Boolean and reified constraints

### 8.7 Boolean and reified constraints

- Boolean FD expressions
- `(#``\``)/1` - constraint NOT, `(#``<``=``>``)/2` - constraint equivalent,
  `(#``\``<``=``>``)/2` - constraint different, `(##)/2` - constraint XOR,
  `(#==``>``)/2` - constraint imply, `(#``\``==``>``)/2` - constraint not imply,
  `(#/``\``)/2` - constraint AND, `(#``\``/``\``)/2` - constraint NAND,
  `(#``\``/)/2` - constraint OR, `(#``\``\``/)/2` - constraint NOR
- `fd_cardinality/2`, `fd_cardinality/3`, `fd_at_least_one/1`, `fd_at_most_one/1`,
  `fd_only_one/1`

#### 8.7.1 Boolean FD expressions

An boolean FD expression is a Prolog term built from integers (0 for false, 1 for true), variables (Prolog or FD variables), partial AC arithmetic constraints (section 8.6.2), full AC arithmetic constraints (section 8.6.3) and functors (or operators) that represent boolean functions. When a sub-expression of a boolean expression is an arithmetic constraint `c`, it is reified. Namely, as soon as the solver detects that `c` is true (i.e. entailed) the sub-expression has the value `1`. Similarly when the solver detects that `c` is false (i.e. disentailed) the sub-expression evaluates as `0`. While neither the entailment nor the disentailment can be detected the sub-expression is evaluated as a domain `0..1`. The following table details the components of an FD boolean expression:

FD Expression | Result
Prolog variable | domain `0..1`
FD variable `X` | domain of `X`, `X` is constrained to be in `0..1`
`0` (integer) | `0` (false)
`1` (integer) | `1` (true)
`#\ E` | not `E`
`E1 #<=> E2` | `E1` equivalent to `E2`
`E1 #\<=> E2` | `E1` not equivalent to `E2` (i.e. `E1` different from `E2)`
`E1 ## E2` | `E1` exclusive OR `E2` (i.e. `E1` not equivalent to `E2)`
`E1 #==> E2` | `E1` implies `E2`
`E1 #\==> E2` | `E1` does not imply `E2`
`E1 #/\ E2` | `E1` AND `E2`
`E1 #\/\ E2` | `E1` NAND `E2`
`E1 #\/ E2` | `E1` OR `E2`
`E1 #\\/ E2` | `E1` NOR `E2`

`#<=>`, `#\<=>`, `##`, `#==>`, `#\==>`, `#/\`, `#\/\`, `#\/` and `#\\/` are predefined infix operators. `#\` is a predefined prefix operator (section 7.14.10).

Errors

a sub-expression `E` is neither a variable nor an integer (0 or 1) nor an FD boolean functor nor reified constraint |  | `type_error(fd_bool_evaluable, E)`
an expression is too complex |  | `resource_error(too_big_fd_constraint)`
a sub-expression is an invalid reified constraint |  | an arithmetic constraint error (section 8.6.1)

#### 8.7.2 `(#\)/1` - constraint NOT, `(#<=>)/2` - constraint equivalent,
`(#\<=>)/2` - constraint different, `(##)/2` - constraint XOR,
`(#==>)/2` - constraint imply, `(#\==>)/2` - constraint not imply,
`(#/\)/2` - constraint AND, `(#\/\)/2` - constraint NAND,
`(#\/)/2` - constraint OR, `(#\\/)/2` - constraint NOR

Templates

  `#\(?fd_bool_evaluable)
  #<=>(?fd_bool_evaluable, ?fd_bool_evaluable)
  #\<=>(?fd_bool_evaluable, ?fd_bool_evaluable)
  ##(?fd_bool_evaluable, ?fd_bool_evaluable)
  #==>(?fd_bool_evaluable, ?fd_bool_evaluable)
  #\==>(?fd_bool_evaluable, ?fd_bool_evaluable)
  #/\(?fd_bool_evaluable, ?fd_bool_evaluable)
  #\/\(?fd_bool_evaluable, ?fd_bool_evaluable)
  #\/(?fd_bool_evaluable, ?fd_bool_evaluable)
  #\\/(?fd_bool_evaluable, ?fd_bool_evaluable)`

Description

`#\= FdBoolExpr1` constraints `FdBoolExpr1` to be false.

