Class 17: Higher-Order Procedures

Held: Monday, 17 February 2003

Summary: Today we investigate higher-order procedures; procedures that take other procedures as parameters or return procedures as values. Higher-order procedures are one of the most important design techniques in the functional paradigm (nearly as important as recursion).

Related Pages:

• EBoard.
• Reading on higher-order procedures.
• Lab on higher-order procedures.

Assignments:

• Read more higher-order procedures.
• Scan chapters 1 to 3 of Algorithms for Functional Programming.

Notes:

• Are there questions on exam 1?
• I've rearranged the order of topics for the course.

Overview:

• Design patterns, revisited.
• Key ideas of higher-order procedures.
• Two key higher-order procedures: map and apply.
• Lab.

Design Patterns

• As I've mentioned earlier, the more you proram, the more you find that there are many problems whose solutions have similar structures.
• We might then want to look more abstractly at those structures and see what kinds of problems they apply to.
• These solution structures are typically called patterns or design patterns.
• In some languages, patterns are simply a guide to the programmer, giving techniques for designing solutions.
• In Scheme, you can actually encode many patterns in procedures.
• Today we'll consider a few examples of design patterns (typically involving recursion) and see how they might be coded.
• We'll also learn about some related issues: procedures as values and anonymous procedures.

The Short Version

• What you need to know:
  • You can make procedures parameters to other procedures.
  • You can write and use anonymous procedures (lambda expressions that are not named).
  • Procedures can build and return other procedures.
• Some built-in procedures you should know:
  • (map proc lst) -- build a new list by applying proc to each element of lst.
  • (apply proc lst) -- call proc on the arguments given by lst
• Examples
  • (define square-vals (lambda (lst) (map (lambda (x) (* x x)) lst)))
  • (define sum (lambda (lst) (apply + lst)))
• Are there any other questions on the readings?

Lab

• Do the lab.

The Long Version

• Here's more information about the various topics.

Apply to All

• Let's start with two similar problems, both of which somone might use to process a list of grades.
• Add 5 to every value in a list of numbers.
• Multiply every value in a list of numbers by 10/8.
• How might we code these?
  • Done in class. Also available in the reading.
• Can we generalize? Certainly
  • Done in class. Also available in the reading.
• As you may have noted, this seems a little bit inelegant.
• What else can we do?
  • We can take the procedure to apply to each value as a parameter to the general procedure!
• Note that the apply to all design pattern is so common that Scheme includes it as a built-in procedure (called map).

Common Higher-Order Procedures

• (map proc lst): what we just wrote.
• (apply proc lst): a way to treat the elements of lst as parameters to proc.
• Example:

  (define sum (lambda (lst) (apply + lst)))
  

Anonymous Procedures

• When we use higher-order procedures, we often have to build small helpers that say what to do. For example, consider

  > (let ((square (lambda (x) (* x x))))
      (map square (list 1 2 3 4)))
  

• But what does this say? It tells us that square is another name for (lambda (x) (* x x)) and then uses that name.
• When we use the name, Scheme simply substitutes the thing that's been named.
• We can do the substitution ourselves

  (map (lambda (x) (* x x)) '(1 2 3 4))
  

• We've used a procedure without naming it!
• Such unnamed procedures are called anonymous procedures. You'll find that we often use them in conjunction with design patterns.

Some Other Patterns

• Extract all the elements of a list that meet some criteria.
• Generalization of sum and product?

Returning Procedures

• Given that we can take procedures as parameters, it may also make sense to return procedures as values.
• What does a procedure value look like? It looks like

  (lambda (arguments) body)
  

• So, here's a procedure that takes one parameter, a number, and returns a procedure. The resultant procedure takes one parameter and adds the first number to its parameter.

  (define make-adder
    (lambda (n)
      (lambda (v) (+ n v))))
  

• How did we build that? First we thought about the result. We wanted a procedure of one parameter that added some number to its parameter.

  (lambda (v) (+ some-number v))
  

• Now, we want to build that value, filling in the some-number.