Fundamentals of Computer Science I (CS151 2003F)

Design Patterns and Higher-Order Procedures

Summary: In this reading, you will investigate the so called higher-order procedures, procedures that take procedures as parameters or return procedures as results. In effect you are considering what happens when we allow procedures to serve as values.

Contents:

Background: Design Patterns

One mark of successful programmers is that they identify and remember common techniques for solving problems. Such abstractions of common structures for solving problems are often called patterns or design patterns. You should already have begun to identify some patterns. For example, you know that procedures almost always have the form

(define procname
  (lambda (parameters)
    body))

You may also have a pattern in mind for the typical recursive procedure over lists:

(define procname
  (lambda (lst)
    (if (null? lst) 
        base-case
        (do-something (car lst) (procname (cdr lst))))))

In some languages, these patterns are simply guides to programmers as they design new solutions. In other languages, such as Scheme, you can often encode a design pattern in a separate procedure.

A Simple Pattern: Apply a Procedure to List Values

Let's begin with two similar problems, both of which an instructor might apply to a list of grades after determining that grading was too harsh. That instructor might want to create a new list in which each grade is incremented by 5 or that instructor might want to multiply every value in the list by 5/4.

Here are simple implementations of the two procedures.

;;; Procedure:
;;;   add-five-to-all
;;; Parameters:
;;;   lst, a list of numbers.
;;; Purpose:
;;;   Add 5 to every value in lst.
;;; Returns:
;;;   newlst, a list of numbers.
;;; Preconditions:
;;;   lst contains only numbers. [Unverified]
;;; Postconditions:
;;;   newlst is the same length as lst
;;;   Each element of newlst list is five greater than the
;;;     corresponding element of lst.
(define add-five-to-all
  (lambda (lst)
    ; If no elements remain, we can't add 5 to anything, so
    ; stick with the empty list.
    (if (null? lst) null
        ; Otherwise, add 5 to the first number, add 5 to all
        ; the remaining numbers, and shove 'em together.
        (cons (+ 5 (car lst)) 
              (add-five-to-all (cdr lst))))))

;;; Procedure:
;;;   scale-all-by-five-fourths
;;; Parameters:
;;;   lst, a list of numbers.
;;; Purpose:
;;;   Scales every value in a list of numbers by 5/4
;;; Returns:
;;;   newlst, a list of numbers.
;;; Preconditions:
;;;   lst contains only numbers. [Unverified]
;;; Postconditions:
;;;   newlst is the same length as lst
;;;   Each element of newlst list is 5/4 times the
;;;     corresponding element of lst.
(define scale-all-by-five-fourths
  (lambda (lst)
    ; If no elements remain, we can't do any more multiplications,
    ; so stick with the empty list.
    (if (null? lst) null
        ; Otherwise, scale the first number, scale all
        ; the remaining numbers, and shove 'em together.
        (cons (* (/ 5 4) (car lst)) 
              (scale-all-by-five-fourths (cdr lst))))))

What do these two procedures have in common? Most of the documentation. They also both return null when given the null list. More importantly, both do something to the car of the list, recurse on the cdr of the list, and then cons the two results together.

Hence, we can design a general pattern for apply a procedure to every value in a list.

(define procname
  (lambda (lst)
    (if (null? lst)
        null
        (cons (modify (car lst)) 
              (procname (cdr lst))))))

We can even turn this pattern into a procedure.

;;; Procedure:
;;;   do-something-to-all
;;; Parameters:
;;;   lst, a list of numbers.
;;; Purpose:
;;;   Applies MODIFY to each value in a list.
;;; Returns:
;;;   newlst, a list of numbers.
;;; Preconditions:
;;;   lst contains only numbers. [Unverified]
;;;   MODIFY is defined, takes one number as a parameter
;;;     and returns a number. [Unverified]
;;; Postconditions:
;;;   newlst is the same length as lst
;;;   Each element of newlst list the result of applying MODIFY
;;;     to the corresponding element of lst.
(define do-something-to-all
  (lambda (lst)
    ; If no elements remain, we can't apply anything else,
    ; so stick with the empty list.
    (if (null? lst) null
        ; Otherwise, apply the procedure  to the first number, 
        ; apply the procedure to the remaining numbers,  and
        ; put the results back together into a list.
        (cons (MODIFY (car lst)) 
              (do-something-to-all (cdr lst))))))

Where does MODIFY come from? We could define it first (as the preconditions suggest). For example, here's an application of do-something-to-all in which we add 5 to each value.

> (define MODIFY
    (lambda (val) (+ val 5)))
> (MODIFY 4)
9
> (do-something-to-all (list 1 2 3))
(6 7 8)

Similarly, we can multiply all the values in a list by 4/5 by redefining MODIFY appropriately.

