6.12
Lecture 11: Defining sameness for complex data, part 1
Casting, type-testing, and “customized" type-testing mechanisms for checking sameness
Motivation
How can we test when two values are “the same”? This seemingly simple question has a surprisingly subtle
answer, and we’ll spend the next two lectures constructing a reasonable answer.
For now! As we’ll see in a few weeks, when we add new language features, we’ll have to revisit this and modify our approach.
As with any other operation, we need to determine the form of our code based on the structure of the data.
11.1 What do we mean by “sameness”?
We cannot define a new operation if we don’t know what we want it to do! Intuitively, we’d like “sameness”
to mean “two values are the same if there’s no way we can tell them apart”. In other words, any method that we call
that uses one value could use the other value and compute the exact same answer; there are no “trick questions”
we could ask to distinguish the two values. In order to do that, and to make sameness be useful, there
are a few other properties it must uphold:
Reflexivity: every object should be the same as itself.
Symmetry: if object x is the same as object y, then y is the same as x.
Transitivity: if two objects are both the same as a third object, then they are the same as each other.
Totality: we can compare any two objects of the same type, and obtain a correct answer.
As we define sameness for our various forms of data, keep these properties in mind, as they may indicate flaws in our designs
that we must correct.
11.2 Review: sameness for built-in types
We can use Java’s built-in == operator to compare two integers, or two booleans:
t.checkExpect(4 == 4, true) t.checkExpect(true == false, false)
Technically, we can use == to compare two doubles, but be careful! Doubles are imprecise,
and two numbers that appear the same may not in fact be equal. It’s always safer to subtract
one double from the other, and compare the difference to a small value:
t.checkExpect(4.3333 == 4.3333, true) // BAD idea t.checkExpect(4.3333 - 4.3333 < 0.001, true) // MUCH BETTER
(This latter one is conceptually what checkInexact does:
t.checkInexact(4.3333, 4.3333, 0.001);
But there is a difference between the two: the former line checks whether the absolute difference between
two numbers is less than some tolerance. On the other hand, checkInexact looks at the
relative difference. The distinction between these two is easier to see with larger numbers.
t.checkExpect(1100000 - 1000000 < 0.1, true); // FAILS: the absolute difference is much bigger than 0.1 t.checkInexact(1100000, 1000000, 0.1); // SUCCEEDS: the relative difference is at most 10%
Relative tolerances are interpreted as percentages of the expected value, while absolute tolerances
are interpreted as plain numbers.)
To compare two strings, we use the equals method:
t.checkExpect("hello".equals("hel" + "lo"), true) t.checkExpect("hello".equals("goodbye"), false)
Technically, we can use == to compare two strings, but it may not always give the answer we expect: it
relies on subtle details of Java that we have not yet explored, and it is much safer to use the equals method.
Do Now!
Convince yourself that == for integers and booleans and equals on Strings obey the four properties above.
11.3 Review: sameness of structured data
Recall a simple definition of a book:
class Book { String title; String author; Book(String title, String author) { this.title = title; this.author = author; } }
Do Now!
How might we define sameness for books?
Two books are the same when their titles are the same and their authors are the same:
// In Book boolean sameBook(Book that) { /* Fields: * this.title * this.author * * Methods of fields: * this.title.equals(String) -> Boolean * this.author.equals(String) -> Boolean * * Fields of parameters: * that.title * that.author */ return this.title.equals(that.title) && this.author.equals(that.author); }
This makes sense: we define sameness of two structures in terms of the sameness of their parts.
Also notice that we restrict our comparison to be between two Books —
Do Now!
Define the method samePoint for the CartPt class.
Do Now!
Revise the definition of Book so that its author field is now of type Author, where Authors have first and last names, and two Authors are the same when both names are the same. Revise the sameBook method. What methods must it delegate to?
