74 lines
3.5 KiB
Markdown
74 lines
3.5 KiB
Markdown
# Context-oriented mathematics
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## The problem
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A known problem for implementing mathematics in statically-typed languages (but not only in them) is that different
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sets of mathematical operators can be defined on the same mathematical objects. Sometimes there is no single way to
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treat some operations, including basic arithmetic operations, on a Java/Kotlin `Number`. Sometimes there are different ways to
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define the same structure, such as Euclidean and elliptic geometry vector spaces over real vectors. Another problem arises when
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one wants to add some kind of behavior to an existing entity. In dynamic languages those problems are usually solved
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by adding dynamic context-specific behaviors at runtime, but this solution has a lot of drawbacks.
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## Context-oriented approach
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One possible solution to these problems is to divorce numerical representations from behaviors.
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For example in Kotlin one can define a separate class which represents some entity without any operations,
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ex. a complex number:
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```kotlin
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data class Complex(val re: Double, val im: Double)
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```
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And then to define a separate class or singleton, representing an operation on those complex numbers:
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```kotlin
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object ComplexOperations {
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operator fun Complex.plus(other: Complex) = Complex(re + other.re, im + other.im)
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operator fun Complex.minus(other: Complex) = Complex(re - other.re, im - other.im)
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}
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```
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In Java, applying such external operations could be very cumbersome, but Kotlin has a unique feature which allows us
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implement this naturally: [extensions with receivers](https://kotlinlang.org/docs/reference/extensions.html#extension-functions).
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In Kotlin, an operation on complex number could be implemented as:
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```kotlin
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with(ComplexOperations) { c1 + c2 - c3 }
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```
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Kotlin also allows the creation of functions with receivers:
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```kotlin
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fun ComplexOperations.doSomethingWithComplex(c1: Complex, c2: Complex, c3: Complex) = c1 + c2 - c3
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ComplexOperations.doComethingWithComplex(c1, c2, c3)
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```
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In fact, whole parts of a program may be run within a mathematical context or even multiple nested contexts.
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In KMath, contexts are not only responsible for operations, but also for raw object creation and advanced features.
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## Other possibilities
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### Type classes
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An obvious candidate to get more or less the same functionality is the type class, which allows one to bind a behavior to
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a specific type without modifying the type itself. On the plus side, type classes do not require explicit context
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declaration, so the code looks cleaner. On the minus side, if there are different sets of behaviors for the same types,
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it is impossible to combine them into one module. Also, unlike type classes, context can have parameters or even
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state. For example in KMath, sizes and strides for `NDElement` or `Matrix` could be moved to context to optimize
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performance in case of a large amount of structures.
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### Wildcard imports and importing-on-demand
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Sometimes, one may wish to use a single context throughout a file. In this case, is possible to import all members
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from a package or file, via `import context.complex.*`. Effectively, this is the same as enclosing an entire file
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with a single context. However when using multiple contexts, this technique can introduce operator ambiguity, due to
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namespace pollution. If there are multiple scoped contexts which define the same operation, it is still possible to
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to import specific operations as needed, without using an explicit context with extension functions, for example:
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```
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import context.complex.op1
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import context.quaternion.op2
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```
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