Algebra.NET is a simple library designed to facilitate easy expression and manipulation of algebraic functions. For instance, here's a simple function:

Function<Func<double, double>> a = Algebra.Function(x => 2 * x + 1);

We can compile such a function to efficient IL:

```
Func<double, double> func = a.Compile("times2plus1");
```

Or we can apply some algebraic identities to rewrite it:

```
Identity associative = Algebra.Identity(x => x + 1 == 1 + x);
Identity mulEqAdd = Algebra.Identity(x => 2 * x == x + x);
Console.WriteLine(a);
Console.WriteLine(a.Rewrite(1, associative, mulEqAdd));
// Prints:
// ((2 * x) + 1)
// (1 + (x + x))
```

Rewrites can sometimes loop forever (consider "x + y == y + x"), so the Rewrite method takes a number indicating the maximum number of iterations to perform all the rewrites.

All the usual arithmetic operations are available, including an extension method for exponentiation:

```
var f = Algebra.Function(x => x.Pow(3));
Console.WriteLine(x);
// Prints:
// (x ^ (3))
```

## Design

As of this writing, Algebra.NET is a functional example of a simple term rewriting system. Term rewriting is usually pretty awkward to express in an object-oriented language, and I banged my head against the keyboard to figure out a nice way to do it, until I hit on just doing unification (of course!).

So I reused the term language and added an equality operator to generate an identity that conceptually maps one term to another. I then perform unification on the left hand side, and generate a set of substitutions to transform the matching term into the right hand side of the identity.

It was ultimately quite simple, consisting of 3 methods on Term:

```
Term Rewrite(Identity e, Term[] bindings)
bool TryUnify(Term e, Term[] bindings)
Term Subsitute(Term[] bindings)
```

Rewrite tries to recursively unify the Identity's left hand side with the current term using TryUnify. On success, the 'bindings' array will have been populated by TryUnify with the substitutions to perform, so it substitutes the bindings into the identity's right hand side to generate the new term.

There are only 3 term types: constants, variables and binary operations. Negation is handled as a binary operation "0 - x" for simplicity. The unification methods on each of the term types are only a few lines of code each, but are quite powerful!

So if you want to understand expression compilation to CIL, unification, or term rewriting, this is pretty much as simple as it gets.

Algebra.NET doesn't perform any term simplification at this point, only term rewriting. Some rewrites may of course be simplifications, but a term like "0 - 3" will not be simplified to "-3".

## Future Work

As mentioned, Algebra.NET doesn't perform simplification, so that's a big one. I started developing this to work on symbolic and automatic differentiation for numerical optimization problems. I'm aware of other .NET libraries for this, but I didn't like how clumsy it was to write algebraic expressions, nor did they have nice and extensible facilities for rewriting expressions. So I created this in my spare time and intended to continue fleshing it out as needed.

## Comments

This sort of overlaps that, since equalities are the natural place to start. I was initially motivated to write a small algebraic library to do symbolic and automatic differentiation to port Jules Jacob's Newton optimization blog posts to C#. I just need to define the simplification identities, and then I can tackle differentiation.