A wrong question: Is a Square a Rectangle?

Overview

1. what is Subtyping?
2. what is Liskov Substitution Principle?
3. what's the problem?
4. how to solve this mistake of type system

What is Subtyping?

A common misunderstanding is OOP must have subtyping and generic at the same time, but they are independent features, generic is just a type-level function with takes a type, generates a type. Subtyping is a relationship between type, when we say A <: B, which means A is a subtype of B. In Java, we use concrete syntax extends or implements.

What is Liskov Substitution Principle(LSP)?

LSP said

if S is a subtype of T, then objects of type T in a program may be replaced with objects of type S without altering the program

What's the problem

Square is a special Rectangle where all four sides are equal in length. Thus we might write down:

class Rectangle {
    double x, y;
    Rectangle(double initX, double initY) { x = initX; y = initY; }

    public void setX(double x) { this.x = x; }
    public void setY(double y) { this.y = y; }
}

class Square extends Rectangle {
    Square(double initX, double initY) throws SquareException {
        super(initX, initY);
        if (initX != initY) {
            throw new SquareException(initX, initY);
        }
    }
}

Ok, so we can have following program:

Square s = new Square(2, 2);
Rectangle r = s;
r.setX(10);

Oops, s is not a square anymore, this correctly follows LSP but incorrect generally. To solve this, we revert the relationship between Square and Rectangle, now we have:

class Square {
    double x;
}

class Rectangle extends Square {
    double y;
}

Now setX and setY both will not affect the Square, imagine a method double area(), LSP was broken(now Square returns x*x and Rectangle returns x*y). So now we have more problem.

Solution

Here is a solution: refinement type. By using refinement type, we can claim type Square = Rectangle when (x = y). Then problem solved, when programming if we can't proof x = y this predicate then compiler won't think that is a Square but a Rectangle. Since Square is a Rectangle, method in Rectangle still can be using by Square, thus we didn't lose our power. However, program like s.setX(v), would need to prove v = y, to prove it is a Square. Here we have two choices:

1. change s type to Rectangle, the problem of this solution is if we write r = s; r.setX(v), compiler might not work for this(this is also why I'm designing controllable-refinement), mutable semantic always needs more effort.
2. throw compile error that Square constraint was broke by the statement.

Choose one and work with it, that's all, have a nice day.