Expression Trees

Most generic interface to represent a mathematical expression in Diofant is a tree. Let us take the expression

>>> x*y + x**2
 2
x  + x⋅y

We can see what this expression looks like internally by using repr()

>>> repr(_)
"Add(Pow(Symbol('x'), Integer(2)), Mul(Symbol('x'), Symbol('y')))"

The easiest way to tear this apart is to look at a diagram of the expression tree

digraph{ # Graph style "bgcolor"="transparent" "ordering"="out" "rankdir"="TD" ######### # Nodes # ######### "Add(Pow(Symbol('x'), Integer(2)), Mul(Symbol('x'), Symbol('y')))_()" ["color"="black", "label"="Add", "shape"="ellipse"]; "Pow(Symbol('x'), Integer(2))_(0,)" ["color"="black", "label"="Pow", "shape"="ellipse"]; "Symbol('x')_(0, 0)" ["color"="black", "label"="Symbol('x')", "shape"="ellipse"]; "Integer(2)_(0, 1)" ["color"="black", "label"="Integer(2)", "shape"="ellipse"]; "Mul(Symbol('x'), Symbol('y'))_(1,)" ["color"="black", "label"="Mul", "shape"="ellipse"]; "Symbol('x')_(1, 0)" ["color"="black", "label"="Symbol('x')", "shape"="ellipse"]; "Symbol('y')_(1, 1)" ["color"="black", "label"="Symbol('y')", "shape"="ellipse"]; ######### # Edges # ######### "Add(Pow(Symbol('x'), Integer(2)), Mul(Symbol('x'), Symbol('y')))_()" -> "Pow(Symbol('x'), Integer(2))_(0,)"; "Add(Pow(Symbol('x'), Integer(2)), Mul(Symbol('x'), Symbol('y')))_()" -> "Mul(Symbol('x'), Symbol('y'))_(1,)"; "Pow(Symbol('x'), Integer(2))_(0,)" -> "Symbol('x')_(0, 0)"; "Pow(Symbol('x'), Integer(2))_(0,)" -> "Integer(2)_(0, 1)"; "Mul(Symbol('x'), Symbol('y'))_(1,)" -> "Symbol('x')_(1, 0)"; "Mul(Symbol('x'), Symbol('y'))_(1,)" -> "Symbol('y')_(1, 1)"; }

First, let’s look at the leaves of this tree. We got here instances of the Symbol class and the Diofant version of integers, instance of the Integer class, even technically we input integer literal 2.

What about x*y? As we might expect, this is the multiplication of x and y. The Diofant class for multiplication is Mul

>>> repr(x*y)
"Mul(Symbol('x'), Symbol('y'))"

Thus, we could have created the same object by writing

>>> Mul(x, y)
x⋅y

When we write x**2, this creates a Pow class instance

>>> repr(x**2)
"Pow(Symbol('x'), Integer(2))"

We could have created the same object by calling

>>> Pow(x, 2)
 2
x

Now we get to our final expression, x*y + x**2. This is the addition of our last two objects. The Diofant class for addition is Add, so, as you might expect, to create this object, we also could use

>>> Add(Pow(x, 2), Mul(x, y))
 2
x  + x⋅y

Note

The arguments of Add and the commutative arguments of Mul are stored in an order, which is independent of the order inputted.

There is no subtraction class. x - y is represented as x + (-1)*y

>>> repr(x - y)
"Add(Symbol('x'), Mul(Integer(-1), Symbol('y')))"