`FdBoolExpr1 #<=> FdBoolExpr2` constrains `FdBoolExpr1` to be equivalent to `FdBoolExpr2`.

`FdBoolExpr1 #\<=> FdBoolExpr2` constrains `FdBoolExpr1` to be equivalent to not `FdBoolExpr2`.

`FdBoolExpr1 ## FdBoolExpr2` constrains `FdBoolExpr1` XOR `FdBoolExpr2` to be true

`FdBoolExpr1 #==> FdBoolExpr2` constrains `FdBoolExpr1` to imply `FdBoolExpr2`.

`FdBoolExpr1 #\==> FdBoolExpr2` constrains `FdBoolExpr1` to not imply `FdBoolExpr2`.

`FdBoolExpr1 #/\ FdBoolExpr2` constrains `FdBoolExpr1` AND `FdBoolExpr2` to be true.

`FdBoolExpr1 #\/\ FdBoolExpr2` constrains `FdBoolExpr1` AND `FdBoolExpr2` to be false.

`FdBoolExpr1 #\/ FdBoolExpr2` constrains `FdBoolExpr1` OR `FdBoolExpr2` to be true.

`FdBoolExpr1 #\\/ FdBoolExpr2` constrains `FdBoolExpr1` OR `FdBoolExpr2` to be false.

`FdBoolExpr1` and `FdBoolExpr2` are boolean FD expressions (section 8.7.1).

Note that `#\<=>` (not equivalent) and `##` (exclusive or) are synonymous.

These predicates post boolean constraints that are managed by the FD solver using a partial arc-consistency algorithm to reduce the domain of involved variables. The (dis)entailment of reified constraints is detected using either the bounds (for partial AC arithmetic constraints) or the full domain (for full AC arithmetic constraints).

`#<=>`, `#\<=>`, `##`, `#==>`, `#\==>`, `#/\`, `#\/\`, `#\/` and `#\\/` are predefined infix operators. `#\` is a predefined prefix operator (section 7.14.10).

Errors

Refer to the syntax of boolean FD expressions for possible errors (section 8.7.1).

Portability

GNU Prolog predicates.

#### 8.7.3 `fd_cardinality/2`, `fd_cardinality/3`, `fd_at_least_one/1`, `fd_at_most_one/1`,
`fd_only_one/1`

Templates

  `fd_cardinality(+fd_bool_evaluable_list, ?fd_variable)
  fd_cardinality(+integer, ?fd_variable, +integer)
  fd_at_least_one(+fd_bool_evaluable_list)
  fd_at_most_one(+fd_bool_evaluable_list)
  fd_only_one(+fd_bool_evaluable_list)`

Description

`fd_cardinality(List, Count)` unifies `Count` with the number of constraints that are true in `List`. This is equivalent to post the constraint `B1 + B2 + …+ Bn #= Count` where each variable `Bi` is a new variable defined by the constraint `Bi #<=> Ci` where `Ci` is the `i`th constraint of `List`. Each `Ci` must be a boolean FD expression (section 8.7.1).

`fd_cardinality(Lower, List, Upper)` is equivalent to `fd_cardinality(List, Count), Lower #=< Count, Count #=< Upper`

`fd_at_least_one(List)` is equivalent to `fd_cardinality(List, Count), Count #>= 1`.

`fd_at_most_one(List)` is equivalent to `fd_cardinality(List, Count), Count #=< 1`.

`fd_only_one(List)` is equivalent to `fd_cardinality(List, 1)`.

Errors

`List` is a partial list |  | `instantiation_error`
`List` is neither a partial list nor a list |  | `type_error(list, List)`
`Count` is neither an FD variable nor an integer |  | `type_error(fd_variable, Count)`
`Lower` is a variable |  | `instantiation_error`
`Lower` is neither a variable nor an integer |  | `type_error(integer, Lower)`
`Upper` is a variable |  | `instantiation_error`
`Upper` is neither a variable nor an integer |  | `type_error(integer, Upper)`
an element `E` of the `List` list is an invalid boolean expression |  | an FD boolean constraint (section 8.7.1)

Portability

GNU Prolog predicates.

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