> (define MODIFY
    (lambda (val) (* (/ 5 4) val)))
> (MODIFY 4)
5
> (do-something-to-all (list 1 2 3))
(5/4 5/2 15/4)

You may find this strategy of defining MODIFY first inelegant. I certainly do. It also won't work for all cases. For example, what if we want to add five to all the numbers in a list and then scale by five-fourths? We'd have to redefine MODIFY in the middle of our code. Ugly.

Is there a better solution? Yes! Instead of forcing MODIFY to be defined before we call do-something-to-all, we can make the procedure a parameter to the pattern. For example, here's a renamed version of do-something-to-all that takes the extra parameter.

;;; Procedure:
;;;   apply-all
;;; Parameters:
;;;   proc, a procedure that takes one parameter (a number) and
;;;     returns a number.
;;;   lst, a list of numbers.
;;; Purpose:
;;;   Applies proc to each value in a list.
;;; Returns:
;;;   newlst, a list of numbers.
;;; Preconditions:
;;;   The types of the parameters are correct. [Unverified]
;;; Postconditions:
;;;   newlst is the same length as lst
;;;   Each element of newlst is the result of applying proc
;;;     to the corresponding element of lst.
(define apply-all
  (lambda (proc lst)
    ; If no elements remain, we can't apply anything else,
    ; so stick with the empty list.
    (if (null? lst) null
        ; Otherwise, apply the procedure  to the first number, 
        ; apply the procedure to the remaining numbers,  and
        ; put the results back together into a list.
        (cons (proc (car lst)) 
              (apply-all proc (cdr lst))))))

Here are some simple tests

> (let ((addfive (lambda (v) (+ 5 v))))
    (apply-all addfive (list 1 2 3)))
(6 7 8)
> (let ((square (lambda (v) (* v v))))
    (apply-all square (list 1 2 3)))
(1 4 9)
> (apply-all list (list 1 2 3))
((1) (2) (3))
> (apply-all odd? (list 1 2 3))
(#t #f #t)

You should have observed a very important (and somewhat stunning) moral from this example, procedures can be parameters to other procedures. We call the procedures that take other procedures as parameters higher-order procedures.

List Of Whatever

There are many advantages to encoding design patterns in higher-order procedures. An important one is that it stops us from tediously writing the same thing over and over and over again. Think about writing the predicates list-of-numbers?, list-of-reals?, list-of-pairs?, list-of-symbols?, and so on and so forth. As one colleague says,

Writing and testing one of these definitions is an interesting and instructive exercise for the beginning Scheme programmer. Writing and testing another one is good practice. Writing and testing the third one is, frankly, a little tedious. If we then move on to [others], eventually programming is reduced to typing.

So, how do we avoid the repetitious typing? We begin with one of the procedures.

(define list-of-reals?
  (lambda (val)
    (or (null? val)
        (and (pair? val)
             (real? (car val))
             (list-of-reals? (cdr val))))))

Next, we identify the parts of the procedure that depend on our current type (i.e., that everything is a list).

(define list-of-XXX?
  (lambda (val)
    (or (null? val)
        (and (pair? val)
             (XXX? (car val))
             (list-of-XXX? (cdr val))))))

Finally, we remove the dependent part or parts from the procedure name and make them parameters to the modified procedure.

(define list-of
  (lambda (pred? val)
    (or (null? val)
        (and (pair? val)
             (pred? (car val))
             (list-of pred? (cdr val))))))

Here's how we might test whether something is a list of numbers.

> (list-of number? (list 1 2 3))
#t
> (list-of number? (list 1 'a 3))
#f

We can also define list-of-numbers using this procedure.

(define list-of-numbers?
  (lambda (lst)
    (list-of number? lst)))

The results are the same.

> (list-of-numbers? (list 1 2 3))
#t
> (list-of-numbers? (list 1 'a 3))
#f

Built-in Higher-Order Procedures

We have seen that it is possible to write our own higher-order procedures. Scheme also includes a number of built-in higher-order procedures. You can read about many of them in section 6.4 of the Scheme report (r5rs), which is available through the DrScheme Help Desk. Here are some of the more popular ones.

The map procedure takes as arguments a procedure and one or more lists and builds a new list whose contents are the result of applying the procedure to the corresponding elements of each list. (That is, the ith element of the result list is the result of applying the procedure to the ith element of each source list.) map is essentially the same as our apply-all except that map does not guarantee that it steps through the list in a left-to-right order.

