Introduction/Computer Science - Computers and Computations | COSC 102, Study notes of Computer Science

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Computers & Computation Home computers are being called upon to perform many new functions, including the consumption of homework formerly eaten by the dog. Doug Larson Computer Science is no more about computers than astronomy is about telescopes. Edsger W. Dijkstra What is the central core of computer science? My answer is simple—it is the art of programming a computer. It is the art of designing efficient and elegant methods of getting a computer to solve problems, theoretical or practical, small or large, simple or complex. It is the art of translating this design into an effective and accurate computer program. C.A.R. Hoare Opposite page: The XO Laptop Photo courtesy of OLPC Project (www.olpc.org)

Computers & Computation

you are reading this, that the ploy worked! But remember that the reason you

are holding a robot of your own in your hand is also because of computers.

Your robot itself is a computer. Whether it was a ploy or not you have

assimilated many key ideas in computing and computer science. In this

chapter, we will make some of these ideas more explicit and also give you a

flavor for what computer science is really all about. As Dijkstra puts it,

computer science is no more about computers than astronomy is about

telescopes.

Computers are dumb

Say you are hosting an international exchange student in your home. Soon

after her arrival you teach her the virtues of PB&J (Peanut Butter & Jelly)

sandwiches. After keenly listening to you, her mouth starts to water and she

politely asks you if you can share the recipe with her. You write it down on a

piece of paper and hand it to her.

Do This: Go ahead; write down the recipe to make a PB&J sandwich.

Seriously, do try to write it down. We insist!

OK, now you have a recipe for making PB&J sandwiches.

Do This: Go ahead, use your recipe from above to make yourself a PB&J

sandwich. Try to follow the instructions exactly as literally as you can. If you

successfully managed to make a PB&J sandwich, congratulations! Go ahead

and enjoy it. Do you think your friend will be able to follow your recipe and

enjoy PB&J sandwiches?

You have to agree, writing a recipe for PB&J sandwiches seemed trivial at

first, but when you sit down to write it you have no choice but to question

several assumptions: does she know what peanut butter is? Should I

recommend a specific brand? Ditto for jelly? By the way, did you forget to

mention it was grape jelly? What kind of bread to use? Will it be pre-sliced?

If not, you need a knife for slicing the loaf? Did you specify how thick the

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slices should be? Should she use the same knife for spreading the peanut

butter and the jelly? Etc. The thing is, at such a level of detail you can go on

and on...does your friend know how to use a knife to spread butter or jelly on

a slice? Suddenly, a seemingly trivial task becomes a daunting exercise. In

reality, in writing down your recipe, you make several assumptions: she

knows what a sandwich is, it involves slices of bread, spreading things on the

slices, and slapping them together. There, you have a sandwich!

Think of the number of recipes that have been published in cookbooks all

over the world. Most good cookbook authors start with a dish, write down a

recipe, try it a number of times and refine it in different ways. Prior to

publication, the recipe is tested by others. After all, the recipe is for others to

follow and recreate a dish. The recipe is revised and adjusted based on

feedback by recipe testers. The assumption that cookbook authors make is that

you will have the competence to follow the recipe and recreate the dish to

your satisfaction. It may never result in the same dish that the author prepared.

But it will give you a base to improvise upon. Wouldn't it be nice if there was

an exact way of following a recipe so that you end up with the exact same

result as the cookbook author every time you made that dish? Would it? That

may depend on your own tastes and preferences. Just for fun, here is a recipe

for some yummy Saffron Chicken Kabobs.

Saffron Chicken Kabobs

Ingredients

1 lb boneless chicken breast, cubed into 1-2 inch pieces

1 medium onion, sliced

1 tsp saffron threads

1 lime

1 tbsp olive oil

Salt and black pepper to taste

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void wiggle(double speed, double waitTime) { robot.motors(-speed, speed); wait(waitTime); robot.motors(speed, - speed); wait(waitTime); robot.stop(); } // The main dance program int main() { connect(); cout << "Running the dance routine..." << endl; yoyo(0.5, 0.5); wiggle(0.5, 0.5); yoyo(1, 1); wiggle(1, 1); cout << "...Done" << endl; disconnect(); }

In many ways, this program above is like a recipe:

To do a robot dance

Ingredients

1 function yoyo for the robot to go back and forth at a given speed

1 function wiggle that enables the robot to wiggle at a given speed

Preparation

1. yoyo at speed 0.5, wait 0.

2. wiggle at speed 0.5, wait 0.

3. yoyo at speed 1, wait 1

4. wiggle at speed 1, wait 1

Further, you could similarly specify the steps involved in doing the yoyo and

wiggle motions as a recipe. This may seem like a trivial example, but it

makes two very important points: a computer program is like a recipe in that it

Computers & Computation

lays out the list of ingredients and a method or steps for accomplishing the

given task; and, like a recipe, its ingredients and the steps require careful pre-

planning and thought. Importantly, computer programs are different from

recipes in one aspect: they are designed to be followed by a computer!

