Due
Friday, 1 February 2019
Summary
We explore why and how to define your own procedures in Racket.
Prerequisites
Algorithm building blocks. An abbreviated introduction to Racket.

## Introduction

As you may recall from the introduction to algorithms, the ability to write subroutines is one of the key components of algorithm design. Subroutines have a name that we use to refer to the subroutine and zero or more parameters that provide the input to the subroutine.

For example, we might want to define a procedure, square, that takes as input a number and computes the square of that number.

> (square 5)
25
> (square 1.5)
2.25
> (square 1/3)
1/9
> (+ (square 1/2) (square 1/3))
13/36
> (square (square 2))
16


Of course, square can have multiple meanings. If we’re making drawings, it could also mean “make a square”. Let’s consider an example.

> (color-square 10 "red")

> (color-square 5 "blue")

> (above (color-square 12 "red")
(beside (color-square 8 "blue")
(color-square 8 "purple")))



The square is a relatively simple example. Consider, for example, the following definition of a simple drawing of a house.

> (overlay/align "center" "bottom"
(overlay/align "left" "center"
(circle 3 "solid" "yellow")
(rectangle 15 25 "solid" "brown"))
(above (triangle 50 "solid" "red")
(rectangle 40 50 "solid" "black")))



What if we want to make different size or different color houses? We could copy and paste the code. However, if we changed our mind about how to structure our houses, we’d then have to update every copy. We’d be better off writing a procedure that takes the size and color as parameters.

> (house 50 "black")

> (house 30 "blue")

> (beside (house 30 "blue") (house 30 "green") (house 30 "yellow"))



## Defining procedures with lambda

Racket provides a variety of mechanisms for defining procedures. We will start with the most general, which uses the keyword lambda. This mechanism is relatively straightforward.

Typically, we think of a procedure as having three main aspects: The name we use to refer to the procedure, the names of the parameters (inputs) to the procedure, and the instructions the procedure executes.

Here is the general form of procedure definitions in Racket, at least as we will use them in this class. Racket does not require the indentation, but it makes it much easier to read and we will require it.

(define procedure-name
(lambda (formal-parameters)
instructions))


You’ve already seen the define; we use define to name things. In this case, we’re naming a procedure, rather than a value. The procedure-name part is straightforward; it’s the name we will use to refer to the procedure. The “lambda” is a special keyword in Racket to indicate that “Hey! This is a procedure!”. (Lambda has a special place in the history of mathematical logic and programming language design and has always meant “function” or “procedure”. It’s special enough that the designers of DrRacket chose it for the icon.) The formal-parameters are the names that we give to the inputs. For example, we might call our input to the color-square procedure side-length and color. Finally, the instructions are a series of Racket expressions that explain how to do the associated work.

Let’s look at a simple example, that of squaring a number.

(define my-square
(lambda (x)
(* x x)))


Mentally, most Racket programmers read this as something like

my-square names a procedure that takes one input, x, and computes its result by multiplying x by itself.

While you will normally define procedures in the definitions pane, you can also create them in the interactions pane. Let’s see how this procedure works.

> (define my-square
(lambda (x)
(* x x)))
> (my-square 5)
25
> (my-square 1/2)
1/4
> (my-square (my-square 2))
16
> my-square
#<procedure:my-square>


You may note in the last line that when we asked DrRacket for the “value” of my-square, it told us that it’s a procedure named my-square. Compare that for other values we might define.

> (define x 5)
> x
5
> (define phrase "All mimsy were the borogoves")
> phrase
"All mimsy were the borogoves"
> (define red-square (rectangle 15 15 "solid" "red"))
> red-square

> (define multiply *)
> multiply
#<procedure:*>


In every case, DrRacket is showing us the value associated with the name. In some cases, it’s a number. In some cases, it’s a string. In some cases, it’s an image. And in some cases, it’s a procedure.

How does the procedure we’ve just defined work? Here’s the easiest way to think about it: When you call a procedure you’ve defined with lambda, DrRacket substitutes in the arguments in the procedure call for the corresponding parameters within the instructions. Next, it evaluates the updated instructions.

For example, when you call (my-square 5), DrRacket substitutes 5 for x in (* x x), giving (* 5 5). It then evaluates the (* 5 5), computing 25.

What about a nested call, such as (my-square (my-square 2))? As you may recall, Racket evaluates nested expressions from the inside out. So, it first computes (my-square 2). Substituting 2 in for x, it arrives at (* 2 2). The multiplication gives a value of 4. The (my-square 2) is then replaced by 4. DrRacket is then left to evaluate (my-square 4). This time, it substitutes 4 in for the x, giving it (* 4 4). It does the multiplication to arrive at a result of 16.

We might show the steps as follows, with the => symbol representing the conversion that happens at each step.