digraph{ # Graph style "bgcolor"="transparent" "ordering"="out" "rankdir"="TD" ######### # Nodes # ######### "Add(Symbol('x'), Mul(Integer(-1), Symbol('y')))_()" ["color"="black", "label"="Add", "shape"="ellipse"]; "Symbol('x')_(0,)" ["color"="black", "label"="Symbol('x')", "shape"="ellipse"]; "Mul(Integer(-1), Symbol('y'))_(1,)" ["color"="black", "label"="Mul", "shape"="ellipse"]; "Integer(-1)_(1, 0)" ["color"="black", "label"="Integer(-1)", "shape"="ellipse"]; "Symbol('y')_(1, 1)" ["color"="black", "label"="Symbol('y')", "shape"="ellipse"]; ######### # Edges # ######### "Add(Symbol('x'), Mul(Integer(-1), Symbol('y')))_()" -> "Symbol('x')_(0,)"; "Add(Symbol('x'), Mul(Integer(-1), Symbol('y')))_()" -> "Mul(Integer(-1), Symbol('y'))_(1,)"; "Mul(Integer(-1), Symbol('y'))_(1,)" -> "Integer(-1)_(1, 0)"; "Mul(Integer(-1), Symbol('y'))_(1,)" -> "Symbol('y')_(1, 1)"; }

Similarly to subtraction, there is no division class

>>> repr(x/y)
"Mul(Symbol('x'), Pow(Symbol('y'), Integer(-1)))"

digraph{ # Graph style "bgcolor"="transparent" "ordering"="out" "rankdir"="TD" ######### # Nodes # ######### "Mul(Symbol('x'), Pow(Symbol('y'), Integer(-1)))_()" ["color"="black", "label"="Mul", "shape"="ellipse"]; "Symbol('x')_(0,)" ["color"="black", "label"="Symbol('x')", "shape"="ellipse"]; "Pow(Symbol('y'), Integer(-1))_(1,)" ["color"="black", "label"="Pow", "shape"="ellipse"]; "Symbol('y')_(1, 0)" ["color"="black", "label"="Symbol('y')", "shape"="ellipse"]; "Integer(-1)_(1, 1)" ["color"="black", "label"="Integer(-1)", "shape"="ellipse"]; ######### # Edges # ######### "Mul(Symbol('x'), Pow(Symbol('y'), Integer(-1)))_()" -> "Symbol('x')_(0,)"; "Mul(Symbol('x'), Pow(Symbol('y'), Integer(-1)))_()" -> "Pow(Symbol('y'), Integer(-1))_(1,)"; "Pow(Symbol('y'), Integer(-1))_(1,)" -> "Symbol('y')_(1, 0)"; "Pow(Symbol('y'), Integer(-1))_(1,)" -> "Integer(-1)_(1, 1)"; }

We see that x/y is represented as x*y**(-1).

But what about x/2? Following the pattern of the previous example, we might expect to see Mul(x, Pow(Integer(2), -1)). But instead, we have

>>> repr(x/2)
"Mul(Rational(1, 2), Symbol('x'))"

Rational numbers are always combined into a single term in a multiplication, so that when we divide by 2, it is represented as multiplying by 1/2.

Walking the Tree

Let’s look at how to dig our way through an expression tree, using a very generic interface — attributes func and args.

The head of the object is encoded in the func attribute

>>> expr = 2 + x*y; expr
x⋅y + 2
>>> expr.func
<class 'diofant.core.add.Add'>

The class of an object need not be the same as the one used to create it

>>> Add(x, x)
2⋅x
>>> _.func
<class 'diofant.core.mul.Mul'>

Note

Diofant classes heavy use of the __new__() class constructor, which, unlike __init__(), allows a different class to be returned from the constructor.

The children of a node in the tree are held in the args attribute

>>> expr.args
(2, x⋅y)

Note

Every expression with non-empty args can be reconstructed, using

>>> expr.func(*expr.args)
x⋅y + 2

Empty args signal that we have hit a leaf of the expression tree

>>> x.args
()
>>> Integer(2).args
()

This interface allows us to write recursive generators that walk expression trees either in post-order or pre-order fashion

>>> tuple(preorder_traversal(expr))
(x⋅y + 2, 2, x⋅y, x, y)
>>> tuple(postorder_traversal(expr))
(2, x, y, x⋅y, x⋅y + 2)