Final answer:
The question deals with the definition of a lambda function for computing the hypotenuse of a right triangle in CPS. Only part a provides the correct implementation of the Pythagorean theorem, while the other parts contain incorrect or recursive definitions that either misrepresent the theorem or lead to infinite loops.
Step-by-step explanation:
The question involves the use of CPS (Continuation-Passing Style) in conjunction with the lambda calculus and functions to calculate the length of the hypotenuse of a right triangle using the Pythagorean theorem. In CPS, functions do not return values directly; instead, they pass the result to another function known as a continuation.
In part a, (define hyp (lambda (a b) (sqrt (+ ( a a ) ( b b))))), the lambda function defines the calculation for the hypotenuse 'hyp' using the Pythagorean theorem correctly, which states c = √(a² + b²).
Part b, (define hyp (lambda (a b) (sqrt (hyp a b)))), shows an incorrect definition because it recursively calls the 'hyp' function inside the square root, leading to an infinite loop.
Part c, (define hyp (lambda (a b) (sqrt (* ( a a ) ( b b))))), incorrectly calculates the hypotenuse by multiplying the squares of 'a' and 'b' and then taking the square root, which does not align with the Pythagorean theorem.
Part d, (define hyp (lambda (a b) (sqrt (- ( a a ) ( b b))))), is also incorrect because it involves subtracting the square of 'b' from the square of 'a', which is not a valid transformation of the Pythagorean theorem for finding the hypotenuse.