论弗雷格的丘奇数纲

Question

此程序在 GHC 下正确编译和运行：

type Church a = (a -> a) -> a -> a

ch :: Int -> Church a
ch 0 _ = id
ch n f = f . ch (n-1) f

unch :: Church Int -> Int
unch n = n (+1) 0

suc :: Church a -> Church a
suc n f = f . n f

pre :: Church ((a -> a) -> a) -> Church a
pre n f a = n s z id
    where s g h = h (g f)
          z     = const a

main :: IO ()
main = do let seven = ch 7
              eight = suc seven
              six   = pre seven
          print (unch eight)
          print (unch six)

但是用弗雷格编译时出现了以下错误：

E /home/xgp/work/flab/src/main/frege/flab/fold.fr:23: type error in expression seven
    type is : Int
    expected: (t1→t1)→t1
E /home/xgp/work/flab/src/main/frege/flab/fold.fr:23: type error in expression seven
    type is : (t1→t1)→t1
    expected: Int
E /home/xgp/work/flab/src/main/frege/flab/fold.fr:23: type error in expression seven
    type is : (t1→t1)→t1
    expected: Int
E /home/xgp/work/flab/src/main/frege/flab/fold.fr:23: type error in
    expression seven
    type is apparently Int
    used as function

为什么？是否可以修改程序在Frege下通过编译？

Answer 1

这是 let 绑定变量类型的泛化确实产生影响的罕见情况之一。

关键是，弗雷格在这方面就像 GHC with pragma -XMonoLocalBinds，详情请看这里：https://github.com/Frege/frege/wiki/GHC-Language-Options-vs.-Frege#Let-Generalization and here: https://downloads.haskell.org/~ghc/latest/docs/html/users_guide/other-type-extensions.html#typing-binds（SPJ 的一篇论文也有 link，这解释了基本原理）

简而言之，这意味着所有未注释的 let bound veriabes 都将具有 monomorphic 类型，并且不能用于不同类型。要恢复多态性，需要显式类型签名。

要让你的程序编译通过，注释 seven 与

的绑定就足够了

seven :: Church a

关于print/println：前者不刷新输出。所以你在 REPL 中有：

frege> print 'a'
IO ()
frege> print 'b'
IO ()
frege> println "dammit!"
abdammit!
IO ()

论弗雷格的丘奇数纲

On Church numeral program under Frege

frege