Assignment 7– Scheme Programming

CS314 
Assignment 7
Submission: pdf file through sakai.rutgers.edu
Problem 1 – Scheme Programming
1. As we discussed in class, let and let* do not add anything to the expressiveness of the language, i.e., they are only a convenient shorthand.
For instance,
(let ((x v1) (y v2)) e) can be rewritten as
((lambda (x y) e) v1 v2).
How can you rewrite (let* ((x v1) (y v2) (z v3)) e) in terms
of λ-abstractions and function applications?
2. Use the map and reduce functions we learned in class to implement
function maxAbsoluteVal that determines the maximal absolute value
of a list of integer numbers. Example
(define maxAbsoluteVal
(lambda (l)
... ))
...
(maxAbsoluteVal ’(-5 -3 -7 -10 12 8 7)) -- 12
Problem 2 – Lambda Calculus
Use α/β-reductions to compute the final answer for the following λ-terms.
Your computation ends with a final result if no more reductions can be applied. Does the order in which you apply the β-reduction make a difference
whether you can compute a final result? Justify your answer.
1. (((λx.x) (λx.28)) (λz.z))
1
2. ((λx.((λz.((λx.(z x)) 2)) (λy.(* x y)))) 6)
3. ((λz. ((λy.z) ((λx.(x x))(λx.(x x))))) 11)
Problem 3 – Programming in Lambda Calculus
In lecture 16 and 17, we discussed the encoding of logical constants true
and false in lambda calculus, together with the implementation of logical
operators.
1. Compute the value of ((and true) true) using β-reductions.
2. Define the or operator in lambda calculus. Prove that your definition
is correct, i.e., your lambda term for or implements the logical or operation.
3. Define the exor (exclusive or) operator in lambda calculus. Prove that
your definition is correct, i.e., your lambda term for exor “implements”
the logical exor operation.
Problem 4 – Lambda Calculus and Combinators S & K
Let’s assume the S and K combinators:
• K ≡ λxy.x
• S ≡ λxyz.((xz)(yz))
Prove that the identify function I ≡ λx.x is equivalent to ((S K) K),
i.e.,
I ≡ SKK
2