Assignment 3

Due on Oct 9 before lecture.

Please bring a hardcopy of your solutions to lecture. Or submit a hardcopy of your solutions to Paul Darga before lecture.

Problem 1

Recall Church numerals from Chapter 5 of the Pierce book. They use unary representation for natural numbers. Suppose we want to use binary representation instead. One way to encode natural numbers is as follows:

0 = \i. \o. \t. ot
1 = \i. \o. \t. it
2 = \i. \o. \t. i(ot)
3 = \i. \o. \t. i(it)
4 = \i. \o. \t. i(o(ot))
5 = \i. \o. \t. i(o(it))
6 = \i. \o. \t. i(i(ot))
7 = \i. \o. \t. i(i(it))
8 = \i. \o. \t. i(o(o(ot)))
9 = \i. \o. \t. i(o(o(it)))
10 = \i. \o. \t. i(o(i(ot)))
11 = \i. \o. \t. i(o(i(it)))
...

Intuitively, a natural number n is a function that takes three parameters, i as the bit 1, o as the bit 0, and t as the sequence terminator, and returns the sequence of applications representing the binary encoding of n.

Problem 1a (4 Points)

Write a combinator double n that multiplies a natural number n by 2. (A combinator is a lambda term without free variables.)

Problem 1b (4 Points)

Recall the definition of Church booleans from Chapter 5 of the Pierce book:

true = \t. \f. t
false = \t. \f. f

Write a combinator iszero n that returns the Church boolean true if n is zero, and returns the Church boolean false otherwise.