`Constraints.USUAL_CONSTRAINTS`

— Constant`USUAL_CONSTRAINTS::Dict`

Dictionary that contains all the usual constraints defined in Constraint.jl. It is based on XCSP3-core specifications available at https://arxiv.org/abs/2009.00514

Adding a new constraint is as simple as defining a new function with the same name as the constraint and using the `@usual`

macro to define it. The macro will take care of adding the new constraint to the `USUAL_CONSTRAINTS`

dictionary.

**Example**

`@usual concept_all_different(x; vals=nothing) = xcsp_all_different(list=x, except=vals)`

`Constraints.USUAL_SYMMETRIES`

— Constant`USUAL_SYMMETRIES`

A Dictionary that contains the function to apply for each symmetry to avoid searching a whole space.

`Constraints.Constraint`

— Type`Constraint`

Parametric structure with the following fields.

`concept`

: a Boolean function that, given an assignment`x`

, outputs`true`

if`x`

satisfies the constraint, and`false`

otherwise.`error`

: a positive function that works as preferences over invalid assignments. Return`0.0`

if the constraint is satisfied, and a strictly positive real otherwise.

`ConstraintCommons.extract_parameters`

— Function`extract_parameters(s::Symbol, constraints_dict=USUAL_CONSTRAINTS; parameters=ConstraintCommons.USUAL_CONSTRAINT_PARAMETERS)`

Return the parameters of the constraint `s`

in `constraints_dict`

.

**Arguments**

`s::Symbol`

: the constraint name.`constraints_dict::Dict{Symbol,Constraint}`

: dictionary of constraints. Default is`USUAL_CONSTRAINTS`

.`parameters::Vector{Symbol}`

: vector of parameters. Default is`ConstraintCommons.USUAL_CONSTRAINT_PARAMETERS`

.

**Example**

`extract_parameters(:all_different)`

`Constraints.args`

— Method`args(c::Constraint)`

Return the expected length restriction of the arguments in a constraint `c`

. The value `nothing`

indicates that any strictly positive number of value is accepted.

`Constraints.concept`

— Method`concept(c::Constraint)`

Return the concept (function) of constraint `c`

. concept(c::Constraint, x...; param = nothing) Apply the concept of `c`

to values `x`

and optionally `param`

.

`Constraints.concept`

— Method`concept(s::Symbol, args...; kargs...)`

Return the concept of the constraint `s`

applied to `args`

and `kargs`

. This is a shortcut for `concept(USUAL_CONSTRAINTS[s])(args...; kargs...)`

.

**Arguments**

`s::Symbol`

: the constraint name.`args...`

: the arguments to apply the concept to.`kargs...`

: the keyword arguments to apply the concept to.

**Example**

`concept(:all_different, [1, 2, 3])`

`Constraints.concept_vs_error`

— Method`concept_vs_error(c, e, args...; kargs...)`

Compare the results of a concept function and an error function for the same inputs. It is mainly used for testing purposes.

**Arguments**

`c`

: The concept function.`e`

: The error function.`args...`

: Positional arguments to be passed to both the concept and error functions.`kargs...`

: Keyword arguments to be passed to both the concept and error functions.

**Returns**

- Boolean: Returns true if the result of the concept function is not equal to whether the result of the error function is greater than 0.0. Otherwise, it returns false.

**Examples**

`concept_vs_error(all_different, make_error(:all_different), [1, 2, 3]) # Returns false`

`Constraints.constraints_descriptions`

— Function`constraints_descriptions(C=USUAL_CONSTRAINTS)`

Return a pretty table with the descriptions of the constraints in `C`

.

**Arguments**

`C::Dict{Symbol,Constraint}`

: dictionary of constraints. Default is`USUAL_CONSTRAINTS`

.

**Example**

`constraints_descriptions()`

`Constraints.constraints_parameters`

— Function`constraints_parameters(C=USUAL_CONSTRAINTS)`

Return a pretty table with the parameters of the constraints in `C`

.

**Arguments**

`C::Dict{Symbol,Constraint}`

: dictionary of constraints. Default is`USUAL_CONSTRAINTS`

.

