- Deadline: April 28. Please put your answer into Xiaoming's mailbox (tagged by X.GU) in CS department mailroom.

1. Draw the syntax tree for the following lambda expression.

For call-by-value and call-by-name semantics, give the complete
content of the environment at the time when sub-expressions
`(x a)`

and `(+ a a)`

are evaluated. If an environment has
more than one binding, indicate the order in which they are
searched for a particular name. Give the result of the evaluation.

((lambda (a) ((lambda(x) ((lambda (a) (x a)) (+ 2 1))) (lambda(y) (+ a a)))) (+ 1 1))

2. Recursion construction using lambda calculus (455 only)

By recursion we mean a function is defined in terms of itself. Unfortunately lambda calculus does not allow this. However, a function can call some special function and then regenerate itself. By this means, we can achieve recursion using lambda calculus. An instance of this powerful function is defined as

Y := (lambda (y) ((lambda (x) (y x x)) (lambda (x) (y x x))))

It's easy to verify that for any function `F`

, `YF = F(YF)`

.
Now Suppose we want to define a recursive function, something like

F(n) = 1, if n == 0; else n*F(n-1)

Since we can't use `F`

in defining `F`

, we use `f`

instead as
a place-holder argument for the function to be passed to itself.
Therefore the lambda calculus representation of the function is

F := (lambda (f n) (if (zero n) 1 (* n (f (- n 1)))))

Apply `Y`

to `F`

, then we get `(YF)n = F(YF)n = F(F(YF))n = ...`

By expanding `YF`

to `F(YF)`

level by level, the function gets evaluated
in a recursive way.

Now you need to define a recursive function that counts the number
of elements from a list. You may use `if`

conditional test as following
syntax

`(if <test-expr> <true-expr> <false-expr>)`

which means if TEST-EXPR is evaluated true then evaluate TRUE-EXPR,
o/w evaluate FALSE-EXPR. You may also use `empty`

as a predicate
testing whether a list is empty or not, and `rest`

returning the list
with the first element popped off.

To answer this question, you need to put down your `F`

function,
then show how `(YF)(list)`

expands.

-- XiaomingGu - 14 Apr 2008

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Topic revision: r2 - 2008-04-16 - XiaomingGu

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