Create encodings.md

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+1. [Introduction](introduction.md)  
+2. [Fitness functions](fitness-functions.md)  
+3. **Encodings**  
+4. [Algorithms](algorithms.md)  
+5. [Genetic operators](genetic-operators.md)  
+6. [Stop conditions](stop-conditions.md)  
+7. [Metrics](metrics.md)    
+8. [Miscellaneous](miscellaneous.md)
+
+------------------------------------------------------------------------------------------------
+
+# Encodings
+
+The library defines several GA classes instead of using just a single one.
+The difference between these is the encoding type used to represent the
+problems and their solutions. Each of the GA classes is for a particular
+encoding type, and it can only be used for objective functions using the
+same encoding type.
+
+The below table shows each of the GA classes in the library, the encoding
+(or gene) type used by them, and the problem (or fitness function) type
+they can be used for:
+
+  |     GA class    |  Encoding type  |      Problem type      |
+  |:---------------:|:---------------:|:----------------------:|
+  |   `BinaryGA`    |   BinaryGene    |    Binary-encoded      |
+  |     `RCGA`      |    RealGene     | Floating-point-encoded |
+  | `PermutationGA` | PermutationGene |     Combinatorial      |
+  |   `IntegerGA`   |   IntegerGene   |    Integer-encoded     |
+
+The encoding also determines what kind of crossover and mutation methods
+can be used in the GA as these genetic operators also depend on the
+encoding, and are generally defined for a particular gene type.
+
+All of the GA classes are in the main `gapp` namespace.
+
+```cpp
+// the fitness function uses permutation encoding
+PermutationGA{}.solve(problems::TSP52{});
+
+// the fitness function uses real-encoding
+RCGA{}.solve(problems::Sphere{}, Bounds{ -10.0, 10.0 });
+
+// the fitness function uses binary-encoding
+BinaryGA{}.solve(problems::Sphere{});
+```
+
+## Solution representation
+
+The gene type determines how the candidate solutions are encoded in
+the population. The representation of the solutions will be a vector
+of the gene type used in all cases. Currently there is no way to
+change this to use some other data structure. This should be taken
+into account when defining new encodings.
+
+```cpp
+template<typename GeneType>
+using Chromosome = std::vector<GeneType>;
+```
+
+The candidates contain some more information in addition to their
+chromosomes, for example their fitness vectors, but these are
+independent of the gene type.
+
+The population is then made up of several candidates encoded in
+this way.
+
+## Custom encodings
+
+It is also possible to use a different encoding type by defining a
+new GA class. In order to do this, you have to:
+
+ - define the gene type that will be used
+ - define a specialization for `GaTraits<GeneType>`
+ - specialize `is_bounded<GeneType>` if needed
+ - define the GA class, derived from `GA<GeneType>`
+ - define crossover and mutation operators for this encoding
+
+The gene type may be anything, with one restriction: the types
+already used for the existing encodings are reserved and can't
+be used to define new encodings. See `<encoding/gene_types.hpp>`
+for the types that are already in use.
+
+```cpp
+using MyGeneType = std::variant<double, int>;
+```
+
+The specialization of `GaTraits<T>` for the gene type is
+required to define some attributes of the GA. These are the
+default crossover and mutation operators that will be used
+by the GA if none are specified explicitly, and the default
+mutation probability for the mutation operators:
+
+```cpp
+namespace gapp
+{
+    struct GaTraits<MyGeneType>
+	{
+	    using DefaultCrossover = MyCrossover;
+		using DefaultMutation  = MyMutation;
+		static Probability defaultMutationRate(size_t chrom_len) { return 1.0 / chrom_len; }
+	};
+
+}; // namespace gapp
+```
+
+Specializing the `is_bounded<T>` variable is only needed if
+the gene type used will have it's lower and upper bounds specified
+for each gene in the `solve` method. The value should be `true` in
+this case, and `false` otherwise.
+
+```cpp
+namespace gapp
+{
+    // This isn't technically needed, since false is the default value
+    template<>
+    inline constexpr bool is_bounded<MyGeneType> = false;
+
+} // namespace gapp
+```
+
+The actual GA class should be derived from `GA<T>` using the desired gene
+type for the type parameter `T`. The derived class only has to implement
+the `generateCandidate` method, and optionally the `initialize` method:
+
+```cpp
+class MyGA : public GA<MyGeneType>
+{
+public:
+	// Generate a random candidate solution. This is used to
+	// create the initial population.
+	Candidate<GeneType> generateCandidate() const override
+	{
+	    Candidate<GeneType> candidate;
+		// ...
+		return candidate;
+	}
+};
+```
+
+In addition to everything above, crossover and mutation operators will
+also have to be defined for this encoding type, as these operators depend
+on the encoding type, and the operators included in the library will not
+work for new encodings.
+
+------------------------------------------------------------------------------------------------