kmath/doc/contexts.md

Context-oriented mathematics

The problem

A known problem for implementing mathematics in statically-typed languages (but not only in them) is that different sets of mathematical operators can be defined on the same mathematical objects. Sometimes there is no single way to treat some operations, including basic arithmetic operations, on a Java/Kotlin Number. Sometimes there are different ways to define the same structure, such as Euclidean and elliptic geometry vector spaces over real vectors. Another problem arises when one wants to add some kind of behavior to an existing entity. In dynamic languages those problems are usually solved by adding dynamic context-specific behaviors at runtime, but this solution has a lot of drawbacks.

Context-oriented approach

One possible solution to these problems is to divorce numerical representations from behaviors. For example in Kotlin one can define a separate class which represents some entity without any operations, ex. a complex number:

data class Complex(val re: Double, val im: Double)

And then to define a separate class or singleton, representing an operation on those complex numbers:

object ComplexOperations {
    operator fun Complex.plus(other: Complex) = Complex(re + other.re, im + other.im)
    operator fun Complex.minus(other: Complex) = Complex(re - other.re, im - other.im)
}

In Java, applying such external operations could be very cumbersome, but Kotlin has a unique feature which allows us implement this naturally: extensions with receivers. In Kotlin, an operation on complex number could be implemented as:

with(ComplexOperations) { c1 + c2 - c3 }

Kotlin also allows the creation of functions with receivers:

fun ComplexOperations.doSomethingWithComplex(c1: Complex, c2: Complex, c3: Complex) = c1 + c2 - c3

ComplexOperations.doComethingWithComplex(c1, c2, c3)

In fact, whole parts of a program may be run within a mathematical context or even multiple nested contexts.

In KMath, contexts are not only responsible for operations, but also for raw object creation and advanced features.

Other possibilities

An obvious candidate to get more or less the same functionality is the type class, which allows one to bind a behavior to a specific type without modifying the type itself. On the plus side, type classes do not require explicit context declaration, so the code looks cleaner. On the minus side, if there are different sets of behaviors for the same types, it is impossible to combine them into one module. Also, unlike type classes, context can have parameters or even state. For example in KMath, sizes and strides for NDElement or Matrix could be moved to context to optimize performance in case of a large amount of structures.