Bailout Fold

Posted on 2018-01-09 by nbloomf

Tags: arithmetic-made-difficult, literate-haskell

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This page is part of a series on Arithmetic Made Difficult.

This post is literate Haskell; you can load the source into GHCi and play along.

{-# LANGUAGE NoImplicitPrelude #-}
module BailoutFold (
  bfoldr, _test_bfoldr, main_bfoldr
) where

import Testing
import Booleans
import Tuples
import DisjointUnions
import NaturalNumbers
import Lists
import HeadAndTail

In this post we’ll develop a list-flavored version of bailout recursion.

Let \(A\), \(B\), and \(C\) be sets. Suppose we have mappings \[\delta : B \rightarrow C,\] \[\beta : A \times \lists{A} \rightarrow B \rightarrow \bool,\] \[\psi : A \times \lists{A} \times B \rightarrow C,\] and \[\omega : A \times \lists{A} \times B \rightarrow B.\] Then there is a unique mapping \(\Theta : \lists{A} \times B \rightarrow C\) such that \[\Theta(\nil,u) = \delta(u)\] and \[\Theta(\cons(a,x),u) = \bif{\beta(a,x,u)}{\psi(a,x,u)}{\Theta(x,\omega(a,x,u))}.\] We denote this \(\Theta\) by \(\bfoldr(\delta)(\beta)(\psi)(\omega)\).

Define \(\varepsilon : B \times \lists{A} \rightarrow C\) by \[\varepsilon(u,x) = \delta(u)\] and \(\varphi : A \times C^{B \times \lists{A}} \rightarrow C^{B \times \lists{A}}\) by \[\varphi(a,g)(u,x) = \bif{\beta(a,\tail(x),u)}{\psi(a,\tail(x),u)}{g(\omega(a,\tail(x),u),\tail(x))},\] and let \(\Theta : \lists{A} \times B \rightarrow C\) be given by \[\Theta(x,u) = \foldr(\varepsilon)(\varphi)(x)(u,x).\] Note that \[\begin{eqnarray*} & & \Theta(\nil,u) \\ & = & \foldr(\varepsilon)(\varphi)(\nil)(u,\nil) \\ & \href{/posts/arithmetic-made-difficult/Lists.html#def-foldr-nil} = & \varepsilon(u,\nil) \\ & = & \delta(u) \end{eqnarray*}\] and \[\begin{eqnarray*} & & \Theta(\cons(a,x),u) \\ & = & \foldr(\varepsilon)(\varphi)(\cons(a,x))(u,\cons(a,x)) \\ & \href{/posts/arithmetic-made-difficult/Lists.html#def-foldr-cons} = & \varphi(a,\foldr(\varepsilon)(\varphi)(x))(u,\cons(a,x)) \\ & = & \bif{\beta(a,\tail(\cons(a,x)),u)}{\psi(a,\tail(\cons(a,x)),u)}{\foldr(\varepsilon)(\varphi)(x)(\omega(a,\tail(\cons(a,x)),u),\tail(\cons(a,x)))} \\ & = & \bif{\beta(a,x,u)}{\psi(a,x,u)}{\foldr(\varepsilon)(\varphi)(x)(\omega(a,x,u),x)} \\ & = & \bif{\beta(a,x,u)}{\psi(a,x,u)}{\Theta(x,\omega(a,x,u))} \end{eqnarray*}\] as needed. Suppose \(\Psi : \lists{A} \times B \rightarrow C\) also satisfies these equations; we show that \(\Psi = \Theta\) by list induction on \(x\). For the base case \(x = \nil\), we have \[\begin{eqnarray*} & & \Psi(\nil,u) \\ & = & \delta(u) \\ & = & \Theta(\nil,u) \end{eqnarray*}\] as needed. For the inductive step, suppose the equality holds for all \(u\) for some \(x\), and let \(a \in A\). Now \[\begin{eqnarray*} & & \Psi(\cons(a,x),u) \\ & = & \bif{\beta(a,x,u)}{\psi(a,x,u)}{\Psi(x,\omega(a,x,u))} \\ & = & \bif{\beta(a,x,u)}{\psi(a,x,u)}{\Theta(x,\omega(a,x,u))} \\ & = & \Theta(\cons(a,x),u) \end{eqnarray*}\] so that \(\Psi = \Theta\).

Implementation

We can implement \(\bfoldr\) using the definition or the universal property.

bfoldr, bfoldr'
  :: (List t)
  => (b -> c)
  -> (a -> t a -> b -> Bool)
  -> (a -> t a -> b -> c)
  -> (a -> t a -> b -> b)
  -> t a -> b -> c


bfoldr' delta beta psi omega x u = foldr epsilon phi x (tup u x)
  where
    epsilon (Pair b _) = delta b
    phi a g (Pair b y) = if beta a (tail y) b
      then psi a (tail y) b
      else g (tup (omega a (tail y) b) (tail y))


bfoldr delta beta psi omega z u = case uncons z of
  Left () -> delta u
  Right (Pair a x) -> if beta a x u
    then psi a x u
    else bfoldr delta beta psi omega x (omega a x u)

We should check that these are equivalent.

_test_bfoldr_equiv :: (List t, Equal (t a), Equal a, Equal c)
  => t a -> b -> c
  -> Test ((b -> c) -> (a -> t a -> b -> Bool) -> (a -> t a -> b -> c) -> (a -> t a -> b -> b) -> t a -> b -> Bool)
_test_bfoldr_equiv _ _ _ =
  testName "bfoldr(delta,beta,psi,omega)(x,u) == bfoldr'(delta,beta,psi,omega)(x,u)" $
  \delta beta psi omega x u -> eq
    (bfoldr delta beta psi omega x u)
    (bfoldr' delta beta psi omega x u)

Testing

Suite:

_test_bfoldr ::
  ( TypeName a, Show a, Equal a, Arbitrary a, CoArbitrary a
  , TypeName (t a), List t, Equal (t a), Arbitrary (t a), Show (t a), CoArbitrary (t a)
  , CoArbitrary b, CoArbitrary c, Arbitrary b, Arbitrary c, Show b, Equal c
  , TypeName b, TypeName c
  ) => Int -> Int -> t a -> b -> c -> IO ()
_test_bfoldr size cases t b c = do
  testLabel3 "bfoldr" t b c

  let args = testArgs size cases

  runTest args (_test_bfoldr_equiv t b c)

Main:

main_bfoldr :: IO ()
main_bfoldr = do
  _test_bfoldr 50 1000 (nil :: ConsList Bool)  (zero :: Unary) (zero :: Unary)
  _test_bfoldr 50 1000 (nil :: ConsList Unary) (zero :: Unary) (zero :: Unary)
  _test_bfoldr 50 1000 (nil :: ConsList Bool)  (true :: Bool)  (zero :: Unary)
  _test_bfoldr 50 1000 (nil :: ConsList Unary) (true :: Bool)  (zero :: Unary)