Arithmetic constraints

9.6 Arithmetic constraints

FD arithmetic expressions
Partial AC: (#=)/2 - constraint equal, (#\=)/2 - constraint not equal,
(#<)/2 - constraint less than, (#=<)/2 - constraint less than or equal,
(#>)/2 - constraint greater than, (#>=)/2 - constraint greater than or equal
Full AC: (#=#)/2 - constraint equal, (#\=#)/2 - constraint not equal,
(#<#)/2 - constraint less than, (#=<#)/2 - constraint less than or equal,
(#>#)/2 - constraint greater than, (#>=#)/2 - constraint greater than or equal
fd_prime/1, fd_not_prime/1

9.6.1 FD arithmetic expressions

An FD arithmetic expression is a Prolog term built from integers, variables (Prolog or FD variables), and functors (or operators) that represent arithmetic functions. The following table details the components of an FD arithmetic expression:

FD Expression

Result

Prolog variable

domain 0..fd_max_integer

FD variable X

domain of X

integer number N

domain N..N

+ E

same as E

- E

opposite of E

E1 + E2

sum of E1 and E2

E1 - E2

subtraction of E2 from E1

E1 * E2

multiplication of E1 by E2

E1 / E2

integer division of E1 by E2 (only succeeds if the remainder is 0)

E1 ** E2

E1 raised to the power of E2 (E1 or E2 must be an integer)

min(E1,E2)

minimum of E1 and E2

max(E1,E2)

maximum of E1 and E2

dist(E1,E2)

distance, i.e. |E1 - E2|

E1 // E2

quotient of the integer division of E1 by E2

E1 rem E2

remainder of the integer division of E1 by E2

quot_rem(E1,E2,R)

quotient of the integer division of E1 by E2
(R is the remainder of the integer division of E1 by E2)

FD expressions are not restricted to be linear. However non-linear constraints usually yield less constraint propagation than linear constraints.

+, -, *, /, //, rem and ** are predefined infix operators. + and - are predefined prefix operators (section 8.14.10).

Errors

a sub-expression is of the form _ ** E and E is a variable instantiation_error

a sub-expression E is neither a variable nor an integer nor an FD arithmetic functor type_error(fd_evaluable, E)

an expression is too complex resource_error(too_big_fd_constraint)

9.6.2 Partial AC: (#=)/2 - constraint equal, (#\=)/2 - constraint not equal,
(#<)/2 - constraint less than, (#=<)/2 - constraint less than or equal,
(#>)/2 - constraint greater than, (#>=)/2 - constraint greater than or equal

Templates

#=(?fd_evaluable, ?fd_evaluable)
#\=(?fd_evaluable, ?fd_evaluable)
#<(?fd_evaluable, ?fd_evaluable)
#=<(?fd_evaluable, ?fd_evaluable)
#>(?fd_evaluable, ?fd_evaluable)
#>=(?fd_evaluable, ?fd_evaluable)

Description

FdExpr1 #= FdExpr2 constrains FdExpr1 to be equal to FdExpr2.

FdExpr1 #\= FdExpr2 constrains FdExpr1 to be different from FdExpr2.

FdExpr1 #< FdExpr2 constrains FdExpr1 to be less than FdExpr2.

FdExpr1 #=< FdExpr2 constrains FdExpr1 to be less than or equal to FdExpr2.

FdExpr1 #> FdExpr2 constrains FdExpr1 to be greater than FdExpr2.

FdExpr1 #>= FdExpr2 constrains FdExpr1 to be greater than or equal to FdExpr2.

FdExpr1 and FdExpr2 are arithmetic FD expressions (section 9.6.1).

#=, #\=, #<, #=<, #> and #>= are predefined infix operators (section 8.14.10).

These predicates post arithmetic constraints that are managed by the solver using a partial arc-consistency algorithm to reduce the domain of involved variables. In this scheme only the bounds of the domain of variables are updated. This leads to less propagation than full arc-consistency techniques (section 9.6.3) but is generally more efficient for arithmetic. These arithmetic constraints can be reified (section 9.7).

