty 'Gustav 1988 'brown))

(make-parent (list Gustav) 'Fred 1950 'yellow)

To create a parent structure for Fred, we first define one for Gustav so that we can form (list Gustav), the list of children for Fred.

完整定义:

;; Youngest Generation:
(define Gustav (make-parent empty 'Gustav 1988 'brown))

(define Fred&Eva (list Gustav))

;; Middle Generation:
(define Adam (make-parent empty 'Adam 1950 'yellow))
(define Dave (make-parent empty 'Dave 1955 'black))
(define Eva (make-parent Fred&Eva 'Eva 1965 'blue))
(define Fred (make-parent Fred&Eva 'Fred 1966 'pink))

(define Carl&Bettina (list Adam Dave Eva))

;; Oldest Generation:
(define Carl (make-parent Carl&Bettina 'Carl 1926 'green))
(define Bettina (make-parent Carl&Bettina 'Bettina 1926 'green))
Figure :  A Scheme representation of the descendant family tree

现在我们来研究blue-eyed-descendent？的开发，他是blue-eyed-ancestor的自然对应物，该函数读入一个parent结构体，判断他或货代是否有眼睛是蓝色的。下面的有点难，我最开始试图只用一个函数来搞定，发现始终不行，看了答案才知道：

;; blue-eyed-descendant? : parent  ->  boolean
;; to determine whether a-parent any of the descendants (children,
;; grandchildren, and so on) have 'blue in the eyes field
(define (blue-eyed-descendant? a-parent)
  (cond
    [(symbol=? (parent-eyes a-parent) 'blue) true]
    [else (blue-eyed-children? (parent-children a-parent))]))

;; blue-eyed-children? : list-of-children  ->  boolean
;; to determine whether any of the structures in aloc is blue-eyed
;; or has any blue-eyed descendant
(define (blue-eyed-children? aloc)
  (cond
    [(empty? aloc) false]
    [else
      (cond
        [(blue-eyed-descendant? (first aloc)) true]
        [else (blue-eyed-children? (rest aloc))])]))
;; blue-eyed-descendant? : parent  ->  boolean
;; to determine whether a-parent any of the descendants (children,
;; grandchildren, and so on) have 'blue in the eyes field
(define (blue-eyed-descendant? a-parent)
  (or (symbol=? (parent-eyes a-parent) 'blue)
      (blue-eyed-children? (parent-children a-parent))))

;; blue-eyed-children? : list-of-children  ->  boolean
;; to determine whether any of the structures in aloc is blue-eyed
;; or has any blue-eyed descendant
(define (blue-eyed-children? aloc)
  (cond
    [(empty? aloc) false]
    [else (or (blue-eyed-descendant? (first aloc))
              (blue-eyed-children? (rest aloc)))]))
Figure :  Two programs for finding a blue-eyed descendant

Exercise 15.1.2.   Develop the function how-far-removed. It determines how far a blue-eyed descendant, if one exists, is removed from the given parent. If the given parent has blue eyes, the distance is 0; if eyes is not blue but at least one its children's eyes are, the distance is 1; and so on. If no descendant of the given parent has blue eyes, the function returns false when it is applied to the corresponding family tree.    Solution

;; A parent is a structure: (make-parent loc n d e)
;; where loc is a list of children,
;; n and e are symbols, and d is a number.
(define-struct parent (children name date eyes))

;; A list of children is either
;;   .  empty or
;;   .  (cons p loc) where p is a parent and loc is a list of children. 