**Example**

`constraints_parameters()`

`Constraints.describe`

— Function`describe(constraints::Dict{Symbol,Constraint}=USUAL_CONSTRAINTS; width=150)`

Return a pretty table with the description of the constraints in `constraints`

.

**Arguments**

`constraints::Dict{Symbol,Constraint}`

: dictionary of constraints to describe. Default is`USUAL_CONSTRAINTS`

.`width::Int`

: width of the table.

**Example**

`describe()`

`Constraints.error_f`

— Method`error_f(c::Constraint)`

Return the error function of constraint `c`

. error_f(c::Constraint, x; param = nothing) Apply the error function of `c`

to values `x`

and optionally `param`

.

`Constraints.make_error`

— Method`make_error(symb::Symbol)`

Create a function that returns an error based on the predicate of the constraint identified by the symbol provided.

**Arguments**

`symb::Symbol`

: The symbol used to determine the error function to be returned. The function first checks if a predicate with the prefix "icn*" exists in the Constraints module. If it does, it returns that function. If it doesn't, it checks for a predicate with the prefix "error*". If that exists, it returns that function. If neither exists, it returns a function that evaluates the predicate with the prefix "concept_" and returns the negation of its result cast to Float64.

**Returns**

- Function: A function that takes in a variable
`x`

and an arbitrary number of parameters`params`

. The function returns a Float64.

**Examples**

```
e = make_error(:all_different)
e([1, 2, 3]) # Returns 0.0
e([1, 1, 3]) # Returns 1.0
```

`Constraints.params_length`

— Method`params_length(c::Constraint)`

Return the expected length restriction of the arguments in a constraint `c`

. The value `nothing`

indicates that any strictly positive number of parameters is accepted.

`Constraints.shrink_concept`

— Method`shrink_concept(s)`

Simply delete the `concept_`

part of symbol or string starting with it. TODO: add a check with a warning if `s`

starts with something different.

`Constraints.symmetries`

— Method`symmetries(c::Constraint)`

Return the list of symmetries of `c`

.

`Constraints.xcsp_all_different`

— Method`xcsp_all_different(list::Vector{Int})`

Return `true`

if all the values of `list`

are different, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.

**Variants**

`:all_different`

: Global constraint ensuring that the values in`x`

are all different.

```
concept(:all_different, x; vals)
concept(:all_different)(x; vals)
```

**Examples**

```
c = concept(:all_different)
c([1, 2, 3, 4])
c([1, 2, 3, 1])
c([1, 0, 0, 4]; vals=[0])
c([1, 0, 0, 1]; vals=[0])
```

`Constraints.xcsp_all_equal`

— Method`xcsp_all_equal(list::Vector{Int}, val::Int)`

Return `true`

if all the values of `list`

are equal to `val`

, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`val::Int`

: value to compare to.

**Variants**

`:all_equal`

: Global constraint ensuring that the values in`x`

are all equal.

```
concept(:all_equal, x; val=nothing, pair_vars=zeros(x), op=+)
concept(:all_equal)(x; val=nothing, pair_vars=zeros(x), op=+)
```

**Examples**

```
c = concept(:all_equal)
c([0, 0, 0, 0])
c([1, 2, 3, 4])
c([3, 2, 1, 0]; pair_vars=[0, 1, 2, 3])
c([0, 1, 2, 3]; pair_vars=[0, 1, 2, 3])
c([1, 2, 3, 4]; op=/, val=1, pair_vars=[1, 2, 3, 4])
c([1, 2, 3, 4]; op=*, val=1, pair_vars=[1, 2, 3, 4])
```

`Constraints.xcsp_cardinality`

— Method`xcsp_cardinality(list, values, occurs, closed)`

Return `true`

if the number of occurrences of the values in `values`

in `list`

satisfies the given condition, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`values::Vector{Int}`

: list of values to check.`occurs::Vector{Int}`

: list of occurrences to check.`closed::Bool`

: whether the constraint is closed or not.

