What math do I need to learn to create a general equation solver?

Question

I'm interested in creating a general equation solving engine, akin to wolfram alpha (though it doesn't need to be quite as advanced). I'd like it to be able to solve any of the equations students encounter from grade school to early college; so including...

Simple real number polynomials
Equations of complex variables
Vector and Matrix equations
Equations with differentiation and integration
Differential equations
(Possibly) Boolean equations

Also, I'd like to be able to solve these equations axiomatically. For example if you wanted to solve $x + 3 = 5$, you could find the steps like...

Subtract $3$ on the right hand side

$$(x + 3) - 3 = 5 - 3$$

Associativity of addition

$$x + (3 - 3) = 5 - 3$$

Additive inverse

$$x + 0 = 5 - 3$$

Identity element

$$x = 5 - 3$$

Subtraction

$$x = 2$$

So I realize this is a lot, and before I get myself in too deep, I'm looking to know what math I should know before I would be able to create something like this.

So I'm assuming it'd be mostly algebra. I've studied group theory a bit, and touched on rings. Could someone with more knowledge, point me in the right direction? Is this even feasible for a single person?

If I were you I would start out with just one of these, as the methods used will be fairly different (though you should of course make sure to keep your code flexible so you can reuse as much as possible later parts). All in all, this sort of thing will be very complicated, but also probably a great way to learn a lot of neat things. — Tobias Kildetoft, Feb 22 '18 at 06:48
Then there is the problem of what programming language to use. — marty cohen, Feb 22 '18 at 06:53
Previously: Computer Algebra: Algorithms for solving equations symbolically — , Feb 22 '18 at 07:09
You will need to consider the map between the language of mathematics and the symbolic computer representation - https://en.wikipedia.org/wiki/Semantics_(computer_science). Also, ways to equate different symbolic statements - https://en.wikipedia.org/wiki/Proof_theory . — Damien, Feb 22 '18 at 07:54

score -1 · Answer 1 · answered Feb 22 '18 at 07:11

Think for approaches yourself and then have a look at how others did it.

Back in my early undergraduate days I thought "Why not write a program that produces theorems in an endless fashion and then go pick some new to publish them?" Given that cute idea I wondered how math worked at core and how to formalize it. I noticed two things:

Math objects have a class hierarchy pattern. A group is a pair of a set and a binary operation on it. A pair is a two-element set etc. pp.
Functionals in a boarder sense, i.e. objects that assign objects to objects play a central role and just using functions/methods (in the programming sense) capsuled in classes would not be enough to imitate this.

So in the end I came to believe I would need an object-oriented functional language to bring math into the computer. Imagine my surprise when I found out about Coq and that it does exactly that.

Notice that, as opposite to Wolfram Alpha, Coq is open source, so you can learn a lot from it and also use it as backend.

In the end, I became more interested with formally proofing theorems than to find them (I also realized my idea is close to read interesting information out of $\pi$), so now I'm a Mizar enthusiast. Mizar is not open source, and it doesn't proof things for you but checks your proofs, very rigorously.

What math do I need to learn to create a general equation solver?

1 Answers1