;; EXAMPLES OF DATA
(define robby (make-parent empty "Robby"  'blue))
(define ted (make-parent empty "Ted"  'brown))
(define pat (make-parent empty "Pat"  'brown))
(define pete (make-parent empty "Pete"  'brown))
(define alice (make-parent (list robby ted pat pete) "Alice"  'blue))
(define bill (make-parent (list robby ted pat pete) "Bill"  'brown))
(define lolly (make-parent empty "Lolly"  'blue))
(define tutu (make-parent (list alice lolly) "Tutu"  'brown))

;; how-far-removed : parent -> number or false
;; to find the distance to the nearest blue eyed descendent
;; or return false if there is no such descendant
(define (how-far-removed a-parent)
  (cond
    [(symbol=? 'blue (parent-eyes a-parent))
     ]
    [else
     (add1/false
      (how-far-removed-children
       (parent-children a-parent)))]))

;; how-far-removed-children : list of children -> number or false
;; to find the nearest blue eyed descent for a list of children,
;; assuming that one exists
(define (how-far-removed-children children)
  (cond
    [(empty? children) false]
    [else (min/false (how-far-removed (first children))
                     (how-far-removed-children (rest children)))]))

;; add1/false : number or false -> number or false
;; to add one to a number, or return false
(define (add1/false n)
  (cond
    [(number? n) (+ n )]
    [else false]))

;; min/false : (number or false) (number or false) -> (number or false)
;; to compute the minimum number of two numbers
;; if both are numbers, find the minimum.
;; if only one is a number, return that.
;; if both are false, return false.
(define (min/false n m)
  (cond
    [(and (number? n) (number? m)) (min n m)]
    [(and (number? n) (boolean? m)) n]
    [(and (boolean? n) (number? m)) m]
    [(and (boolean? n) (boolean? m)) false]))

;; EXAMPLES AS TESTS
(how-far-removed ted) "should be" false
(how-far-removed robby) "should be"
(how-far-removed alice) "should be"
(how-far-removed bill) "should be"
(how-far-removed tutu) "should be" 

Exercise 15.1.3.   Develop the function count-descendants, which consumes a parent and produces the number of descendants, including the parent.

Develop the function count-proper-descendants, which consumes a parent and produces the number of proper descendants, that is, all nodes in the family tree, not counting the parent.    Solution

(define (count-descendants a-parent)
  (cond
    [(empty? (parent-children a-parent)) ]
    [else (+  (count-children (parent-children a-parent)))]
    )
  )

;; count-children : list-of-children -> number
;; consumes a list-of-children c
;; produces the number of parents in c
(define (count-children a-loc)
  (cond
    [(empty? a-loc) ]
    [else (+ (count-descendants (first a-loc))
             (count-children (rest a-loc)))]))

(count-descendants Eva)
(count-descendants Carl) 

(define (count-proper-descendants a-parent)
  (count-children (parent-children a-parent))
  )
 

Exercise 15.1.4.   Develop the function eye-colors, which consumes a parent and produces a list of all eye colors in the tree. An eye color may occur more than once in the list.

Hint: Use the Scheme operation append, which consumes two lists and produces the concatenation of the two lists.    Solution

;; eye-colors : parent -> list
;; consumes a parent and produces a list of all eye colors in the tree
(define (eye-colors a-parent)
  (cons (parent-eyes a-parent) (children-eyes (parent-children a-parent)))
  )

(define (children-eyes a-loc)
  (cond
    [(empty? a-loc) empty]
    [else (append (eye-colors (first a-loc))
                  (children-eyes (rest a-loc))
                  )
          ]
    )
  )
(eye-colors Carl)

相互递归

为相互引用的定义设计函数：
在上述例子中，我们需要2个相关的定义：

Templates:The templates are created in parallel, following the advice concerning compound data, mixed data, and self-referential data. Finally, we must determine for each selector expression in each template whether it corresponds to a cross-reference to some definition. If so, we annotate it in the same way we annotate cross-references.

Here are the templates for our running example:

模板被并行地创建。

fun-parent模板中没有条件，因为parent的数据定义中并不包含任何子句，而是包含对第二个模板的相互引用：处理parent结构体中的chidlren字段。按照同样的规则，fun-children是一个条件式，第2个cond子句中包含了一处自引用，处理表的rest部分，以及对表的first元素，即parent结构体-的一处相互引用。

对数据定义和模板的比较表明了他们是多么的类似。

15.3 网页再谈