**Variants**

`:cardinality`

: Global constraint that restricts the number of times specific values in a list`values`

can appear in`x`

.

```
concept(:cardinality, x; bool=false, vals)
concept(:cardinality)(x; bool=false, vals)
```

`:cardinality_closed`

: Global constraint that restricts the number of times in a list`values`

can appear in`x`

. It is closed, meaning that the variables in`x`

cannot have values outside the ones in`list`

.

```
concept(:cardinality_closed, x; vals)
concept(:cardinality_closed)(x; vals)
```

`:cardinality_open`

: Global constraint that restricts the number of times in a list`values`

can appear in`x`

. It is open, meaning that the variables in`x`

can have values outside the ones in`list`

.

```
concept(:cardinality_open, x; vals)
concept(:cardinality_open)(x; vals)
```

**Examples**

```
c = concept(:cardinality)
c([2, 5, 10, 10]; vals=[2 0 1; 5 1 3; 10 2 3])
c([8, 5, 10, 10]; vals=[2 0 1; 5 1 3; 10 2 3], bool=false)
c([8, 5, 10, 10]; vals=[2 0 1; 5 1 3; 10 2 3], bool=true)
c([2, 5, 10, 10]; vals=[2 1; 5 1; 10 2])
c([2, 5, 10, 10]; vals=[2 0 1 42; 5 1 3 7; 10 2 3 -4])
c([2, 5, 5, 10]; vals=[2 0 1; 5 1 3; 10 2 3])
c([2, 5, 10, 8]; vals=[2 1; 5 1; 10 2])
c([5, 5, 5, 10]; vals=[2 0 1 42; 5 1 3 7; 10 2 3 -4])
cc = concept(:cardinality_closed)
cc([8, 5, 10, 10]; vals=[2 0 1; 5 1 3; 10 2 3])
co = concept(:cardinality_open)
co([8, 5, 10, 10]; vals=[2 0 1; 5 1 3; 10 2 3])
```

`Constraints.xcsp_channel`

— Method`xcsp_channel(; list)`

Return `true`

if the channel constraint is satisfied, `false`

otherwise. The channel constraint ensures that if the i-th element of `list`

is assigned the value j, then the j-th element of `list`

must be assigned the value i.

**Arguments**

`list::Union{AbstractVector, Tuple}`

: list of values to check.

**Variants**

`:channel`

: Ensures that if the i-th element of`x`

is assigned the value j, then the j-th element of`x`

must be assigned the value i.

```
concept(:channel, x; dim=1, id=nothing)
concept(:channel)(x; dim=1, id=nothing)
```

**Examples**

```
c = concept(:channel)
c([2, 1, 4, 3])
c([1, 2, 3, 4])
c([2, 3, 1, 4])
c([2, 1, 5, 3, 4, 2, 1, 4, 5, 3]; dim=2)
c([2, 1, 4, 3, 5, 2, 1, 4, 5, 3]; dim=2)
c([false, false, true, false]; id=3)
c([false, false, true, false]; id=1)
```

`Constraints.xcsp_circuit`

— Method`xcsp_circuit(; list, size)`

Return `true`

if the circuit constraint is satisfied, `false`

otherwise. The circuit constraint is a global constraint ensuring that the values of `x`

form a circuit, i.e., a sequence where each value is the index of the next value in the sequence, and the sequence eventually loops back to the start.

**Arguments**

`list::AbstractVector`

: list of values to check.`size::Int`

: size of the circuit.

**Variants**

`:circuit`

: Global constraint ensuring that the values of`x`

form a circuit, i.e., a sequence where each value is the index of the next value in the sequence, and the sequence eventually loops back to the start. Often used for routing problems.

```
concept(:circuit, x; op, val)
concept(:circuit)(x; op, val)
```

**Examples**

```
c = concept(:circuit)
c([1, 2, 3, 4])
c([2, 3, 4, 1])
c([2, 3, 1, 4]; op = ==, val = 3)
c([4, 3, 1, 3]; op = >, val = 0)
```

`Constraints.xcsp_count`

— Method`xcsp_count(list, values, condition)`

Return `true`

if the number of occurrences of the values in `values`

in `list`

satisfies the given condition, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`values::Vector{Int}`

: list of values to check.`condition`

: condition to satisfy.

