As another example of a user-defined type, we’ll define a class called MyTime that records the time of day. We’ll provide an __init__ method to ensure that every instance is created with appropriate attributes and initialization. The class definition looks like this:
1 2 3 4 5 6 7 class MyTime: def __init__(self, hrs=0, mins=0, secs=0): """ Create a MyTime object initialized to hrs, mins, secs """ self.hours = hrs self.minutes = mins self.seconds = secs
We can instantiate a new MyTime object:
1 tim1 = MyTime(11, 59, 30)
The state diagram for the object looks like this:
We’ll leave it as an exercise for the readers to add a __str__ method so that MyTime objects can print themselves decently.
In the next few sections, we’ll write two versions of a function called add_time, which calculates the sum of two MyTime objects. They will demonstrate two kinds of functions: pure functions and modifiers.
The following is a rough version of add_time:
1 2 3 4 5 6 def add_time(t1, t2): h = t1.hours + t2.hours m = t1.minutes + t2.minutes s = t1.seconds + t2.seconds sum_t = MyTime(h, m, s) return sum_t
The function creates a new MyTime object and returns a reference to the new object. This is called a pure function because it does not modify any of the objects passed to it as parameters and it has no side effects, such as updating global variables, displaying a value, or getting user input.
Here is an example of how to use this function. We’ll create two MyTime objects: current_time, which contains the current time; and bread_time, which contains the amount of time it takes for a breadmaker to make bread. Then we’ll use add_time to figure out when the bread will be done.
>>> current_time = MyTime(9, 14, 30) >>> bread_time = MyTime(3, 35, 0) >>> done_time = add_time(current_time, bread_time) >>> print(done_time) 12:49:30
The output of this program is 12:49:30, which is correct. On the other hand, there are cases where the result is not correct. Can you think of one?
The problem is that this function does not deal with cases where the number of seconds or minutes adds up to more than sixty. When that happens, we have to carry the extra seconds into the minutes column or the extra minutes into the hours column.
Here’s a better version of the function:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 def add_time(t1, t2): h = t1.hours + t2.hours m = t1.minutes + t2.minutes s = t1.seconds + t2.seconds if s >= 60: s -= 60 m += 1 if m >= 60: m -= 60 h += 1 sum_t = MyTime(h, m, s) return sum_t
This function is starting to get bigger, and still doesn’t work for all possible cases. Later we will suggest an alternative approach that yields better code.
There are times when it is useful for a function to modify one or more of the objects it gets as parameters. Usually, the caller keeps a reference to the objects it passes, so any changes the function makes are visible to the caller. Functions that work this way are called modifiers.
increment, which adds a given number of seconds to a MyTime object, would be written most naturally as a modifier. A rough draft of the function looks like this:
1 2 3 4 5 6 7 8 9 10 def increment(t, secs): t.seconds += secs if t.seconds >= 60: t.seconds -= 60 t.minutes += 1 if t.minutes >= 60: t.minutes -= 60 t.hours += 1
The first line performs the basic operation; the remainder deals with the special cases we saw before.
Is this function correct? What happens if the parameter seconds is much greater than sixty? In that case, it is not enough to carry once; we have to keep doing it until seconds is less than sixty. One solution is to replace the if statements with while statements:
1 2 3 4 5 6 7 8 9 10 def increment(t, seconds): t.seconds += seconds while t.seconds >= 60: t.seconds -= 60 t.minutes += 1 while t.minutes >= 60: t.minutes -= 60 t.hours += 1
This function is now correct when seconds is not negative, and when hours does not exceed 23, but it is not a particularly good solution.
Once again, OOP programmers would prefer to put functions that work with MyTime objects into the MyTime class, so let’s convert increment to a method. To save space, we will leave out previously defined methods, but you should keep them in your version:
1 2 3 4 5 6 7 8 9 10 11 12 13 class MyTime: # Previous method definitions here... def increment(self, seconds): self.seconds += seconds while self.seconds >= 60: self.seconds -= 60 self.minutes += 1 while self.minutes >= 60: self.minutes -= 60 self.hours += 1
The transformation is purely mechanical — we move the definition into the class definition and (optionally) change the name of the first parameter to self, to fit with Python style conventions.
