# 程序代写代做代考 interpreter Haskell junit scheme algorithm THE BASIC DESIGN RECIPE

THE BASIC DESIGN RECIPE
Susan Older and Jim Royer

January 2013 revision for CIS 252

We shall use a general design recipe1 when developing programs in this course. The intent of
the design recipe is to provide you with three things:

1. A series of questions that help guide you through the design process

2. A way to tie up the initial program specification with the testing of the completed (but
maybe not correct) program

3. A scheme for explaining what the program does, together with pre-packaged tests to

Below we discuss the basic design recipe. We shall revise the recipe over time to cover new
ways to construct programs. Right now we are presuming only the “high-school algebra”
part of Haskell.2 Here is a tiny, complete example of the end result of following the basic
design recipe. (Recall that anything following a “–” is a comment in Haskell.)

— CONTRACT

— areaOfRing : : number -> number -> number

— PURPOSE

— Takes two numbers, outer and inner, and computes the area of

— a ring with radius outer and with a hole with radius inner.

— Assumes that outer and inner are both nonnegative,

— with outer >= inner.

— EXAMPLES

— (areaOfRing 5 3) should return 50.24.

— (areaOfRing 5 4) should return 28.26.

— DEFINITION

areaOfRing outer inner = (areaOfDisk outer) – (areaOfDisk inner)

— TESTS

ans1 = areaOfRing 5 3 — should be 50.24

ans2 = areaOfRing 5 4 — should be 28.26

We shall use the above as a running example in discussing the parts of the design recipe
below.

1 This design recipe is adapted from the text How to Design Programs by Felleisen, Findler, Flatt, and Krish-
namurthi [FFFK01]. Their book is on the web here, and their first discussion of the design recipe is here. Note
that, instead of Haskell, they are the language Scheme (a second cousin to Haskell), so the finer details of that
section will not apply here. We shall also use their “area of a ring” example in explaining the ingredients of
the design recipe.

2Which is just like ordinary high school algebra with an enlightened view of syntax.

http://www.htdp.org/
http://www.htdp.org/2003-09-26/Book/curriculum-Z-H-5.html#node_sec_2.5

The Basic Design Recipe CIS 252: Introduction to Computer Science

Parts of the design recipe

The design recipe is essentially a programming version of Polya’s How to Solve It scheme
[Pol45]. Polya tells us that our first step is to understand the problem. With a program, this
usually comes down to answering the following questions:

1. What kind of inputs should the program take?
2. What kind of output should the program produce?
3. How does the output depend on the input? (That is, what’s computed?)

Question 3 is the interesting one, but first let’s deal with 1 and 2.

Recipe step 1: The contract

Example. We want a program to compute the area of a ring. A ring consists of a
disk with a circular hole in it. [Polya: Draw the picture!] Here is a picture: }v
Our givens are: the radius of the outer disk and the radius of the inner disk. The
area of the ring is going to be a number too.

So the answer to Question 1 is that the inputs are two numbers, and the answer to
Question 2 is that the output is a number.

In the program text we state our answer to Questions 1 and 2 with a contract. For our ring-
area example, the contract would be:

— areaOfRing : : number -> number -> number

The general form of a contract for an n-input program is:

— 〈program-name〉 : : 〈type1〉 -> … -> 〈typen〉 -> 〈typeout〉

(Later on, we shall generalize the stuffings out of this.) For the time being think of “: :” and
“->” just as Haskell punctuation. The funny things of the form 〈name〉 are “fill in the blank”
variables. The name of your program goes in the 〈program-name〉 slot. The type of the i-th
input goes in the 〈typei〉 slot, and the type of the output goes in the 〈typeout〉 slot. Type is
the formal name of “the kind of data.” Types can describe numbers, character strings, lists,
pictures, trees, and on and on.

It turns out that Haskell cares intensely about contracts. We will later place our contracts
outside of comments so the Haskell interpreter can check them for us.

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What is the point of contracts?

1. If you do not understand what sort of inputs your program should take and what
sort of output it produces, you most definitely do not understand the problem your
program is trying to solve.

2. Programs are typically constructed by putting together smaller programs in various
ways. The contracts let us (and Haskell) check whether we are trying to “place a square
peg in a round hole.”3

Recipe step 2: Analysis and statement of the purpose

Now let us deal with Question 3: How does the output depend on the input?

The example, continued. In this example, we are working with a problem that is
reasonably well stated. (Life is usually not so easy.) So we can use the problem
statement as the purpose statement.

— PURPOSE

— Takes two numbers, outer and inner, and computes the

— area of a ring with radius outer and with a hole

— Assumes that outer and inner are both nonnegative,

— with outer >= inner.

Let’s dig a bit deeper into the “what is computed” problem. We know from high-
school algebra that the area of a disk of radius r is: π · r2. So the ring’s area is:

(the big ring’s area) − (the hole’s area) = π · r2o − π · r
2
i ,

where ro is the radius of the outer disk and ri is the radius of the inner disk. So the
problem we will have in the definition step is translating this math into Haskell.