**Variants**

`:count`

: Constraint ensuring that the number of occurrences of the values in`vals`

in`x`

satisfies the given condition.

```
concept(:count, x; vals, op, val)
concept(:count)(x; vals, op, val)
```

`:at_least`

: Constraint ensuring that the number of occurrences of the values in`vals`

in`x`

is at least`val`

.

```
concept(:at_least, x; vals, val)
concept(:at_least)(x; vals, val)
```

`:at_most`

: Constraint ensuring that the number of occurrences of the values in`vals`

in`x`

is at most`val`

.

```
concept(:at_most, x; vals, val)
concept(:at_most)(x; vals, val)
```

`:exactly`

: Constraint ensuring that the number of occurrences of the values in`vals`

in`x`

is exactly`val`

.

```
concept(:exactly, x; vals, val)
concept(:exactly)(x; vals, val)
```

**Examples**

```
c = concept(:count)
c([2, 1, 4, 3]; vals=[1, 2, 3, 4], op=≥, val=2)
c([1, 2, 3, 4]; vals=[1, 2], op==, val=2)
c([2, 1, 4, 3]; vals=[1, 2], op=≤, val=1)
```

`Constraints.xcsp_cumulative`

— Method`xcsp_cumulative(; origins, lengths, heights, condition)`

Return `true`

if the cumulative constraint is satisfied, `false`

otherwise. The cumulative constraint is a global constraint operating on a set of tasks, defined by `origin`

(starting times), `length`

, and `height`

. This constraint ensures that, at each point in time, the sum of the `height`

of tasks that overlap that point, respects a numerical condition.

**Arguments**

`origins::AbstractVector`

: list of origins of the tasks.`lengths::AbstractVector`

: list of lengths of the tasks.`heights::AbstractVector`

: list of heights of the tasks.`condition::Tuple`

: condition to check.

**Variants**

`:cumulative`

: Global constraint operating on a set of tasks, defined by`origin`

(starting times),`length`

, and`height`

. This constraint ensures that, at each point in time, the sum of the`height`

of tasks that overlap that point, respects a numerical condition.

```
concept(:cumulative, x; pair_vars, op, val)
concept(:cumulative)(x; pair_vars, op, val)
```

**Examples**

```
c = concept(:cumulative)
c([1, 2, 3, 4, 5]; val = 1)
c([1, 2, 2, 4, 5]; val = 1)
c([1, 2, 3, 4, 5]; pair_vars = [3 2 5 4 2; 1 2 1 1 3], op = ≤, val = 5)
c([1, 2, 3, 4, 5]; pair_vars = [3 2 5 4 2; 1 2 1 1 3], op = <, val = 5)
```

`Constraints.xcsp_element`

— Method`xcsp_element(; list, index, condition)`

Return `true`

if the element constraint is satisfied, `false`

otherwise. The element constraint is a global constraint specifying that a variable in `x`

indexed by `id`

should be equal to a `value`

.

**Arguments**

`list::Union{AbstractVector, Tuple}`

: list of values to check.`index::Int`

: index of the value to check.`condition::Tuple`

: condition to check.

**Variants**

`:element`

: Global constraint specifying that a variable in`x`

indexed by`id`

should be equal to a`value`

.

```
concept(:element, x; id=nothing, op===, val=nothing)
concept(:element)(x; id=nothing, op===, val=nothing)
```

**Examples**

```
c = concept(:element)
c([1, 2, 3, 4, 5]; id=1, val=1)
c([1, 2, 3, 4, 5]; id=1, val=2)
c([1, 2, 3, 4, 2])
c([1, 2, 3, 4, 1])
```

`Constraints.xcsp_extension`

— Method`xcsp_extension(; list, supports=nothing, conflicts=nothing)`

Global constraint enforcing that `x`

matches a configuration within the supports set `pair_vars[1]`

or does not match any configuration within the conflicts set `pair_vars[2]`

. It embodies the logic: `x ∈ pair_vars[1] || x ∉ pair_vars[2]`

, providing a comprehensive way to define valid (supported) and invalid (conflicted) configurations for constraint satisfaction problems.

**Arguments**

`list::Vector{Int}`

: A list of variables`supports::Vector{Vector{Int}}`

: A set of supported tuples. Default to nothing.`conflicts::Vector{Vector{Int}}`

: A set of conflicted tuples. Default to nothing.