Now we can invoke increment using the syntax for invoking a method.
Again, the object on which the method is invoked gets assigned to the first parameter, self. The second parameter, seconds gets the value 500.
Often a high-level insight into the problem can make the programming much easier.
In this case, the insight is that a MyTime object is really a three-digit number in base 60! The second component is the ones column, the minute component is the sixties column, and the hour component is the thirty-six hundreds column.
When we wrote add_time and increment, we were effectively doing addition in base 60, which is why we had to carry from one column to the next.
This observation suggests another approach to the whole problem — we can convert a MyTime object into a single number and take advantage of the fact that the computer knows how to do arithmetic with numbers. The following method is added to the MyTime class to convert any instance into a corresponding number of seconds:
1 2 3 4 5 6 7 8 class MyTime: # ... def to_seconds(self): """ Return the number of seconds represented by this instance """ return self.hours * 3600 + self.minutes * 60 + self.seconds
Now, all we need is a way to convert from an integer back to a MyTime object. Supposing we have tsecs seconds, some integer division and mod operators can do this for us:
1 2 3 4 hrs = tsecs // 3600 leftoversecs = tsecs % 3600 mins = leftoversecs // 60 secs = leftoversecs % 60
You might have to think a bit to convince yourself that this technique to convert from one base to another is correct.
In OOP we’re really trying to wrap together the data and the operations that apply to it. So we’d like to have this logic inside the MyTime class. A good solution is to rewrite the class initializer so that it can cope with initial values of seconds or minutes that are outside the normalized values. (A normalized time would be something like 3 hours 12 minutes and 20 seconds. The same time, but unnormalized could be 2 hours 70 minutes and 140 seconds.)
Let’s rewrite a more powerful initializer for MyTime:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 class MyTime: # ... def __init__(self, hrs=0, mins=0, secs=0): """ Create a new MyTime object initialized to hrs, mins, secs. The values of mins and secs may be outside the range 0-59, but the resulting MyTime object will be normalized. """ # Calculate total seconds to represent totalsecs = hrs*3600 + mins*60 + secs self.hours = totalsecs // 3600 # Split in h, m, s leftoversecs = totalsecs % 3600 self.minutes = leftoversecs // 60 self.seconds = leftoversecs % 60
Now we can rewrite add_time like this:
1 2 3 def add_time(t1, t2): secs = t1.to_seconds() + t2.to_seconds() return MyTime(0, 0, secs)
This version is much shorter than the original, and it is much easier to demonstrate or reason that it is correct.
In some ways, converting from base 60 to base 10 and back is harder than just dealing with times. Base conversion is more abstract; our intuition for dealing with times is better.
But if we have the insight to treat times as base 60 numbers and make the investment of writing the conversions, we get a program that is shorter, easier to read and debug, and more reliable.
It is also easier to add features later. For example, imagine subtracting two MyTime objects to find the duration between them. The naive approach would be to implement subtraction with borrowing. Using the conversion functions would be easier and more likely to be correct.
Ironically, sometimes making a problem harder (or more general) makes the programming easier, because there are fewer special cases and fewer opportunities for error.
Specialization versus Generalization
Computer Scientists are generally fond of specializing their types, while mathematicians often take the opposite approach, and generalize everything.
What do we mean by this?
If we ask a mathematician to solve a problem involving weekdays, days of the century, playing cards, time, or dominoes, their most likely response is to observe that all these objects can be represented by integers. Playing cards, for example, can be numbered from 0 to 51. Days within the century can be numbered. Mathematicians will say “These things are enumerable — the elements can be uniquely numbered (and we can reverse this numbering to get back to the original concept). So let’s number them, and confine our thinking to integers. Luckily, we have powerful techniques and a good understanding of integers, and so our abstractions — the way we tackle and simplify these problems — is to try to reduce them to problems about integers.”