One of the most important built-in higher-order procedures is apply, which takes a procedure and a list as arguments and invokes the procedure, giving it the elements of the list as its arguments:

> (apply string=? (list "foo" "foo"))
#t
> (apply * (list 3 4 5 6))
360
> (apply append (list (list 'a 'b 'c) (list 'd) (list 'e 'f)
                       null (list 'g 'h 'i)))
(a b c d e f g h i)

Anonymous Procedures

Recall that in the examples above we wrote something like

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

Let's take this code apart. The let expression makes square the name of a procedure of one parameter that squares the parameter. The code then applies the square procedure to a list. Scheme substitutes the procedure for square (just as it substitutes a value for a named value). Hence, to Scheme, the code above is essentially equivalent to

(map (lambda (v) (* v v)) (list 1 2 3))

Because Scheme does this substitution, we can also do it. That is, we can write the previous code and have Scheme execute it.

> (map (lambda (v) (* v v)) (list 1 2 3))
(1 4 9)

So, what does this new code say? It says Apply a procedure that squares its argument to ever element of the list (1 2 3). Do we care what that procedure is named? No. We describe procedures without names as anonymous procedures. You will find that you frequently use anonymous procedures with design patterns and higher-order procedures.

In effect, you can read lambda as a procedure. Hence, (lambda (params) body) can be considered a shorthand for a procedure of parameters params that computes body.

Other Common Design Patterns

At this point, you've seen many other design patterns that typically involve recursion. You may find it valuable to design corresponding procedures to encapsulate those patterns. Here are some of the patterns you may wish to think about:

insert, which inserts a binary (two parameter) operation between all of values in a list. For example, sum inserts a plus between neighboring values in a list (i.e., (sum (list 1 2 3 4)) is 1+2+3+4). Similarly, product inserts a times between neighboring values. If we had defined insert (see the lab), we might define sum as

(define sum
  (lambda (lst) (insert + lst)))

tally, which counts the number of values in a list that meet a particular predicate. For example, to count the number of odd values in a list, we'd use

(tally odd? lst)

Similarly, to count the number of sevens or elevens in a list, we'd use

(tally (lambda (v) (or (= 7 v) (= 11 v))) lst)

select, which selects all elements of a list that match a particular predicate. For example,

> (select odd? (list 1 2 3 4 5 6 7))
(1 3 5 7)

remove, which removes all elements of a list that match a particular predicate. For example,

> (remove odd? (list 1 2 3 4 5 6 7))
(2 4 6)

Returning Procedures

Just as it is possible to use procedures as parameters to procedures, it is also possible to return new procedures from procedures. For example, here is a procedure that takes one parameter, a number, and creates a procedure that multiplies its parameters by that number.

;;; Procedure:
;;;   make-multiplier
;;; Parameters:
;;;   n, a number
;;; Purpose:
;;;   Creates a new procedure which multiplies its parameter by n.
;;; Produces:
;;;   proc, a procedure of one parameter
;;; Preconditions:
;;;   n must be a number
;;; Postconditions:
;;;   (proc v) gives v times n.
(define make-multiplier
  (lambda (n) ; Parameter
    ; Return value: A procedure
    (lambda (v) (* v n))))

Let's test it out

> (make-multiplier 5)
#<procedure>
> (define timesfive (make-multiplier 5))
> (timesfive 4)
20
> (timesfive 101)
505
> (map (make-multiplier 3) (list 1 2 3))
(3 6 9)

We can use the same technique to build the legendary compose operation that given two functions, f and g, builds a function that applies g and then f.

;;; Procedure:
;;;   compose
;;; Parameters:
;;;   f, a unary function
;;;   g, a unary function
;;; Purpose:
;;;   Functionally compose f and g.
;;; Produces:
;;;   fun, a unary function.
;;; Preconditions:
;;;   f can be applied to any values g generates.
;;; Postconditions:
;;;   fun can be applied to any values g can be applied to.
;;;   fun generates values of the type that f generates.
;;;   (fun x) = (f (g x))
(define compose
  (lambda (f g) ; Parameters to compose
    ; Build a procedure of one parameter
    (lambda (x)
      (f (g x)))))

Here are some fun tests.

> (define add2 (lambda (x) (+ 2 x)))
> (define mul5 (lambda (x) (* 5 x)))
> (define fun1 (compose add2 mul5))
> (fun1 5)
27
> (fun1 3)
17
> (define fun2 (compose mul5 add2))
> (fun2 5)
35
> (fun2 3)
25

We can use a more advanced technique to build a procedure that builds list predicates. We'll write a procedure, make-list-pred that takes one parameter, a predicate, and returns a procedure of one parameter, a list, and determines whether the predicate holds for every element of the list. Since we need to build a recursive procedure, we'll use letrec to build that procedure.