**Variants**

`:extension`

: Global constraint enforcing that`x`

matches a configuration within the supports set`pair_vars[1]`

or does not match any configuration within the conflicts set`pair_vars[2]`

. It embodies the logic:`x ∈ pair_vars[1] || x ∉ pair_vars[2]`

, providing a comprehensive way to define valid (supported) and invalid (conflicted) configurations for constraint satisfaction problems.

```
concept(:extension, x; pair_vars)
concept(:extension)(x; pair_vars)
```

`:supports`

: Global constraint ensuring that`x`

matches a configuration listed within the support set`pair_vars`

. This constraint is derived from the extension model, specifying that`x`

must be one of the explicitly defined supported configurations:`x ∈ pair_vars`

. It is utilized to directly declare the configurations that are valid and should be included in the solution space.

```
concept(:supports, x; pair_vars)
concept(:supports)(x; pair_vars)
```

`:conflicts`

: Global constraint ensuring that`x`

does not match any configuration listed within the conflict set`pair_vars`

. This constraint, originating from the extension model, stipulates that`x`

must avoid all configurations defined as conflicts:`x ∉ pair_vars`

. It is useful for specifying configurations that are explicitly forbidden and should be excluded from the solution space.

```
concept(:conflicts, x; pair_vars)
concept(:conflicts)(x; pair_vars)
```

**Examples**

```
c = concept(:extension)
c([1, 2, 3, 4, 5]; pair_vars=[[1, 2, 3, 4, 5]])
c([1, 2, 3, 4, 5]; pair_vars=([[1, 2, 3, 4, 5]], [[1, 2, 1, 4, 5], [1, 2, 3, 5, 5]]))
c([1, 2, 3, 4, 5]; pair_vars=[[1, 2, 1, 4, 5], [1, 2, 3, 5, 5]])
c = concept(:supports)
c([1, 2, 3, 4, 5]; pair_vars=[[1, 2, 3, 4, 5]])
c = concept(:conflicts)
c([1, 2, 3, 4, 5]; pair_vars=[[1, 2, 1, 4, 5], [1, 2, 3, 5, 5]])
```

`Constraints.xcsp_instantiation`

— Method`xcsp_instantiation(; list, values)`

Return `true`

if the instantiation constraint is satisfied, `false`

otherwise. The instantiation constraint is a global constraint ensuring that `x`

takes on a specific set of `values`

in a specific order.

**Arguments**

`list::AbstractVector`

: list of values to check.`values::AbstractVector`

: list of values to check against.

**Variants**

`:instantiation`

: Global constraint ensuring that`x`

takes on a specific set of`values`

in a specific order.

```
concept(:instantiation, x; pair_vars)
concept(:instantiation)(x; pair_vars)
```

**Examples**

```
c = concept(:instantiation)
c([1, 2, 3, 4, 5]; pair_vars=[1, 2, 3, 4, 5])
c([1, 2, 3, 4, 5]; pair_vars=[1, 2, 3, 4, 6])
```

`Constraints.xcsp_intension`

— Method`xcsp_intension(list, predicate)`

An intensional constraint is usually defined from a `predicate`

over `list`

. As such it encompass any generic constraint.

**Arguments**

`list::Vector{Int}`

: A list of variables`predicate::Function`

: A predicate over`list`

**Variants**

`:dist_different`

: Given a 4-dimensional vector`x`

, ensures that the absolute difference between`x[1]`

and`x[2]`

, and between`x[3]`

and`x[4]`

, are different. This constraint is fundamental in ensuring the validity of a Golomb ruler, where no two pairs of marks should have the same distance between them.

```
concept(:dist_different, x)
concept(:dist_different)(x)
```

**Examples**

`2 + 2`

`2 + 2`

```
using Constraints # hide
c = concept(:dist_different)
c([1, 2, 3, 3]) && !c([1, 2, 3, 4])
```

```
using Constraints # hide
c = concept(:dist_different)
c([1, 2, 3, 3]) && !c([1, 2, 3, 4])
```

`Constraints.xcsp_maximum`

— Method`xcsp_maximum(; list, condition)`

Return `true`

if the maximum constraint is satisfied, `false`

otherwise. The maximum constraint is a global constraint specifying that a certain condition should hold for the maximum value in a list of variables.