Computer Scientists tend to do the opposite. We will argue that there are many integer operations that are simply not meaningful for dominoes, or for days of the century. So we’ll often define new specialized types, like MyTime, because we can restrict, control, and specialize the operations that are possible. Object-oriented programming is particularly popular because it gives us a good way to bundle methods and specialized data into a new type.
Both approaches are powerful problem-solving techniques. Often it may help to try to think about the problem from both points of view — “What would happen if I tried to reduce everything to very few primitive types?”, versus “What would happen if this thing had its own specialized type?”
The after function should compare two times, and tell us whether the first time is strictly after the second, e.g.
>>> t1 = MyTime(10, 55, 12) >>> t2 = MyTime(10, 48, 22) >>> after(t1, t2) # Is t1 after t2? True
This is slightly more complicated because it operates on two MyTime objects, not just one. But we’d prefer to write it as a method anyway — in this case, a method on the first argument:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 class MyTime: # Previous method definitions here... def after(self, time2): """ Return True if I am strictly greater than time2 """ if self.hours > time2.hours: return True if self.hours < time2.hours: return False if self.minutes > time2.minutes: return True if self.minutes < time2.minutes: return False if self.seconds > time2.seconds: return True return False
We invoke this method on one object and pass the other as an argument:
1 2 if current_time.after(done_time): print("The bread will be done before it starts!")
We can almost read the invocation like English: If the current time is after the done time, then...
The logic of the if statements deserve special attention here. Lines 11-18 will only be reached if the two hour fields are the same. Similarly, the test at line 16 is only executed if both times have the same hours and the same minutes.
Could we make this easier by using our “Aha!” insight and extra work from earlier, and reducing both times to integers? Yes, with spectacular results!
1 2 3 4 5 6 class MyTime: # Previous method definitions here... def after(self, time2): """ Return True if I am strictly greater than time2 """ return self.to_seconds() > time2.to_seconds()
This is a great way to code this: if we want to tell if the first time is after the second time, turn them both into integers and compare the integers.
Some languages, including Python, make it possible to have different meanings for the same operator when applied to different types. For example, + in Python means quite different things for integers and for strings. This feature is called operator overloading.
It is especially useful when programmers can also overload the operators for their own user-defined types.
For example, to override the addition operator +, we can provide a method named __add__:
1 2 3 4 5 class MyTime: # Previously defined methods here... def __add__(self, other): return MyTime(0, 0, self.to_seconds() + other.to_seconds())
As usual, the first parameter is the object on which the method is invoked. The second parameter is conveniently named other to distinguish it from self. To add two MyTime objects, we create and return a new MyTime object that contains their sum.
Now, when we apply the + operator to MyTime objects, Python invokes the __add__ method that we have written:
>>> t1 = MyTime(1, 15, 42) >>> t2 = MyTime(3, 50, 30) >>> t3 = t1 + t2 >>> print(t3) 05:06:12
The expression t1 + t2 is equivalent to t1.__add__(t2), but obviously more elegant. As an exercise, add a method __sub__(self, other) that overloads the subtraction operator, and try it out.
For the next couple of exercises we’ll go back to the Point class defined in our first chapter about objects, and overload some of its operators. Firstly, adding two points adds their respective (x, y) coordinates:
1 2 3 4 5 class Point: # Previously defined methods here... def __add__(self, other): return Point(self.x + other.x, self.y + other.y)
There are several ways to override the behavior of the multiplication operator: by defining a method named __mul__, or __rmul__, or both.
If the left operand of * is a Point, Python invokes __mul__, which assumes that the other operand is also a Point. It computes the dot product of the two Points, defined according to the rules of linear algebra:
1 2 def __mul__(self, other): return self.x * other.x + self.y * other.y
If the left operand of * is a primitive type and the right operand is a Point, Python invokes __rmul__, which performs scalar multiplication:
1 2 def __rmul__(self, other): return Point(other * self.x, other * self.y)
The result is a new Point whose coordinates are a multiple of the original coordinates. If other is a type that cannot be multiplied by a floating-point number, then __rmul__ will yield an error.