The goals of this step is to understand the problem well enough so as to do the following:

1. Give a cogent English4 advertisement for what the program is supposed to do.

2. Be able to compute, by ad hoc means, what the program should produce on some par-
ticular inputs.

3 When an application program crashes on you, it is often because the language in which the program is
written (e.g., C or C++) permitted “a square peg in a round hole.”

4 Or French, Chinese, Hindi, Etruscan, . . . as the case may be.

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What is the point of the purpose analysis and statement? The intent of the first goal is pretty
clear—especially if you have ever tried to read someone else’s uncommented program. The
contract gives you the general shape of the program. The purpose statement fills in the
contents of that shape.

The second goal may seem silly since you are shortly going to write the program anyhow.
The point of doing some hand calculations is to play computer with concrete data and to try
to think about possible ways of doing the computations.

Recipe step 3: Examples

The example, continued. We decide to use 3.14 as an approximation to π. So we
can compute some values of what areaOfRing should produce. Specifically:

— EXAMPLES

— (areaOfRing 5 3) should return 50.24.

— (areaOfRing 5 4) should return 28.26.

The goal here is to produce concrete examples of what the program should do. In particular,
you want to identify key examples that illustrate tricky cases that the program needs to
handle.

What is the point of the examples?

1. Working through some cases, particularly tricky ones, with concrete data makes you
think through these cases. This helps refine your understanding from step 2.

2. Making these examples explicit helps set a standard with which you can measure the
success of program in the testing step. Coming up with the examples before the pro-
gram gets written helps keep the examples unbiased.5

Recipe step 4: The definition

The example, continued. To keep our story short, we suppose we just happen to
have lying around the program

areaOfDisk r = 3.14 * r * r

that computes the area of a disk with radius r. So, based on step 2,

areaOfRing outer inner = (areaOfDisk outer) – (areaOfDisk inner)

will serve as our definition of areaOfRing.

5 This approach to programming goes under the name testing-first programming and is much beloved by
the extreme programming folks. For example, see here and here.

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http://www.extremeprogramming.org/rules/testfirst.html
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The Basic Design Recipe CIS 252: Introduction to Computer Science

The goal here is to produce a correct definition of our program based on the analysis of the
previous three steps.

What is the point of the definition? Well, we are doing computing, not (just) mathematics!
Our definition is a purported algorithmic solution to the original problem.

N.B.6 This step of the design recipe will be greatly expanded in our refinements of the recipe.
When dealing with “high-school algebra” programs, there isn’t much to say here.

Recipe step 5: Tests

This step corresponds to Polya’s question Can you check your result? in his looking-back step.

The example, concluded. Now we are ready to run some tests of our definition.
All of the examples from step 3 become test examples.

— TESTS

ans1 = areaOfRing 5 3 — should be 50.24

ans2 = areaOfRing 5 4 — should be 28.26

When we look at the values of ans1 and ans2 they agree with the target values. So,
we declare success—we (more or less) trust our definition.

The goal here is to put your program through its paces on some well-chosen data.

What is the point of the tests? If there are difficulties with your definition and if you chose
good test cases, then those difficulties are likely to show up here. (The goodness of your test
conditions needs to be established in steps 2 and 3.) So if your program passes your test,
you have good evidence for the correctness of your program.

N.B. To guarantee correctness with just testing, you have to check all possible inputs, and
there are almost always too many inputs to do this. In the case of our pitiful little example,
if we assume our inputs are 64-bit floating point numbers, then a little calculation shows us
that there are

340, 282, 366, 920, 938, 463, 463, 374, 607, 431, 768, 211, 456

possible test cases.7

The after-dinner step: Can we do better?

Producing a well-documented, well-tested program is just one of the end results of the de-
sign process. Another less obvious one is that we’ve learned something about the algorith-
mic problem we’ve just solved—and you have often learned more than you have realized.
So, as Polya suggests, we can look back and see whether we can solve the problem in a

6“N.B.” abbreviates the latin phrase nota bene which means “note well” (that is, pay special attention).
7 As a point of comparison, the age of the Earth in seconds is about 200,000,000,000,000,000 (= 2× 1017).

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The Basic Design Recipe CIS 252: Introduction to Computer Science

different way or use what we’ve learned here in another place. This can be where the fun
starts.

In the case of our little example, we’ll leave well enough alone.

A summary

The design recipe has five basic steps.

1. The contract
What kind of inputs should the program take?
What kind of output should the program produce?

2. Analysis and statement of the purpose
How does the output depend on the inputs?
Can I compute what the result should be for some sample inputs?

3. Examples
What are some typical inputs and their results?
What are some tricky inputs to handle?

4. Definition
How can I apply what I learned from steps 2 and 3 in program form?
. . . (More to come.) . . .
Have I handled all the cases?

5. Testing
What (if anything) went wrong?
Suppose the program returns unexpected results.

Is the program wrong?
Are some target values wrong?
What is the source of the confusion?

Am I testing all the cases I should?

References
[FFFK01] M. Felleisen, R. Findler, M. Flatt, and S. Krishnamurthi, How to design programs: An intro-

duction to computing and programming, MIT Press, 2001.

[Pol45] G. Polya, How to solve it, Princeton University Press, 1945.

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