Arithmetic constraints
Author: Daniel Diaz
Keywords: GNU Prolog, manual, Prolog, compiler, constraints, finite domains

### 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.

Copyright (C) 1999-2021 Daniel Diaz Verbatim copying and distribution of this entire article is permitted in any medium, provided this notice is preserved. More about the copyright