;;; Procedure:
;;;   make-list-predicate
;;; Parameters:
;;;   pred?, a unary predicate
;;; Purpose:
;;;   Creates a predicate that takes one parameter, and verifies
;;;   that (1) the parameter is a list and (2) pred? holds for
;;;   each element of the list.
;;; Produces:
;;;   list-pred?, a unary predicate.
;;; Preconditions:
;;;   pred? is a unary predicate [Unverified]
;;;   pred? can be applied to any kind of value [Unverified]
;;; Postconditions:
;;;   list-pred? is a unary predicate (takes one parameter, returns
;;;     #t or #f)
;;;   (list-pred? val) returns #t if (1) val is a list and (2) pred?
;;;      holds for each element of val.
;;;   (list-pred? val) returns #f otherwise.
(define make-list-predicate
  (lambda (pred?) ; Parameter to list-pred
    ; Build an appropriate procedure with one parameter
    (letrec ((list-pred? 
              (lambda (val)
                (or (null? val)
                    (and (pair? val)
                         (pred? (car val))
                         (list-pred? (cdr val)))))))
      ; And return it
      list-pred?)))

It's now even simpler to define list predicates.

> (define list-of-numbers? (make-list-predicate number?))
> (define list-of-odds? (make-list-predicate odd?))
> (define list-of-symbols? (make-list-predicate symbol?))
> (list-of-numbers? null)
#t
> (list-of-numbers? (list 1 2 3))
#t
> (list-of-numbers? (list 3+4i 5 2.4))
#t
> (list-of-numbers? (list 3 2 'a))
#f
> (list-of-numbers? (list 'a 2 3))
#f
> (list-of-symbols? (list 'a 'b 'c))
#t
> (list-of-symbols? (list 'a 'b 1))
#f
> (list-of-symbols? null)
#t
> (list-of-odds? (list 1 2 3))
#f
> (list-of-odds? (list 1 3 5))
#t
> (list-of-odds? (list 'a))
odd? expects argument of type <integer>; given a

That last error message is a direct result of the author ignoring the preconditions of make-list-predicate. Since odd? can't be applied to all values, (make-list-predicate odd?) is not guaranteed to work.

Common Procedure Builders

Here are some common procedures that you might use to build other procedures. Not all are predefined in Scheme. We've included some definitions. You should be able to write the others.

(left-section binary-proc arg1) -- Creates a new procedure by filling in the first argument of a binary procedure. For example, we might define a procedure that multiplies a value by 2 with

(define mul2 (left-section * 2))

The left-section procedure is surprisingly powerful. Note that with a little more work, we could even use it to define a variant of make-list-predicate.

(define make-list-predicate
  (lambda (pred?) 
    (left-section list-of pred?)))

The code for left-section is relatively straightforward.

(define left-section
  (lambda (binproc arg1)
    ; Build a new procedure of one argument
    (lambda (arg2)
      ; That calls binproc on the appropriate arguments
      (binproc arg1 arg2))))

(right-section binary-proc arg2) -- Creates a new procedure by filling in the second argument of a binary procedure. For example, we might define a procedure that subtracts 3 from a value with

(define sub2 (right-section - 3))

(make-filter pred?) -- Creates a new filter that uses pred? to determine whether or not to keep elements. Note that make-filter is very similar to remove except that remove takes two parameters (the predicate and the list) and returns a list, while make-filter takes one parameter (the predicate) and returns a function whose purpose is to read a list and remove elements that match the predicate.

(conjunction pred1? pred2? ... predn?) -- Create a new predicate that holds only if all of pred1?, pred2?, through predn? hold (the predicates should be tested in order). We can use conjunction to build a better version of even.

(define better-even? (conjunction number? even?))

(disjunction pred1? pred2? ... predn?) -- Create a new predicate that holds if pred1? holds or pred2? holds or ... or predn? hods. The predicates should be tested in order.

(complement pred?) -- Create a new predicate whose result is the opposite of the result of pred?. This procedure can be defined as

(define complement
  (lambda (pred?)
    ; Result procedure
    (lambda (val)
      (not (pred? val)))))

Some people refer to conjunction as ^and (for higher and), disjunction as ^or (for higher or) and complement as ^not (for higher not).

 

History

Wednesday, 14 March 2001 [Samuel A. Rebelsky]

Tuesday, 15 October 2002 [Samuel A. Rebelsky]

Wednesday, 16 October 2002 [Samuel A. Rebelsky]

Thursday, 13 February 2003 [Samuel A. Rebelsky]

Friday, 14 February 2003 [Samuel A. Rebelsky]

Monday, 11 March 2003 [Samuel A. Rebelsky]

 

Disclaimer: I usually create these pages on the fly, which means that I rarely proofread them and they may contain bad grammar and incorrect details. It also means that I tend to update them regularly (see the history for more details). Feel free to contact me with any suggestions for changes.

This document was generated by Siteweaver on Tue Dec 9 14:00:12 2003.
The source to the document was last modified on Mon Sep 1 13:26:32 2003.
This document may be found at http://www.cs.grinnell.edu/~rebelsky/Courses/CS151/2003F/Readings/higher-order.html.

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Samuel A. Rebelsky, rebelsky@grinnell.edu