Overview

Mathy's Computer Algebra System creates beautiful trees out of math expressions. To modify trees mathy exposes a rules-based transformation system.

Overview

Rules-based systems are known for their flexibility and can have tens or hundreds of rules that define how a system should act.

Contrary to that, Mathy tries to implement as few rules as it can. It does this by building rules based on the Properties of Numbers.

Mathy agents effectively start tabula rasa like AlphaZero by using such basic actions.

Tabula rasa means that Mathy must learn from first principles, in this case an understanding of how to apply the Properties of Numbers.

Properties of Numbers

So you don't recall what the properties of numbers are? Or maybe you just want a refresher? Dummies.com has a decent cheat sheet

Basic Example

To see learning from first principles works, consider adding two like terms together: 4x + 2x

Rather than write a rule for combining like terms, Mathy agents must use multiple simpler rules.

Specifically, the distributive property is used to factor out the common x variable:

Input = 4x + 2x Output = (4 + 2) * x
4 x * 2 x * + 4 2 + x *

This leaves the 4 + 2 to be added using constant arithmetic.

Input = (4 + 2) * x Output = 6x
4 2 + x * 6 x *

It can help to think of this as challenging the model to learn a new skill.

Mathy has to learn how to deploy two actions in a sequence to combine like terms.

What Rules Do

Rules have two primary responsibilities:

  1. Determine if the rule can be applied to a given node in an expression
  2. Apply a transformation to an applicable node, and return information about what has changed

Visual Example

Consider the application of constant arithmetic to simplify the expression 12 * 10 + x so that it results in a tree of 120 + x

Before After
12 10 * x + 120 x +

The constant simplify rule takes the two constant nodes 12 and 10 then evaluates them and replaces them inside the tree with a single constant value of 120

Info

When you want to apply a rule to a tree the find_nodes helper can be used to return a list of nodes that the current rule can be applied to.

Code Example

Consider an example where we want to swap the position of two nodes, so that like terms are adjacent to one another for combination:

Open Example In Colab 

from mathy import CommutativeSwapRule, ExpressionParser

input = "x + y + x"
output = "x + x + y"
parser = ExpressionParser()

input_exp = parser.parse(input)
output_exp = parser.parse(output)

# Verify that the rule transforms the tree as expected
change = CommutativeSwapRule().apply_to(input_exp)
assert str(change.result) == output

After using commutative swap, the tree is now arranged with like terms being adjacent, which enables the application of other rules like the distributive factor out.