11.4 Sameness of union data: Warmup
Recall a simplified form of our shapes examples:
interface IShape { } class Circle implements IShape { int x, y; int radius; Circle(int x, int y, int radius) { this.x = x; this.y = y; this.radius = radius; } } class Rect implements IShape { int x, y; int w, h; Rect(int x, int y, int w, int h) { this.x = x; this.y = y; this.w = w; this.h = h; } }
We can define simple methods to check if two Circles are the same, or two Rects,
by temporarily ignoring the fact that these classes implement IShape, and just defining methods as we did above with Book:
// In Circle public boolean sameCircle(Circle that) { /* Template: * Fields: * this.x, this.y, this.radius * * Fields of parameters: * that.x, that.y, that.radius */ return this.x == that.x && this.y == that.y && this.radius == that.radius; } // In Rect public boolean sameRect(Rect that) { /* Template: * Fields: * this.x, this.y, this.w, this.h * * Fields of parameters: * that.x, that.y, that.w, that.h */ return this.x == that.x && this.y == that.y && this.w == that.w && this.h == that.h; }
We can write tests for these methods, and they work correctly:
// In test method in an Examples class Circle c1 = new Circle(3, 4, 5); Circle c2 = new Circle(4, 5, 6); Circle c3 = new Circle(3, 4, 5); Rect r1 = new Rect(3, 4, 5, 5); Rect r2 = new Rect(4, 5, 6, 7); Rect r3 = new Rect(3, 4, 5, 5); t.checkExpect(c1.sameCircle(c2), false) t.checkExpect(c2.sameCircle(c1), false) t.checkExpect(c1.sameCircle(c3), true) t.checkExpect(c3.sameCircle(c1), true) t.checkExpect(r1.sameRect(r2), false) t.checkExpect(r2.sameRect(r1), false) t.checkExpect(r1.sameRect(r3), true) t.checkExpect(r3.sameRect(r1), true)
11.5 Sameness of union data: flawed attempt #1 using “casting” and type-testing
Ultimately the data type we care about here is IShape; we can’t ignore it forever.
We don’t just want to compare Circles to Circles; we want to compare any IShape to any other IShape.
If we follow the pattern we’ve started above, we ought to define a method
boolean sameShape(IShape that)
on the IShape interface, and implement it in our classes. Let’s try it for Circle first:
// In Circle: public boolean sameShape(IShape that) { /* * Fields: * this.x, this.y, this.radius * * Methods: * this.sameShape(IShape) -> boolean * this.sameCircle(Circle) -> boolean * * Methods on fields: * * Fields on parameters: * * Methods on parameters * that.sameShape(IShape) -> boolean */ ??? }
Our template is exceedingly useless here: if all we know is that that is an IShape,
we have no way of knowing whether that is a Circle or a Rect (or something else).
So we can’t call sameCircle, even though it would be perfect here.
11.5.1 Casting
One naive approach would be to try to somehow force Java to treat that as a Circle —
Do Now!
What could go wrong with taking this position?
Java actually gives us a mechanism to achieve this “forcing”: we can cast the value to the Circle class.
A type cast, or just a cast, is written as ((SomeTypeName)someValue), and it has two aspects to its meaning.
First, it statically (that is, in the source code of our program) lets us treat someValue as if it had been
defined with the type SomeTypeName, even if we don’t know for sure that its value is really of that type,
and so it will let us access the fields and methods of that type without giving a type error. This sounds great; let’s try it:
// In Circle public boolean sameShape(IShape that) { return this.sameCircle((Circle)that); // cast that to a Circle, and use the sameCircle helper }
If we try writing tests for this, it doesn’t go well:
// In the Examples class t.checkExpect(c1.sameShape(c3), true) // works t.checkExpect(c1.sameShape(c2), false) // works t.checkExpect(r1.sameShape(r3), true) // works t.checkExpect(r1.sameShape(r2), false) // works t.checkExpect(c1.sameShape(r1), false) // CRASH! with a ClassCastException
Quite literally, we’re trying to fit a square peg into a round hole...
