Programming Language A 学习笔记（二）

1. 语法糖——元组的“名称引用”与“位置引用”：

(e1,...,en) <=> {1=e1,...,n=en}

类型：t1 * … * tn <=> {1:t1,...,n:tn}

2. 自定义数据类型绑定：

datatype mytype = TwoInts of int * int

| Str of string

| Pizza

3. 访问自定义数据类型的值：

fun f x = (* f has type mytype -> int *)

case x of

Pizza => 3

| TwoInts(i1, i2) => i1 + i2

| Strs=>String.sizes

4. 数据类型绑定与Case表达式简化描述：

datatype t = C1 of t1 | C2 of t2 | … | Cn of tn

case e of p1 => e1 | p2 => e2 | … | pn => en

5. 类型别名（Type Synonyms）

type foo = int

6. 自定义列表类型

datatype my_int_list = Empty

| Cons of int * my_int_list

7. 多态类型（Polymorphic Datatypes）

datatype 'a option = NONE | SOME of 'a

二叉树定义：

datatype ('a, 'b) tree = Node of 'a * ('a, 'b) tree * ('a, 'b) tree

| Leaf of 'b

8. Val-Binding的真相——模式匹配（Pattern-Matching for Each-Of Types）

例8-1-1：

fun sum_triple (triple : int * int * int) =

case triple of

(x, y, z) => z + y + x

例8-2-1：

fun full_name (r : {first : string, middle : string, last : string}) =

case r of

{first = x, middle = y, last = z} => x ^ "" ^ y ^ "" ^ z

例8-2-2：

fun full_name (r : {first : string, middle : string, last : string}) =

let val {first = x, middle = y, last = z} = r

in

x ^ "" ^ y ^ "" ^ z

end

例8-1-2：

fun sum_triple (triple : int * int * int) =

let val (x, y, z) = triple

in

x + y + z

end

例8-2-3：

fun full_name {first = x, middle = y, last = z} =

x ^ "" ^ y ^ "" ^ z

例8-1-3：

fun sum_triple (x, y, z) =

x + y + z

9. 题外话——类型推导（Type inference）

In ML,every variable and function has a type (or your program fails to type-check)—

type inference only means you do not need to write down the type.

10. 题外话——多态类型与等价类型

'a list * 'a list -> 'a list

可以替换为：string list * string list -> string list

不能替换为：string list *int list -> string list

'a 必须替换为同样的数据类型

11. 嵌套模式（Nested Patterns）

a::(b::(c::d))    包含至少3个元素的list

a::(b::(c::[])) 只包含3个元素的list

模式匹配的递归定义（the elegant recursive denition of pattern matching）

1	A variable pattern(x) matches any value v and introduces one binding (from x to v).
2	The pattern C matches the value C,if C is a constructor that carries no data.
3	The pattern C p where C is a constructor and p is a pattern matches a value of the form C v (notice the constructors are the same) if p matches v (i.e., the nested pattern matches the carried value). It introduces the bindings that p matching v introduces.
4	The pattern (p1,p2,...,pn) matches a tuple value (v1,v2,...,vn) if p1 matches v1 and p2 matches v2, ..., and pn matches vn. It introduces all the bindings that the recursive matches introduce.
5	(A similar case for record patterns of the form {f1=p1, … , fn=pn} ...)

例11-1-1：

fun len xs =

case xs of

[] => 0

| x::xs' => 1 + len xs'

例11-1-2：

fun len xs =

case xs of

[] => 0

| _::xs' => 1 + len xs'

通配符（wildcard） (_) 指代任意没有定义数据类型的值

12. 可用的嵌套模式范例：

例12-1：

exception BadTriple

fun zip3 list_triple =

case list_triple of

([], [], []) => []

| (hd1::tl1, hd2::tl2, hd3::tl3) => (hd1, hd2, hd3)::zip3(tl1, tl2, tl3)

| _ => raiseBadTriple

fun unzip3 lst =

case lst of

[] => ([], [], [])

| (a, b, c)::tl => let val (l1, l2, l3) = unzip3 tl

in

(a::l1, b::l2, c::l3)

end

例12-2：

datatype sgn = P | N | Z

fun multsign (x1, x2) =

let fun sign x = if x = 0 then Z else if x > 0 then P else N

in

case(sign x1,sign x2) of

(Z, _) => Z

| (_, Z) => Z

| (P, P) => P

| (N, N) => P

| _ => N (* many say bad style; I am okay with it *)

end

13. 多重选择的函数定义（Multiple Cases in a Function Binding）

例13-1：

datatype exp = Constant of int | Negate of exp | Add of exp * exp | Multiply of exp * exp

fun eval(Constant i) = i

| eval(Negate e2) = ~(eval e2)

| eval(Add(e1, e2)) = (eval e1) + (eval e2)

| eval(Multiply(e1, e2))=(eval e1) * (eval e2)

fun append ([], ys) = ys

| append (x::xs', ys) = x::append(xs', ys)

一般形态（语法糖形式）：

fun f p1 = e1

| f p2 = e2

...

| f pn = en

普通写法：

fun f x =

case x of

p1 => e1

| p2 => e2

...

| pn => en

14. 异常（Exception）

输出异常（关键字） raise：raise List.Empty

定义异常（关键字）exception：exception MyUndesirableCondition

15. 尾递归和累加器

fun sum1 xs =

case xs of

[] => 0

| i::xs' => I + sum1 xs'

16. 尾递归的定义

递归的调用出现在尾位置（saying a call is a tail call if it is in tail position.）

尾位置定义：

1	In fun f(x) = e, e is in tail position.
2	If an expression is not in tail position, then none of its sub expressions are in tail position.
3	If if e1 then e2 else e3 is in tail position, then e2 and e3 are in tail position (but not  e1).(Case-expressions are similar.)
4	If let b1 … bn in e end is in tail position,then e  is in tail position (but no expressions in the bindings are).
5	Function-call arguments are not in tail position.