**Arguments**

`list::Union{AbstractVector, Tuple}`

: list of values to check.`condition::Tuple`

: condition to check.

**Variants**

`:maximum`

: Global constraint ensuring that a certain numerical condition holds for the maximum value in`x`

.

```
concept(:maximum, x; op, val)
concept(:maximum)(x; op, val)
```

**Examples**

```
c = concept(:maximum)
c([1, 2, 3, 4, 5]; op = ==, val = 5)
c([1, 2, 3, 4, 5]; op = ==, val = 6)
```

`Constraints.xcsp_mdd`

— Method`xcsp_mdd(; list, diagram)`

Return a function that checks if the list of values `list`

satisfies the MDD `diagram`

.

**Arguments**

`list::Vector{Int}`

: list of values to check.`diagram::MDD`

: MDD to check.

**Variants**

`:mdd`

: Multi-valued Decision Diagram (MDD) constraint.

The MDD constraint is a constraint that can be used to model a wide range of problems. It is a directed graph where each node is labeled with a value and each edge is labeled with a value. The constraint is satisfied if there is a path from the first node to the last node such that the sequence of edge labels is a valid sequence of the value labels.

```
concept(:mdd, x; language)
concept(:mdd)(x; language)
```

**Examples**

```
c = concept(:mdd)
states = [
Dict( # level x1
(:r, 0) => :n1,
(:r, 1) => :n2,
(:r, 2) => :n3,
),
Dict( # level x2
(:n1, 2) => :n4,
(:n2, 2) => :n4,
(:n3, 0) => :n5,
),
Dict( # level x3
(:n4, 0) => :t,
(:n5, 0) => :t,
),
]
a = MDD(states)
c([0,2,0]; language = a)
c([1,2,0]; language = a)
c([2,0,0]; language = a)
c([2,1,2]; language = a)
c([1,0,2]; language = a)
c([0,1,2]; language = a)
```

`Constraints.xcsp_minimum`

— Method`xcsp_minimum(; list, condition)`

Return `true`

if the minimum constraint is satisfied, `false`

otherwise. The minimum constraint is a global constraint specifying that a certain condition should hold for the minimum value in a list of variables.

**Arguments**

`list::Union{AbstractVector, Tuple}`

: list of values to check.`condition::Tuple`

: condition to check.

**Variants**

`:minimum`

: Global constraint ensuring that a certain numerical condition holds for the minimum value in`x`

.

```
concept(:minimum, x; op, val)
concept(:minimum)(x; op, val)
```

**Examples**

```
c = concept(:minimum)
c([1, 2, 3, 4, 5]; op = ==, val = 1)
c([1, 2, 3, 4, 5]; op = ==, val = 0)
```

`Constraints.xcsp_no_overlap`

— Method`xcsp_no_overlap(; origins, lengths, zero_ignored)`

Return `true`

if the no*overlap constraint is satisfied, false otherwise. The no*overlap constraint is a global constraint used in constraint programming, often in scheduling problems. It ensures that tasks do not overlap in time, i.e., for any two tasks, either the first task finishes before the second task starts, or the second task finishes before the first task starts.

**Arguments**

`origins::AbstractVector`

: list of origins of the tasks.`lengths::AbstractVector`

: list of lengths of the tasks.`zero_ignored::Bool`

: whether to ignore zero-length tasks.

**Variants**

`:no_overlap`

: Global constraint operating on a set of tasks, defined by`origin`

(starting times), and`length`

. This constraint ensures that tasks do not overlap in time, i.e., for any two tasks, either the first task finishes before the second task starts, or the second task finishes before the first task starts. Often used in scheduling problems.

```
concept(:no_overlap, x; pair_vars, bool)
concept(:no_overlap)(x; pair_vars, bool)
```

`:no_overlap_no_zero`

: Global constraint operating on a set of tasks, defined by`origin`

(starting times), and`length`

. This constraint ensures that tasks do not overlap in time, i.e., for any two tasks, either the first task finishes before the second task starts, or the second task finishes before the first task starts. This variant ignores zero-length tasks. Often used in scheduling problems.

```
concept(:no_overlap_no_zero, x; pair_vars)
concept(:no_overlap_no_zero)(x; pair_vars)
```

