Local bindings and recursion

Exercise 1

Write a letrec-expression in which (a) the identifier last-of-list is locally bound to a procedure that finds and returns the last element of a given list, and (b) the body of the expression computes the sum of the last elements of the lists (3 8 2), (7), and (8 5 9 8), invoking last-of-list three times.

Exercise 2

A non-empty list is an s-n-alternator if its elements are alternately symbols and numbers, beginning with a symbol. It is an n-s-alternator if its elements are alternately numbers and symbols, beginning with a number.

Write a letrec-expression in which (a) the identifiers s-n-alternator? and n-s-alternator? are bound to mutually recursive predicates, each of which determines whether a given non-empty list has the indicated characteristic, and (b) the body invokes each of these predicates to determine whether the value of the expression '(2 a 3 b 4 c 5) fits either description.

Exercise 3

Develop a version of the iota procedure that uses letrec to pack a tail-recursive kernel inside a husk that performs precondition testing. (Iota, you'll recall, takes a natural number as argument and returns a list of all the lesser natural numbers, in ascending order.)

Exercise 4

Develop a version of the iota procedure that uses a named let.

Exercise 5

Develop a procedure named take that takes a list ls and a non-negative integer len as arguments and returns a list consisting of the first len elements of ls, in their original order. The procedure should signal an error if ls is not a list, if len is not an exact integer, if len is negative, or if len is greater than the length of ls.

Exercise 6

Develop a procedure named intersection that takes two lists of symbols, left and right, as arguments and returns a list of which the elements are precisely those symbols that are elements of both left and right.