This document discusses representing information of arbitrary size recursively by defining data in terms of itself. It uses the example of defining pizzas, which can be either a plain crust or another pizza with a topping added. Functions for pizzas are defined recursively to decompose their structure. For instance, counting toppings on a pizza recursively decomposes it and adds one to the count of toppings on its base. Recursive data definitions lead to recursive templates and code.

1. Recursive Data DefinitionsCS 5010 Program Design Paradigms “Bootcamp”Week 02, Lesson 2TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA1

2. How to represent info of arbitrary size?phone bookpixels in an imagebombs dropped by a UFOslides in a presentationApproach: think about how such information is constructed.

3. Example: PizzaHow do you construct a pizza? Two possibilities:grab a pizza crusttake a pizza and add a topping...and so on

4. Data Definition(define-struct topped-pizza (topping base))A Topping is a String.A Pizza is either-- the string "plain crust"-- (make-topped-pizza Topping Pizza)

5. Examples(define plain-pizza "plain crust")(define cheese-pizza (make-topped-pizza "cheese" plain-pizza))(define anchovies-cheese (make-topped-pizza "anchovies" cheese-pizza))(define onions-anchovies-cheese (make-topped-pizza "onions" anchovies-cheese))

6. This data definition is self-referential(define-struct topped-pizza (topping base))A Topping is a String.A Pizza is either-- the string "plain crust"-- (make-topped-pizza Topping Pizza)We also call this a recursive data definition

7. Template for pizza functionspizza-fn : Pizza -> ??Given a Pizza, produce ....(define (pizza-fn pizza) (cond [(plain-pizza? pizza) ...] [else (... (topped-pizza-topping pizza) (pizza-fn (topped-pizza-base pizza)))]))Add to wishlist:plain-pizza? : Pizza -> Booleanreturns true iff the given pizza is a plain pizza

8. This template is self-referentialpizza-fn : Pizza -> ??Given a Pizza, produce ....(define (pizza-fn pizza) (cond [(plain-pizza? pizza) ...] [else (... (topped-pizza-topping pizza) (pizza-fn(topped-pizza-base pizza)))]))We also call this a recursive template

9. Example: number of toppingsnumber-of-toppings : Pizza -> NumberGiven a Pizza, produce the number of toppingsExamples:(number-of-toppings plain-pizza) = 0(number-of-toppings cheese-pizza) = 1(number-of-toppings anchovies-cheese) = 2(number-of-toppings onions-anchovies-cheese) = 3

10. Example: number of toppingsnumber-of-toppings : Pizza -> NumberGiven a Pizza, produce the number of toppingsStrategy: structural decomposition(define (number-of-toppings pizza) (cond [(plain-pizza? pizza) ...] [else (... (topped-pizza-topping pizza) (number-of-toppings(topped-pizza-base pizza)))]))How many toppings does the plain pizza have?Answer: 0

11. Example: number of toppingsnumber-of-toppings : Pizza -> NumberGiven a Pizza, produce the number of toppingsStrategy: structural decomposition(define (number-of-toppings pizza) (cond [(plain-pizza? pizza) 0] [else (... (topped-pizza-topping pizza) (number-of-toppings(topped-pizza-base pizza)))]))How many toppings does the plain pizza have?Answer: 0

12. Example: number of toppingsnumber-of-toppings : Pizza -> NumberGiven a Pizza, produce the number of toppingsStrategy: structural decomposition(define (number-of-toppings pizza) (cond [(plain-pizza? pizza) 0] [else (... (topped-pizza-topping pizza) (number-of-toppings(topped-pizza-base pizza)))]))Q: If we knew the number of toppings on the pizza base, how many toppings would there on the whole pizza?Answer: one more topping!

13. Example: number of toppingsnumber-of-toppings : Pizza -> NumberGiven a Pizza, produce the number of toppings(define (number-of-toppings pizza) (cond [(plain-pizza? pizza) 0] [else (+ 1 (number-of-toppings(topped-pizza-base pizza)))]))Self-reference in the data definition leads to self-reference in the template;Self-reference in the template leads to self-reference in the code.Self-reference in the data definition leads to self-reference in the template;Self-reference in the template leads to self-reference in the code.Self-reference in the data definition leads to self-reference in the template;Self-reference in the template leads to self-reference in the code.Self-reference in the data definition leads to self-reference in the template;Self-reference in the template leads to self-reference in the code.

14. Don't forget the tests!;; plain pizza(check-expect (number-of-toppings plain-pizza) 0);; one topping(check-expect (number-of-toppings cheese-pizza) 1)...etc...

15. Steps for the design strategyWhat's the answer for the empty pizza?If you knew the answer for the base pizza, how would you get the answer for the whole pizza?

16. Many other examplesadd-anchovies : Pizza -> Pizzaremove-all-anchovies : Pizza -> PizzaProduce a Pizza like the given one, but with all anchovies removed.replace-all-anchovies-with-onions : Pizza -> Pizzareplace-all-anchovies : Pizza Topping -> Pizzareplace-topping : Pizza Topping Topping -> PizzaProduce a Pizza like the given one, but with all instances of the first topping replaced by the second one.

17. Check the wishlistplain-pizza? : Pizza -> Booleanreturns true iff the given pizza is a plain pizza;; examples: ...;; strategy: function composition(define (plain-pizza? pizza) (and (string? pizza) (string=? pizza "plain crust")));; tests: ....

18. SummaryRepresent arbitrary-sized information using a self-referential (or recursive) data definition.Self-reference in the data definition leads to self-reference in the template.Self-reference in the template leads to self-reference in the code.Follow the recipe!