[基础软件理论与实践] 第二节作业实现 yhgu2000

作业信息

• 课程信息：https://bbs.csdn.net/topics/608593392
• 第二节作业提交帖：https://bbs.csdn.net/topics/608988912

1. Implement the substitution function N[v/x] : subst (N: lambda, x: string, v: value) : lambda
2. Think about how substitution works on arbitrary terms, i.e. N[M/x] where M could contain free variables.
3. Implement Church numberals and arithmetic functions using lambda calculus

作业实现

本次作业基于课程信息给出的示例代码实现。

才知道示例代码里已经留处了写作业的 TODO，回想起自己第一次傻傻地从 PPT 里拷代码结果报了很多编译错误……

任务1、任务2

补充 src/Lambda.res 中 subst 函数：

// Homework: implement the substitution
let rec subst = (x:string, v, a:lambda) => {
  switch a {
  | Var(y) => if x==y {v} else {a}
  | Fn(y, body) => if x==y {a} else { Fn(y, subst(x, v, body)) }
  | App(f, arg) => App(subst(x, v, f), subst(x, v, arg))
  }
}

测试代码：

// \x.x
let test_1 = Fn("x", Var("x"))
Js.Console.log(print_lambda(eval1(test_1)))

// \x.x \x.x = \x.x
let test_2 = App(Fn("x", Var("x")), Fn("x", Var("x")))
Js.Console.log(print_lambda(eval1(test_2)))

// \y.(\x.x y) \y.(\x.x y) = \x.(x \y.(\x.x y))
let test_3 = App(
  Fn("y", Fn("x", App(Var("x"), Var("y")))),
  Fn("y", Fn("x", App(Var("x"), Var("y"))))
)
Js.Console.log(print_lambda(eval1(test_3)))

运行输出如下：

fun x -> x
fun x -> x
fun x -> x (fun y -> fun x -> x y)

任务3

补充 src/Nat.res 中四个函数定义如下：

// Homework: implement the arithmetic functions for peano numbers
let rec peano_add = (n: nat, m: nat): nat => {
  switch n {
  | Z => m
  | S(n') => peano_add(n', S(m))
  }
}
let rec peano_mul = (n: nat, m: nat): nat => {
  switch n {
  | Z => Z
  | S(n') => peano_add(peano_mul(n', m), m)
  }
}

// Homework: implement the arithmetic functions for church numbers
let church_add = (n: cnum<_>, m: cnum<_>): cnum<_> => {
  (s, z) => n(s, m(s, z))
}
let church_mul = (n: cnum<_>, m: cnum<_>): cnum<_> => {
  (s, z) => n((z) => m(s, z), z)
}

测试代码：

// 1 + 2 = 3
let peano_three = peano_add(peano_one, peano_two)
Js.Console.log(peano_decode(peano_three))

// 2 * 3 = 6
let peano_six = peano_mul(peano_two, peano_three)
Js.Console.log(peano_decode(peano_six))

// 1 + 2 = 3
let church_three = church_add(church_one, church_two)
Js.Console.log(church_decode(church_three))

// 2 * 3 = 6
let church_six = church_mul(church_two, church_three)
Js.Console.log(church_decode(church_six))

运行输出如下：

3
6
3
6

作业完成

上周比较忙，这次迟交了，不好意思 😆