(my-square (my-square 2))
=> (my-square (* 2 2))
=> (my-square 4)
=> (* 4 4)
=> 16


You may recall that our second example involved writing a procedure that makes squares of a specified side length and color. A square is just a rectangle with both sides the same size. So, if we want a procedure to make squares, we’ll just call the rectangle procedure, using the same value for the width and height.

(define color-square
(lambda (side color)
(rectangle side side "solid" color)))


What happens if we call color-square on inputs of 15 and "red"? It substitutes 15 for side and "red" for color, giving us (rectangle 15 15 "solid" "red"). And, as we saw in the examples above, that’s a red square of side-length 15.

What about the house example? Let’s look at a simpler version, one that does not include the door. If we did not care about resizing the house, we might just write an expression like

(above (triangle 50 "solid" "red")
(rectangle 40 50 "solid" "black"))


But we’d like to “parameterize” the code to take the size as an input. Let’s say that the size corresponds to the side-length of the triangle (or the size of the main body of the house). We will replace each 50 by size and replace 40 by (* 4/5 size). Let’s see how that works.

> (define simple-house
(lambda (size)
(above (triangle size "solid" "red")
(rectangle (* 4/5 size) size "solid" "black"))))
> (simple-house 20)

> (simple-house 30)



## Zero-parameter procedures

We’ve written procedures so that they take an input. However, there are also advantages to writing procedures that take no inputs. In those cases, a procedure just tells us how to compute a value. Is that different than naming just naming the value? A bit. Let’s explore those differences.

Consider the following two definitions, each of which creates a series of houses.

> (define house-grid
(above (beside (simple-house 20) (simple-house 20))
(beside (simple-house 20) (simple-house 20))))
> (define house-seq
(lambda ()
(beside/align "bottom"
(simple-house 10) (simple-house 15)
(simple-house 20) (simple-house 25))))


The first definition, for house-grid defines a value. (It’s a value which is an image, but it’s still a value.) In contrast, the definition for house-seq defines a procedure. We refer to values and procedures differently.

> house-grid

> house-seq
#<procedure:house-seq>
> (house-seq)

> (house-grid)
application: not a procedure;
expected a procedure that can be applied to arguments
given: (object:image% ...)
arguments...: [none]


But the differences go beyond the particular syntax in which we use them. When you define a named value, DrRacket evaluates the instructions immediately and then stores the result. When you ask for a named value, DrRacket looks up the value and uses it. When you ask DrRacket to evaluate a zero-parameter procedure, it runs the instructions again. You can see the effects when we change the definition of simple-house

> (define simple-house
(lambda (size)
(above (triangle size "solid" "darkred")
(rectangle (* 4/5 size) size "solid" "gray"))))
> house-grid

> (house-seq)



That is, because we’d created house-grid while the old version of simple-house was in effect, the image associated with house-grid does not change. However, since the new definition of simple-house is in effect when we call house-seq, it uses the new definition and we get a sequence of houses that use the new definition.

## Some benefits of procedures

As you may have figured out by now, there are many benefits to defining your own procedures. One of the most important is clarity or readability. Another programmer will likely spend less effort understanding (simple-house 20) than they will trying to understand the more complex above expression involving triangles and rectangles. The first is clearly intended to be a house. The second could be anything, at least until you see it. The other programmer may also find it easier to write programs using simple-house than the much longer series of expressions.

By using a name for a set of code, we are employing the concept of abstraction. That is, because the person calling the procedure knows what the procedure does rather than how it achieves that result, we have abstracted away some of the details.

There are benefits to abstraction and the use of procedures other than readability. For example, it may be that you discover a more efficient way to do a computation. If you’ve written the same code for the computation throughout your program, you’ll have a lot of code to update. But if you’ve created a procedure, you need only update one place in your code, the place you’ve defined the procedure.

The house-seq example above illustrates a similar point. When we decide to change the color of the house and the roof, we only had to make the change in one place, in the definition of simple-house. If we had, instead, created each house within house-seq, we would have multiple places to update.

As these examples suggest, using procedures to parameterize and name sections of code provide us with a variety of advantages. First, we can more easily reuse code in different places. Rather than copying, pasting, and changing, we can simply call the procedure with new parameters. Second, others can more easily read the code we have written. Third, we can more easily update the procedures we’ve written, either to make them more efficient or to change behavior universally.

## Self Checks

### Check 1: A simple procedure

Write a procedure, (subtract2 val) that takes a number as input and subtracts 2 from that number.

> (subtract2 5)
3
> (subtract2 3.25)
1.25
> (subtract2 "hello")
-: contract violation
expected: number?
given: "hello"
argument position: 1st
other arguments...:
2


### Check 2: Exploring steps

Show the steps involved in computing (square (subtract2 5)) and (subtract2 (square 5)).

## Acknowledgements

This section draws upon a reading entitled “Defining your own procedures” and an earlier reading entitled “Writing your own procedures” from Grinnell College’s CSC 151.

The house drawing was inspired by a more sophisticated house drawing from the Racket Image Guide.