A computer is a dumb device designed to follow instructions/recipes. We will

save the technical details of how a computer does what it does for a later

course. But it is almost common knowledge that everything inside is

represented as 0's and 1's. Starting from 0's and 1's one can design encoding

schemes to represent numbers, letters of the alphabet, documents, images,

movies, music, etc. and whatever other abstract entities you would like to

manipulate using a computer. A computer program is ultimately also

represented as a sequence of 0's and 1's and it is in this form that most

computers like to follow recipes. However limiting or degenerate this might

sound it is the key to the power of computers. Especially when you realize

that it is this simplification that enables a computer to manipulate hundreds of

millions of pieces of information every second. The price we have to pay for

all this power is that we have to specify our recipes as computer programs in a

rather formal and precise manner. So much so that there is no room for

improvisation: no pinch of salt vagaries, as in cooking recipes, is acceptable.

This is where programming languages come in. Computer scientists specify

their computational recipes using programming languages. You have been

using the programming language C++ to write your robot programs. Other

examples of programming languages are Java, Python, and C# (pron.: sea

sharp). There are well over 2000 programming languages in existence!

Do This: Can you find out how many programming languages there are?

What are the ten most commonly used programming languages?

Prior to the existence of programming languages computers were

programmed using long sequences of 0's and 1's. Needless to say it drove

several people crazy! Programming languages, like C++, enable a friendlier

way for programmers to write programs. Programming languages provide

easy access to encodings that represent the kinds of things we, humans, relate

to. For example, the C++ statement:

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This is further refined into specific steps, perhaps even some variables are

identified and named, etc. Once you convince yourself that you have a way of

solving the problem, what you have is an algorithm

The idea of an algorithm is central to computer science so we will spend some

time here developing this notion. Perhaps the best way to relate to it is by an

example. Assume that a robot goes into a

grocery store to buy a dozen fresh eggs.

Assuming it is capable of doing this, how

will it ensure that it has selected the freshest

eggs available?

Your personal robot is probably not up to

this kind of task but imagine that it was.

Better yet, leave the mechanics aside, let us

figure out how you would go and buy the

freshest eggs. Well, you would somehow need to know what today's date is.

Assume it is September 15, 2007 (why this date? it'll become clear soon!).

Now you also know that egg cartons typically carry a freshness date on them.

In fact, USDA (the United States Department of Agriculture) offers voluntary,

no cost, certification programs for egg farms.

An egg farmer can volunteer to participate in

USDA's egg certification program whereby the

USDA does regular inspections and also

provides help in categorizing eggs by various

sizes. For example, eggs are generally

classified as Grade AA, Grade A, or Grade B.

Most grocery stores carry Grade A eggs. They

can also come in various sizes: Extra Large,

Large, Small, etc. What is more interesting is

that the carton labeling system has some very

useful information encoded on it.

Every USDA certified egg carton has at least

three pieces of information (see picture on the

Egg Carton Labeling

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right): a "sell by" date (or a "use by date" or a "best by" date), a code

identifying the specific farm the eggs came from, and a date on which the

eggs were packed in that carton. Most people buy eggs by looking at the "sell

by" date or the "best by" date. However the freshness information is really

encoded in the packed on date. To make things more confusing, this date is

encoded as the day of the year.

For example, take a look at the top carton shown on the previous page. Its

"sell by" date is October 4. "P1107" is the farm code. This carton was packed

on the 248th day of the year. Further, USDA requires that all certified eggs be

packed within 7 days of being laid. Thus, the eggs in the top carton were laid

somewhere between day 241 and day 248 of 2007. What dates correspond to

those dates?

Next, look at the bottom carton. Those eggs have a later "sell by" date

(October 18) but an earlier packed date: 233. That is those eggs were laid

somewhere between day 226 and day 233 of 2007.

Which eggs are fresher?

Even though the "sell by" date on the second carton is two weeks later, the

first carton contains fresher eggs. In fact, the eggs in the upper carton were

laid at least two weeks later!

The packed on date is encoded as a 3-digit number. Thus eggs packed on

January 1 will be labeled: 001; eggs packed on December 31, 2007 will be

labeled: 365.

Do This: Go to the USDA web site (www.usda.gov) and see if you can find

out which farm the two eggs cartons came from.

For a robot, the problem of buying the freshest eggs becomes that of figuring

out, given a packed on date, what the date was when the eggs were packed?

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Input

3 - digit packed on date encoding

Current year

Output

Date the eggs were packed

Example:

Input: 248, 2007

Output: The eggs were packed on September 5, 2007

Any ideas as to how you would solve this problem? It always helps to try and

do it yourself, with pencil and paper. Take the example above, and see how

you would arrive at the output date. While you are working it out, try to write

down your problem solving process. Your algorithm or recipe will be very

similar.

Suppose we are trying to decode the input 248, 2007. If you were to do this by

hand, using a pen and paper, the process might go something like this:

The date is not in January because it has 31 days and 248 is much larger than 31. Lets us subtract 31 out of 248: 248 - 31 = 217 217 is also larger than 28, the number of days in February,

So, let us subtract 28 from 217: 217 - 28 = 189 189 is larger than 31, the number of days in March. Subtract 31 from 189: 189 - 31 = 158 158 is larger than 30, the number of days in April. So: 158 - 30 = 128 128 is larger than 31, the number of days in May. Hence: 128 - 31 = 97 97 is larger than 30, the number of days in June.