`:no_overlap_with_zero`

: Global constraint operating on a set of tasks, defined by`origin`

(starting times), and`length`

. This constraint ensures that tasks do not overlap in time, i.e., for any two tasks, either the first task finishes before the second task starts, or the second task finishes before the first task starts. This variant includes zero-length tasks. Often used in scheduling problems.

```
concept(:no_overlap_with_zero, x; pair_vars)
concept(:no_overlap_with_zero)(x; pair_vars)
```

**Examples**

```
c = concept(:no_overlap)
c([1, 2, 3, 4, 5])
c([1, 2, 3, 4, 1])
c([1, 2, 4, 6, 3]; pair_vars = [1, 1, 1, 1, 1])
c([1, 2, 4, 6, 3]; pair_vars = [1, 1, 1, 3, 1])
c([1, 2, 4, 6, 3]; pair_vars = [1, 1, 3, 1, 1])
c([1, 1, 1, 3, 5, 2, 7, 7, 5, 12, 8, 7]; pair_vars = [2, 4, 1, 4 ,2 ,3, 5, 1, 2, 3, 3, 2], dim = 3)
c([1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4]; pair_vars = [2, 4, 1, 4 ,2 ,3, 5, 1, 2, 3, 3, 2], dim = 3)
```

`Constraints.xcsp_nvalues`

— Method`xcsp_nvalues(list, condition, except)`

Return `true`

if the number of distinct values in `list`

satisfies the given condition, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`condition`

: condition to satisfy.`except::Union{Nothing, Vector{Int}}`

: list of values to exclude. Default is`nothing`

.

**Variants**

`:nvalues`

: Ensures that the number of distinct values in`x`

satisfies a given numerical condition.

The constraint is defined by the following expression: `nValues(x, op, val)`

where `x`

is a list of variables, `op`

is a comparison operator, and `val`

is an integer value.

```
concept(:nvalues, x; op, val)
concept(:nvalues)(x; op, val)
```

**Examples**

```
c = concept(:nvalues)
c([1, 2, 3, 4, 5]; op = ==, val = 5)
c([1, 2, 3, 4, 5]; op = ==, val = 2)
c([1, 2, 3, 4, 3]; op = <=, val = 5)
c([1, 2, 3, 4, 3]; op = <=, val = 3)
```

`Constraints.xcsp_ordered`

— Method`xcsp_ordered(list::Vector{Int}, operator, lengths)`

Return `true`

if all the values of `list`

are in an increasing order, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`operator`

: comparison operator to use.`lengths`

: list of lengths to use. Defaults to`nothing`

.

**Variants**

`:ordered`

: Global constraint ensuring that all the values of`x`

are in a total order defined by a comparison operator.

```
concept(:ordered, x; op=≤, pair_vars=nothing)
concept(:ordered)(x; op=≤, pair_vars=nothing)
```

`:increasing`

: Global constraint ensuring that all the values of`x`

are in an increasing order.

```
concept(:increasing, x; op=≤, pair_vars=nothing)
concept(:increasing)(x; op=≤, pair_vars=nothing)
```

`:decreasing`

: Global constraint ensuring that all the values of`x`

are in a decreasing order.

```
concept(:decreasing, x; op=≥, pair_vars=nothing)
concept(:decreasing)(x; op=≥, pair_vars=nothing)
```

`:strictly_increasing`

: Global constraint ensuring that all the values of`x`

are in a strictly increasing order.

```
concept(:strictly_increasing, x; op=<, pair_vars=nothing)
concept(:strictly_increasing)(x; op=<, pair_vars=nothing)
```

`:strictly_decreasing`

: Global constraint ensuring that all the values of`x`

are in a strictly decreasing order.

```
concept(:strictly_decreasing, x; op=>, pair_vars=nothing)
concept(:strictly_decreasing)(x; op=>, pair_vars=nothing)
```

**Examples**

```
c = concept(:ordered)
c([1, 2, 3, 4, 4]; op=≤)
c([1, 2, 3, 4, 5]; op=<)
!c([1, 2, 3, 4, 3]; op=≤)
!c([1, 2, 3, 4, 3]; op=<)
```

`Constraints.xcsp_regular`

— Method`xcsp_regular(; list, automaton)`

Ensures that a sequence `x`

(interpreted as a word) is accepted by the regular language represented by a given automaton. This constraint verifies the compliance of `x`

with the language rules encoded within the `automaton`

parameter, which must be an instance of `<:AbstractAutomaton`

.