This example demonstrates both kinds of multiplication:
>>> p1 = Point(3, 4) >>> p2 = Point(5, 7) >>> print(p1 * p2) 43 >>> print(2 * p2) (10, 14)
What happens if we try to evaluate p2 * 2? Since the first parameter is a Point, Python invokes __mul__ with 2 as the second argument. Inside __mul__, the program tries to access the x coordinate of other, which fails because an integer has no attributes:
>>> print(p2 * 2) AttributeError: 'int' object has no attribute 'x'
Unfortunately, the error message is a bit opaque. This example demonstrates some of the difficulties of object-oriented programming. Sometimes it is hard enough just to figure out what code is running.
Most of the methods we have written only work for a specific type. When we create a new object, we write methods that operate on that type.
But there are certain operations that we will want to apply to many types, such as the arithmetic operations in the previous sections. If many types support the same set of operations, we can write functions that work on any of those types.
For example, the multadd operation (which is common in linear algebra) takes three parameters; it multiplies the first two and then adds the third. We can write it in Python like this:
1 2 def multadd (x, y, z): return x * y + z
This function will work for any values of x and y that can be multiplied and for any value of z that can be added to the product.
We can invoke it with numeric values:
>>> multadd (3, 2, 1) 7
Or with Points:
>>> p1 = Point(3, 4) >>> p2 = Point(5, 7) >>> print(multadd (2, p1, p2)) (11, 15) >>> print(multadd (p1, p2, 1)) 44
In the first case, the Point is multiplied by a scalar and then added to another Point. In the second case, the dot product yields a numeric value, so the third parameter also has to be a numeric value.
A function like this that can take arguments with different types is called polymorphic.
As another example, consider the function front_and_back, which prints a list twice, forward and backward:
1 2 3 4 5 def front_and_back(front): import copy back = copy.copy(front) back.reverse() print(str(front) + str(back))
Because the reverse method is a modifier, we make a copy of the list before reversing it. That way, this function doesn’t modify the list it gets as a parameter.
Here’s an example that applies front_and_back to a list:
>>> my_list = [1, 2, 3, 4] >>> front_and_back(my_list) [1, 2, 3, 4][4, 3, 2, 1]
Of course, we intended to apply this function to lists, so it is not surprising that it works. What would be surprising is if we could apply it to a Point.
To determine whether a function can be applied to a new type, we apply Python’s fundamental rule of polymorphism, called the duck typing rule: If all of the operations inside the function can be applied to the type, the function can be applied to the type. The operations in the front_and_back function include copy, reverse, and print.
Not all programming languages define polymorphism in this way. Look up duck typing, and see if you can figure out why it has this name.
copy works on any object, and we have already written a __str__ method for Point objects, so all we need is a reverse method in the Point class:
1 2 def reverse(self): (self.x , self.y) = (self.y, self.x)
Then we can pass Points to front_and_back:
>>> p = Point(3, 4) >>> front_and_back(p) (3, 4)(4, 3)
The most interesting polymorphism is the unintentional kind, where we discover that a function we have already written can be applied to a type for which we never planned.
Write a Boolean function between that takes two MyTime objects, t1 and t2, as arguments, and returns True if the invoking object falls between the two times. Assume t1 <= t2, and make the test closed at the lower bound and open at the upper bound, i.e. return True if t1 <= obj < t2.
Turn the above function into a method in the MyTime class.
Overload the necessary operator(s) so that instead of having to write
if t1.after(t2): ...
we can use the more convenient
if t1 > t2: ...
Rewrite increment as a method that uses our “Aha” insight.
Create some test cases for the increment method. Consider specifically the case where the number of seconds to add to the time is negative. Fix up increment so that it handles this case if it does not do so already. (You may assume that you will never subtract more seconds than are in the time object.)
Can physical time be negative, or must time always move in the forward direction? Some serious physicists think this is not such a dumb question. See what you can find on the Internet about this.