Folds Equivalence - Exercise 2.38

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In this article, we will explore the concept of folds equivalence, specifically focusing on exercise 2.38 from the book "Structure and Interpretation of Computer Programs" (SICP). We will delve into the properties of folds, including associativity and commutativity, and provide examples to illustrate these concepts.

Introduction to Folds

Folds are a fundamental concept in functional programming, allowing us to reduce a list of values to a single value by applying a binary operation. There are two types of folds: left-fold and right-fold. The left-fold corresponds to nested (op prev-values-acc curr-el) calls, while the right-fold corresponds to (op curr-el next-values-acc) calls.

Associativity and Commutativity

To establish the equivalence between left-fold and right-fold, we need to show that the operation used in the fold is both associative and commutative.

Commutativity

Commutativity is a property of an operation that states that the order of the operands does not affect the result. In the context of folds, commutativity is necessary because the left-fold corresponds to nested (op prev-values-acc curr-el) calls, while the right-fold corresponds to (op curr-el next-values-acc) calls. To move the init value from (op last-el init) "up to" (op init first-el), we need to swap the order of each op, which requires commutativity.

Associativity

Associativity is a property of an operation that states that the order in which we apply the operation to multiple operands does not affect the result. In the context of folds, associativity is necessary because we need to move the init value from (op last-el init) "up to" (op init first-el). This requires that the operation be associative, so that we can move the init value up the chain of operations without affecting the result.

Visualizing the Equivalence

We can visualize the equivalence between left-fold and right-fold by applying the operations in a step-by-step manner. Here is an example:

(left-fold op init (1 2 3 ... n))
= (op ...(op (op (op init 1) 2) 3)... n)
{ associativity }
= (op ...(op (op (op init 1) 2) n)... 3) ;; elements move to the right
= (op (op ...(op (op init 1) n)... 2) 3)
= (op (op ...(op (op init 1) n)... 3) 2) ;; to their appropriate ordered place
= (op (op (op ...(op init n)... 1) 3) 2)
= (op (op (op ...(op init n)... 3) 1) 2)
= (op (op (op ...(op init n)... 3) 2) 1)
{ commutativity }
= (op (op (op ...(op n init)... 3) 2) 1) ;; argument swaps
= (op (op (op 3 ...(op n init)...) 2) 1)
= (op (op 2 (op 3 ...(op n init)...)) 1)
= (op 1 ( 2 (op 3 ...(op n init)...)))
= (fold-right op init (1 2 3 ... n))

Example: Commutativity Does Not Suffice

While commutativity is necessary for the equivalence between left-fold and right-fold, it is not sufficient. Consider the following example:

(define (f a b) (+ (* 2 a) (* 2 b)))
;; f is commutative
;; f(a,b) = 2a + 2b = 2b + 2a = f(b,a)
;; but not associative
;; f(a,f(b,c)) = 2a + 2f(b,c) = 2a + 2(2b + 2c) = 2a + 4b + 4c
;; !=
;; f(f(a,b),c) = 2f(a,b) + 2c = 2(2a + 2b) = 4a + 4b + 2c

(fold-left f 0 (list 1 2 3 4)) ;; 52
(fold-right f 0 (list 1 2 3 4)) ;; 98

In this example, the operation f is commutative, but not associative. As a result, the left-fold and right-fold produce different results.

Conclusion

In this article, we will continue to explore the concept of folds equivalence, specifically focusing on exercise 2.38 from the book "Structure and Interpretation of Computer Programs" (SICP). We will provide a Q&A section to help clarify any doubts and provide additional insights into the topic.

Q: What is the difference between left-fold and right-fold?

A: The main difference between left-fold and right-fold is the order in which the operation is applied to the elements of the list. In left-fold, the operation is applied from left to right, while in right-fold, the operation is applied from right to left.

Q: Why is associativity necessary for the equivalence between left-fold and right-fold?

A: Associativity is necessary because we need to move the init value from (op last-el init) "up to" (op init first-el). This requires that the operation be associative, so that we can move the init value up the chain of operations without affecting the result.

Q: Why is commutativity necessary for the equivalence between left-fold and right-fold?

A: Commutativity is necessary because the left-fold corresponds to nested (op prev-values-acc curr-el) calls, while the right-fold corresponds to (op curr-el next-values-acc) calls. To move the init value from (op last-el init) "up to" (op init first-el), we need to swap the order of each op, which requires commutativity.

Q: Can you provide an example of an operation that is commutative but not associative?

A: Yes, consider the following example:

(define (f a b) (+ (* 2 a) (* 2 b)))
;; f is commutative
;; f(a,b) = 2a + 2b = 2b + 2a = f(b,a)
;; but not associative
;; f(a,f(b,c)) = 2a + 2f(b,c) = 2a + 2(2b + 2c) = 2a + 4b + 4c
;; !=
;; f(f(a,b),c) = 2f(a,b) + 2c = 2(2a + 2b) = 4a + 4b + 2c

In this example, the operation f is commutative, but not associative.

Q: Can you provide an example of an operation that is associative but not commutative?

A: Yes, consider the following example:

(define (f a b) (cons a b))
;; f is associative
;; f(f(a,b),c) = (cons a (cons b c)) = (cons (cons a b) c) = f(a,f(b,c))
;; but not commutative
;; f(a,b) = (cons a b) != (cons b a) = f(b,a)

In this example, the operation f is associative, but not commutative.

Q: How can I determine if an operation is associative or commutative?

A: To determine if an operation is associative commutative, you can use the following steps:

  • For associativity: Check if (f (f a b) c) = (f a (f b c)) for all a, b, and c.
  • For commutativity: Check if (f a b) = (f b a) for all a and b.

Q: Can you provide a summary of the key points discussed in this article?

A: Yes, here is a summary of the key points discussed in this article:

  • The equivalence between left-fold and right-fold requires that the operation used in the fold be both associative and commutative.
  • Associativity is necessary because we need to move the init value from (op last-el init) "up to" (op init first-el).
  • Commutativity is necessary because the left-fold corresponds to nested (op prev-values-acc curr-el) calls, while the right-fold corresponds to (op curr-el next-values-acc) calls.
  • An example of an operation that is commutative but not associative is f(a,b) = 2a + 2b.
  • An example of an operation that is associative but not commutative is f(a,b) = (cons a b).
  • To determine if an operation is associative or commutative, you can use the following steps: Check if (f (f a b) c) = (f a (f b c)) for all a, b, and c for associativity, and check if (f a b) = (f b a) for all a and b for commutativity.