What went wrong? We passed Circle’s sameShape method a Rect value,
and the code of the method tried to pretend that that Rect was actually a Circle,
and that simply cannot work.
So where did the exception get thrown? It came from the second part of the meaning of casts:
dynamically (that is, at runtime), a cast will check that the given value really is an instance of the specified type.
When it is, the cast succeeds and the program continues; but when it isn’t, the cast will throw a ClassCastException.
Do Now!
Which of our properties for sameness have we violated here?
11.5.2 Type-testing using instanceof
We’ve made some progress, but not enough. We can convince Java to compile our code without any type errors,
but we can’t safely use casts, since they may throw exceptions at runtime. When would casting a value to Circle fail? Precisely
when the value is not a Circle (by definition) —
In Racket, we had type-testing predicates (circle? and rect?) that would let us check what kind of a structure we had. Until now, we have said that Java has no such mechanism. In fact, Java actually does have a type-testing mechanism, but it does not work as well as we might hope. Let’s see how to use it, and how it breaks.
To test whether a value is an instance of a given class or interface, we introduce a new operator, instanceof.
We’ll illustrate how it works by way of examples:
t.checkExpect(new Circle(3, 4, 5) instanceof Circle, true) // because Circle is a Circle. t.checkExpect(new Circle(3, 4, 5) instanceof IShape, true) // because Circle implements IShape t.checkExpect(new Circle(3, 4, 5) instanceof Book, false) // because a Circle is not a Book t.checkExpect(new Rect(3, 4, 5, 6) instanceof Circle, false) // because a Rect is not a Circle
Using instanceof, we can test a value against a class or interface type, and get a boolean answer.
(This is unlike any other operator we have seen so far, which all let us work with two values, not with values and types.)
Notice that we can ask absurd questions, like whether a new Circle value is an instanceof Book, even when we
know the answer must always be false.
Let’s now use this operator to improve our code above:
// In Circle public boolean sameShape(IShape that) { if (that instanceof Circle) { // that is-a Circle -- we can safely cast! return this.sameCircle((Circle)that); } else { // that is not a Circle return false; } }
Now our code works: when that is not a Circle, we just return false. When it is a Circle,
we can safely cast it to Circle and then invoke sameCircle with it.
Notice that we need both the cast and the instanceof: the cast convinces Java to let us use that as a Circle,
and the instanceof test ensures that it actually will be safe for us to do so.
This is a good example of the difference between static and dynamic information available in our code, and goes to show that static types sometimes cause problems (that must then be worked around), even as they help solve others.
If we try our tests, they work now:
// In the Examples class t.checkExpect(c1.sameShape(r1), false) // works
Do Now!
Implement sameShape for the Rect class, following the same pattern as for Circle. What test should you write to confirm that it works?
// In Rect public boolean sameShape(IShape that) { if (that instanceof Rect) { // that is-a Rect -- we can safely cast! return this.sameRect((Rect)that); } else { // that is not a Rect return false; } }
// In the Examples class t.checkExpect(r1.sameShape(c1), false); // works
11.5.3 What goes wrong with casting and instanceof?
Let’s add another type of shape to our classes: let’s bring back the Square class as a subclass of Rect.
Do Now!
Implement sameShape for Square, along with a sameSquare helper method. Write tests to confirm that it works properly.
