Crate discrete

Combinatorial phantom types for discrete mathematics.

All discrete spaces have the following functions:

fn count(dim) -> usize;
fn to_index(dim, pos) -> usize;
fn to_pos(dim, index, &mut pos);

A discrete space is countable and has a one-to-one map with the natural numbers.

For example, a pair of natural numbers is a discrete space. There exists an algorithm that converts each pair of numbers into a number. Likewise there exists an algorithm that takes a number and converts it into a pair.

To construct a pair, you write:

let x: Pair<Data> = Construct::new();

The x above is a phantom variable, which means it does not use memory in the compiled program. The phantom variable represents the discrete space that we have constructed. Now we can call methods on the space to examine its discrete structure.

A pair can be visualized as edges between points. If we have 4 points then we can create 6 edges:

  o---o
  |\ /|
  | X |
  |/ \|
  o---o

To check this we can write:

let dim = 4; // number of points
assert_eq!(x.count(dim), 6); // count edges

Phantom types makes it possible to construct advanced discrete spaces. By using a generic program, the algorithms to examine the structure follows from the construction of the space.

This makes it possible to solve tasks such as:

• Compute upper bounds for many problems
• Store data with a non-trivial structure
• Convert from and to natural numbers
• Iterate through all elements of a space
• Pick a random object of the space

Iterating through all elements of a space can be done simply by counting from zero up to the size of the space. For each number, we convert to a position within the space.

Picking a random object of the space can be done by generating a random number between 0 and the size of the space.

Advanced spaces

Phantom types are used because they represent the general spaces. For example, we can represent a general two-dimensional space, instead of binding the type to the size.

For any constructed space, there is a dimension and position type. The dimension and position types are compositions, given by the type of the constructed space.

Modules

• space
  Helper trait for implementing discrete spaces.

Structs

• BigUint
  A big unsigned integer type.
• Context
  A discrete space that can model spatial operations over arbitrary states, therefore useful for context analysis.
• Data
  Used by the final subspace.
• Dimension
  Dimension is natural number, position is the same as index.
• DimensionN
  Dimension is a list of numbers, position is a list of numbers.
• DirectedContext
  Same as Context, but for directed edges.
• Either
  Selects between two spaces.
• EqPair
  Dimension is natural number, position is (min, max).
• Homotopy
  A discrete space that models undirected homotopy paths at a specified homotopy level.
• NeqPair
  Dimension is natural number, position is (a, b). Represents all directional pairs that has not same element for a and b.
• Of
  Used to combine the dimensional and position types.
• Pair
  Dimension is natural number, position is (min, max).
• Permutation
  Dimension is natural number, position is a list of numbers.
• PowerSet
  Dimension is natural number, position is a list of numbers.
• SqPair
  A discrete space that models a full square of NxN pairs.

Enums

• HPoint
  Stores a higher order point for homotopy spaces.
• Select
  Selects between spaces.

Traits

• Construct
  Constructs a new space.
• Count
  Implemented by spaces that can count the number of objects.
• ToIndex
  Implemented by spaces that can convert position to index.
• ToPos
  Implemented for spaces which can convert an index to position type.
• Zero
  Used to construct an uninitialized element of a discrete space.