Errors

Refer to the syntax of arithmetic FD expressions for possible errors (section 9.6.1).

Portability

GNU Prolog predicates.

9.6.3 Full AC: (#=#)/2 - constraint equal, (#\=#)/2 - constraint not equal,
(#<#)/2 - constraint less than, (#=<#)/2 - constraint less than or equal,
(#>#)/2 - constraint greater than, (#>=#)/2 - constraint greater than or equal

Templates

#=#(?fd_evaluable, ?fd_evaluable)
#\=#(?fd_evaluable, ?fd_evaluable)
#<#(?fd_evaluable, ?fd_evaluable)
#=<#(?fd_evaluable, ?fd_evaluable)
#>#(?fd_evaluable, ?fd_evaluable)
#>=#(?fd_evaluable, ?fd_evaluable)

Description

FdExpr1 #=# FdExpr2 constrains FdExpr1 to be equal to FdExpr2.

FdExpr1 #\=# FdExpr2 constrains FdExpr1 to be different from FdExpr2.

FdExpr1 #<# FdExpr2 constrains FdExpr1 to be less than FdExpr2.

FdExpr1 #=<# FdExpr2 constrains FdExpr1 to be less than or equal to FdExpr2.

FdExpr1 #># FdExpr2 constrains FdExpr1 to be greater than FdExpr2.

FdExpr1 #>=# FdExpr2 constrains FdExpr1 to be greater than or equal to FdExpr2.

FdExpr1 and FdExpr2 are arithmetic FD expressions (section 9.6.1).

#=#, #\=#, #<#, #=<#, #># and #>=# are predefined infix operators (section 8.14.10).

These predicates post arithmetic constraints that are managed by the solver using a full arc-consistency algorithm to reduce the domain of involved variables. In this scheme the full domain of variables is updated. This leads to more propagation than partial arc-consistency techniques (section 9.6.1) but is generally less efficient for arithmetic. These arithmetic constraints can be reified (section 9.7.1).

Errors

Refer to the syntax of arithmetic FD expressions for possible errors (section 9.6.1).

Portability

GNU Prolog predicates.

9.6.4 fd_prime/1, fd_not_prime/1

Templates

fd_prime(?fd_variable)
fd_not_prime(?fd_variable)

Description

fd_prime(X) constraints X to be a prime number between 0..vector_max. This constraint enforces a sparse representation for the domain of X (section 9.1).

fd_not_prime(X) constraints X to be a non prime number between 0..vector_max. This constraint enforces a sparse representation for the domain of X (section 9.1).

Errors

X is neither an FD variable nor an integer type_error(fd_variable, X)

Portability

GNU Prolog predicates.

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9.6 Arithmetic constraints

9.6.1 FD arithmetic expressions

9.6.2 Partial AC: (#=)/2 - constraint equal, (#\=)/2 - constraint not equal, (#<)/2 - constraint less than, (#=<)/2 - constraint less than or equal, (#>)/2 - constraint greater than, (#>=)/2 - constraint greater than or equal

9.6.3 Full AC: (#=#)/2 - constraint equal, (#\=#)/2 - constraint not equal, (#<#)/2 - constraint less than, (#=<#)/2 - constraint less than or equal, (#>#)/2 - constraint greater than, (#>=#)/2 - constraint greater than or equal

9.6.4 fd_prime/1, fd_not_prime/1

9.6.2 Partial AC: (#=)/2 - constraint equal, (#\=)/2 - constraint not equal,
(#<)/2 - constraint less than, (#=<)/2 - constraint less than or equal,
(#>)/2 - constraint greater than, (#>=)/2 - constraint greater than or equal

9.6.3 Full AC: (#=#)/2 - constraint equal, (#\=#)/2 - constraint not equal,
(#<#)/2 - constraint less than, (#=<#)/2 - constraint less than or equal,
(#>#)/2 - constraint greater than, (#>=#)/2 - constraint greater than or equal