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, it does not use memory in the compiled program. It 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 generic program, the algorithms to examine the structure follows from the construction of the space.

This makes it possible to solve tasks as:

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

Iterating through the 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.

Pick 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 represents 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.

Structs

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).
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.

Enums

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.