Skip to main content

Algebra.NET: A Simple Algebra eDSL for .NET

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

John Zabroski said…
I am jealous of how productive you are! ;-) I wish I had your energy to pursue these things. Do you do this in your spare time, or as part of your job? And, may I ask, what is the nature of your need for an optimization library? And are you planning to work on a benchmark facility to compare all of them? I have been truly crazy about eking performance out of my applications in the past 2 years.
Sandro Magi said…
I do this in my spare time, although my job sometimes overlaps, which helps my sanity. I've had to solve some optimization problems, like bin packing for minimizing shipping costs, and Steiner trees for minimum cost layout of air ducts. Many of the best approximations are LP problems which can be expressed as a series of inequalities.

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.

Popular posts from this blog

async.h - asynchronous, stackless subroutines in C

The async/await idiom is becoming increasingly popular. The first widely used language to include it was C#, and it has now spread into JavaScript and Rust. Now C/C++ programmers don't have to feel left out, because async.h is a header-only library that brings async/await to C! Features: It's 100% portable C. It requires very little state (2 bytes). It's not dependent on an OS. It's a bit simpler to understand than protothreads because the async state is caller-saved rather than callee-saved. #include "async.h" struct async pt; struct timer timer; async example(struct async *pt) { async_begin(pt); while(1) { if(initiate_io()) { timer_start(&timer); await(io_completed() || timer_expired(&timer)); read_data(); } } async_end; } This library is basically a modified version of the idioms found in the Protothreads library by Adam Dunkels, so it's not truly ground bre

Building a Query DSL in C#

I recently built a REST API prototype where one of the endpoints accepted a string representing a filter to apply to a set of results. For instance, for entities with named properties "Foo" and "Bar", a string like "(Foo = 'some string') or (Bar > 99)" would filter out the results where either Bar is less than or equal to 99, or Foo is not "some string". This would translate pretty straightforwardly into a SQL query, but as a masochist I was set on using Google Datastore as the backend, which unfortunately has a limited filtering API : It does not support disjunctions, ie. "OR" clauses. It does not support filtering using inequalities on more than one property. It does not support a not-equal operation. So in this post, I will describe the design which achieves the following goals: A backend-agnostic querying API supporting arbitrary clauses, conjunctions ("AND"), and disjunctions ("OR"). Implemen

Easy Automatic Differentiation in C#

I've recently been researching optimization and automatic differentiation (AD) , and decided to take a crack at distilling its essence in C#. Note that automatic differentiation (AD) is different than numerical differentiation . Math.NET already provides excellent support for numerical differentiation . C# doesn't seem to have many options for automatic differentiation, consisting mainly of an F# library with an interop layer, or paid libraries . Neither of these are suitable for learning how AD works. So here's a simple C# implementation of AD that relies on only two things: C#'s operator overloading, and arrays to represent the derivatives, which I think makes it pretty easy to understand. It's not particularly efficient, but it's simple! See the "Optimizations" section at the end if you want a very efficient specialization of this technique. What is Automatic Differentiation? Simply put, automatic differentiation is a technique for calcu