We can start the definition of Square easily enough:
class Square extends Rect { Square(int x, int y, int s) { super(x, y, s, s); } public boolean sameShape(IShape that) { if (that instanceof Square) { return this.sameSquare((Square)that); } else { return false; } } public boolean sameSquare(Square that) { return this.x == that.x && this.y == that.y && this.w == that.w; // No need to check the h field, too... } }
But writing tests for this shows a subtle bug:
// In test method in an Examples class Square s1 = new Square(3, 4, 5); Square s2 = new Square(4, 5, 6); Square s3 = new Square(3, 4, 5); // basic checks comparing two Squares should work t.checkExpect(s1.sameShape(s2), false) t.checkExpect(s2.sameShape(s1), false) t.checkExpect(s1.sameShape(s3), true) t.checkExpect(s3.sameShape(s1), true) // Comparing a Square with a Rect of a different size t.checkExpect(s1.sameShape(r2), false) // Good t.checkExpect(r2.sameShape(s1), false) // Good // Comparing a Square with a Rect of the same size t.checkExpect(s1.sameShape(r1), false) // Good t.checkExpect(r1.sameShape(s1), true) // Not so good
Do Now!
Which property of sameness have we violated?
11.6 Sameness of union data: flawed attempt #2 using “custom” type-testing
What’s going on here? Carefully step through the last two examples. When we test s1.sameShape(r1):
s1 is a Square, so we invoke the sameShape method defined in Square.
This method checks whether that (which is r1) is an instance of Square, which it is not, so the method returns false.
On the other hand, when we test r1.sameShape(s1):
r1 is a Rect, so we invoke the sameShape method defined in Rect.
This method checks whether that (which is s1) is an instance of Square. Since Square is a subclass of Rect, this instanceof test returns true. So the method then casts that to a Rect (which is perfectly fine), and invokes sameRect.
The sameRect method compares all the fields of this (which is r1) to all the fields of that (which is s1), and since they all match, it returns true.
Somehow, our sameShape operation is no longer symmetric. The problem is that instanceof is too lenient:
it returns true when the provided value is an instance of the given class or of any of its subclasses. A Square
is an instanceof Rect, but a Rect is not an instanceof Square, and this asymmetry
is now apparent in the behavior of our sameShape implementation.
To work around this, we can define a set of methods on the IShape interface to tell us when a given shape
is precisely a Circle or a Rect or a Square:
interface IShape { boolean sameShape(IShape that); // is this shape a Circle? boolean isCircle(); // is this shape a Rect? boolean isRect(); // is this shape a Square? boolean isSquare(); }
// in Circle public boolean isCircle() { return true; } public boolean isRect() { return false; } public boolean isSquare() { return false; }
// in Rect public boolean isCircle() { return false; } public boolean isRect() { return true; } public boolean isSquare() { return false; }
// in Square public boolean isCircle() { return false; } public boolean isRect() { return false; } public boolean isSquare() { return true; }
Now if we use these methods instead of instanceof, we’ll get better results:
// in Rect public boolean sameShape(IShape that) { if (that.isRect()) { return this.sameRect((Rect)that); } else { return false; } }
// in Square public boolean sameShape(IShape that) { if (that.isSquare()) { return this.sameSquare((Square)that); } else { return false; } }
Do Now!
Try making these changes, and confirm that our bad test above is now better.
// Comparing a Square with a Rect of the same size t.checkExpect(s1.sameShape(r1), false) // Good t.checkExpect(r1.sameShape(s1), false) // Good
Do Now!
But r1 and s1 both describe rectangles with width and height of 5, at position (3,4). Why are they not equal?
Do Now!
Why must we have Square respond to isRect with false —surely all Squares are Rects?
Remember our original goal for sameness testing: two values are the same if there is no
method we could write, or operation we could use, to distinguish between them. Now that we
have instanceof, there is a way we can distinguish them:
// Distinguishing between a Square and a Rect of the same size t.checkExpect(s1 instanceof Square, true) t.checkExpect(r1 instanceof Square, false)
If there is a way to distinguish between these values, they must not be the same!
Wrapup
We’re still not finished, though: we have these odd isSomeType methods
which don’t have any utility outside of the sameShape methods, and we still have
casts which (if we make any mistakes) will crash our program with an exception. So how can we
implement this functionality without casts and without oddly-specific helper methods?
Exercise
Try to figure this out. Hint: you may want to look at the youngerIAT method from Lecture 7: Accumulator methods, continued, which deals with a similar problem.