forked from kscience/kmath
57 lines
2.8 KiB
Markdown
57 lines
2.8 KiB
Markdown
# 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 completely separate numerical representations from behaviors.
|
|
One solution in Kotlin, is to define a separate class which represents some entity without any operations,
|
|
for example a complex number:
|
|
|
|
```kotlin
|
|
data class Complex(val re: Double, val im: Double)
|
|
```
|
|
|
|
And then define a separate class or singleton, representing an operation on those complex numbers:
|
|
|
|
```kotlin
|
|
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
|
|
to treat this situation: [extensions with receivers](https://kotlinlang.org/docs/reference/extensions.html#extension-functions).
|
|
So in Kotlin, an operation on complex number could be implemented as:
|
|
|
|
```kotlin
|
|
with(ComplexOperations) { c1 + c2 - c3 }
|
|
```
|
|
|
|
Kotlin also allows to create functions with receivers:
|
|
|
|
```kotlin
|
|
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 responsible not only 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. |