Computers & Computation

67 is larger than 31, the number of days in July. 67 - 31 = 36 36 is larger than the number of days in August (31). 36 - 31 = 5 5 is smaller than the number of days in September. Therefore it must be the 5th day of September. The answer is: 248th day of 2007 is September 5, 2007.

That was obviously too repetitious and tedious. But that is where computers

come in. Take a look at the process above and see if there is a pattern to the

steps performed. Sometimes, it is helpful to try another example.

Do This: Suppose the input day and year are: 56, 2007. What is the date?

When you look at the sample computations you have performed, you will see

many patterns. Identifying these is the key to designing an algorithm.

Sometimes, in order to make this easier, it is helpful to identify or name the

key pieces of information being manipulated. Generally, this begins with the

inputs and outputs identified in the problem specification. For example, in this

problem, the inputs are: day of the year, current year. Begin by assigning

these values to specific variable names. That is, let us assign the name day to

represent the day of the year (248 in this example), and year as the name to

store the current year (2007). Notice that we didn't choose to name any of

these variables timbuktu or meaningOfLife!

Also, notice that you have to repeatedly subtract the number of days in a

month, starting from January. Let us assign a variable named, month to keep

track of the month under consideration.

Next, you can substitute the names day and year in the sample computation:

Computers & Computation

day (= 36) is larger than the # of days in month = 8. day = day – 31 ; // next month month = 9 ; day (= 5) is smaller than the # of days in month = 9. Therefore it must be the 5th day of September. The answer is: 9/5/

Notice now how repetitious the above process is. The repetition can be

expressed more concisely as shown below:

Input: day year // start with month = 1, for January month = 1 ; repeat if day is less than number of days in month day = day – number of days in month; // next month month = month + 1 ; else done; Output: day/month/year

It is now starting to look like a recipe or an algorithm. Go ahead and try it

with the sample inputs from above and ensure that you get correct results.

Additionally, make sure that this algorithm will work for boundary cases: 001,

Thirty days hath September

We can refine the algorithm above further: one thing we have left unspecified

above is the computation of the number of days in a month. This information

has to be made explicit for a computer to be able to follow the recipe. So, how

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do we compute the number of days in a month? The answer may seem simple.

Many of you may remember the following poem:

Thirty days hath September

April, June, and November

All the rest have thirty-one

Except for February alone

Which hath twenty-eight days clear

And twenty-nine in each leap year

From a design perspective, we can assume that we have an ingredient, a

function in this case, called daysInMonth that, given a month and a year will

compute and return the number of days in the month. That is, we can refine

our algorithm above to the following:

Ingredients: 1 integer function daysInMotnh(int m, int y): returns the number of days in month m and in year y. Input: an integer, day an intger, year // start with month = 1, for January month = 1 ; repeat if day is less than number of days in month day = day - daysInMonth(month, year); // next month month = month + 1 ; else done; Output: day/month/year (integers)

Now, we do have to solve the secondary problem:

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Ingredients: 1 function bool leapYear(int y) returns true if y is a leap year, false otherwise Input: int m, y; if m is April(4), June(6), September(9), or November(11) days = 30 ; else if m is February (2) if (leapYear(y)) days = 29 ; else days = 28 ; else /* m is January, March, May, July, August, October, December */ days = 31 ; Output: days (an integer)

Most of us have been taught that a leap year is a year that is divisible by 4.

That is the year 2007 is not a leap year, since 2007 is not divisible by 4, but

2008 is a leap year, since it is divisible by 4.

Do This: How do you determine if something is divisible by 4? Try your

solution on the year 1996, 2000, 1900, 2006.

Leap Years: Papal Bull

To design a recipe or an algorithm that determines if a number corresponding

to a year is a leap year or not is straightforward if you accept the definition

from the last section. Thus, we can write:

Input: int y, a year Output: true if y is a leap year, false otherwise

Computers & Computation

Method

if y is divisible by 4 it is a leap year, or true else it is not a leap year, or false

However, this is not the complete story. The western calendar that we follow

is called the Gregorian Calendar which was adopted in 1582 by a Papal Bull

issued by Pope Gregory XIII. The Gregorian Calendar defines a leap year, by

adding an extra day, every fourth year. However, there is a 100-year

correction applied to it that makes the situation a little more complicated:

Century years are not leap years except when they are divisible by 400. That

is the years 1700, 1800, 1900, 2100 are not leap years even though they are

divisible by 4. However, the years 1600, 2000, 2400 are leap years. For more

information on this, see the exercises at the end of the chapter. Our algorithm

for determining if a year is a leap year can be refined as shown below:

Input: int y, a year if y is divisible by 400 it is a leap year, or true else if y is divisible by 100 it is not a leap year, or false else if y is divisible by 4 it is a leap year, or true else it is not a leap year, or false

Finally, we have managed to design all the algorithms or recipes required to

solve the problem. You may have noticed that we used some familiar

constructs to elaborate our recipes or algorithms. Next, let us take a quick

look at the essential constructs that are used in expressing algorithms.