[−][src]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.

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.