From a0f99722a5a10cb939258a6c7eba32a28f2944e8 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Sun, 3 Oct 2010 20:41:34 -0400 Subject: [PATCH 1/1] assignment4 formatting Signed-off-by: Jim Pryor --- assignment4.mdwn | 31 +++++++++++++++++++++++-------- 1 file changed, 23 insertions(+), 8 deletions(-) diff --git a/assignment4.mdwn b/assignment4.mdwn index 54a3bf20..1eec95e1 100644 --- a/assignment4.mdwn +++ b/assignment4.mdwn @@ -14,7 +14,12 @@ can use.
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  1. Suppose you have two lists of integers, `left` and `right`. You want to determine whether those lists are equal: that is, whether they have all the same members in the same order. (Equality for the lists we're working with is *extensional*, or parasitic on the equality of their members, and the list structure. Later in the course we'll see lists which aren't extensional in this way.) +
  2. Suppose you have two lists of integers, `left` and `right`. You want to +determine whether those lists are equal: that is, whether they have all the +same members in the same order. (Equality for the lists we're working with is +*extensional*, or parasitic on the equality of their members, and the list +structure. Later in the course we'll see lists which aren't extensional in this +way.) How would you implement such a list comparison? @@ -36,7 +41,8 @@ First, read this: [[Implementing trees]] #Mutually-recursive functions#
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    1. (Challenging.) One way to define the function `even` is to have it hand off part of the work to another function `odd`: +
    2. (Challenging.) One way to define the function `even` is to have it hand off +part of the work to another function `odd`: let even = \x. iszero x ; if x == 0 then result is @@ -44,7 +50,8 @@ First, read this: [[Implementing trees]] ; else result turns on whether x's pred is odd (odd (pred x)) -At the same tme, though, it's natural to define `odd` in such a way that it hands off part of the work to `even`: +At the same tme, though, it's natural to define `odd` in such a way that it +hands off part of the work to `even`: let odd = \x. iszero x ; if x == 0 then result is @@ -52,7 +59,10 @@ At the same tme, though, it's natural to define `odd` in such a way that it hand ; else result turns on whether x's pred is even (even (pred x)) -Such a definition of `even` and `odd` is called **mutually recursive**. If you trace through the evaluation of some sample numerical arguments, you can see that eventually we'll always reach a base step. So the recursion should be perfectly well-grounded: +Such a definition of `even` and `odd` is called **mutually recursive**. If you +trace through the evaluation of some sample numerical arguments, you can see +that eventually we'll always reach a base step. So the recursion should be +perfectly well-grounded: even 3 ~~> iszero 3 true (odd (pred 3)) @@ -64,24 +74,29 @@ Such a definition of `even` and `odd` is called **mutually recursive**. If you t ~~> iszero 0 false (even (pred 0)) ~~> false -But we don't yet know how to implement this kind of recursion in the lambda calculus. +But we don't yet know how to implement this kind of recursion in the lambda +calculus. The fixed point operators we've been working with so far worked like this: let X = Y T in X <~~> T X -Suppose we had a pair of fixed point operators, `Y1` and `Y2`, that operated on a *pair* of functions `T1` and `T2`, as follows: +Suppose we had a pair of fixed point operators, `Y1` and `Y2`, that operated on +a *pair* of functions `T1` and `T2`, as follows: let X1 = Y1 T1 T2 in let X2 = Y2 T1 T2 in X1 <~~> T1 X1 X2 and X2 <~~> T2 X1 X2 -If we gave you such a `Y1` and `Y2`, how would you implement the above definitions of `even` and `odd`? +If we gave you such a `Y1` and `Y2`, how would you implement the above +definitions of `even` and `odd`? -
    3. (More challenging.) Using our derivation of Y from the [Week3 notes](/week3/#index4h2) as a model, construct a pair `Y1` and `Y2` that behave in the way described. +
    4. (More challenging.) Using our derivation of Y from the [Week3 +notes](/week3/#index4h2) as a model, construct a pair `Y1` and `Y2` that behave +in the way described. (See [[hints/Assignment 4 hint 4]] if you need some hints.) -- 2.11.0