**Arguments**

`list::Vector{Int}`

: A list of variables`automaton<:AbstractAutomaton`

: An automaton representing the regular language

**Variants**

`:regular`

: Ensures that a sequence`x`

(interpreted as a word) is accepted by the regular language represented by a given automaton. This constraint verifies the compliance of`x`

with the language rules encoded within the`automaton`

parameter, which must be an instance of`<:AbstractAutomaton`

.

```
concept(:regular, x; language)
concept(:regular)(x; language)
```

**Examples**

```
c = concept(:regular)
states = Dict(
(:a, 0) => :a,
(:a, 1) => :b,
(:b, 1) => :c,
(:c, 0) => :d,
(:d, 0) => :d,
(:d, 1) => :e,
(:e, 0) => :e,
)
start = :a
finish = :e
a = Automaton(states, start, finish)
c([0,0,1,1,0,0,1,0,0]; language = a)
c([1,1,1,0,1]; language = a)
```

`Constraints.xcsp_sum`

— Method`xcsp_sum(list, coeffs, condition)`

Return `true`

if the sum of the variables in `list`

satisfies the given condition, `false`

otherwise.

**Arguments**

`list::Vector{Int}`

: list of values to check.`coeffs::Vector{Int}`

: list of coefficients to use.`condition`

: condition to satisfy.

**Variants**

`:sum`

: Global constraint ensuring that the sum of the variables in`x`

satisfies a given numerical condition.

```
concept(:sum, x; op===, pair_vars=ones(x), val)
concept(:sum)(x; op===, pair_vars=ones(x), val)
```

**Examples**

```
c = concept(:sum)
c([1, 2, 3, 4, 5]; op===, val=15)
c([1, 2, 3, 4, 5]; op===, val=2)
c([1, 2, 3, 4, 3]; op=≤, val=15)
c([1, 2, 3, 4, 3]; op=≤, val=3)
```

`Constraints.@usual`

— Macro`usual(ex::Expr)`

This macro is used to define a new constraint or update an existing one in the USUAL_CONSTRAINTS dictionary. It takes an expression ex as input, which represents the definition of a constraint.

Here's a step-by-step explanation of what the macro does:

- It first extracts the symbol of the concept from the input expression. This symbol is expected to be the first argument of the first argument of the expression. For example, if the expression is @usual all
*different(x; y=1), the symbol would be :all*different. - It then calls the shrink_concept function on the symbol to get a simplified version of the concept symbol.
- It initializes a dictionary defaults to store whether each keyword argument of the concept has a default value or not.
- It checks if the expression has more than two arguments. If it does, it means that there are keyword arguments present. It then loops over these keyword arguments. If a keyword argument is a symbol, it means it doesn't have a default value, so it adds an entry to the defaults dictionary with the keyword argument as the key and false as the value. If a keyword argument is not a symbol, it means it has a default value, so it adds an entry to the defaults dictionary with the keyword argument as the key and true as the value.
- It calls the make_error function on the simplified concept symbol to generate an error function for the constraint.
- It evaluates the input expression to get the concept function.
- It checks if the USUAL
*CONSTRAINTS dictionary already contains an entry for the simplified concept symbol. If it does, it adds the defaults dictionary to the parameters of the existing constraint. If it doesn't, it creates a new constraint with the concept function, a description, the error function, and the defaults dictionary as the parameters, and adds it to the USUAL*CONSTRAINTS dictionary.

This macro is used to make it easier to define and update constraints in a consistent and possibly automated way.

**Arguments**

`ex::Expr`

: expression to parse.

**Example**

`@usual concept_all_different(x; vals=nothing) = xcsp_all_different(list=x, except=vals)`