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1.4. FD Constraints 13

• Type declaration:

sum :: [int] → (int → int → bool) → int → bool

• Deﬁnition: sum L Op V is true if the sum of all elements in L is related with

V via the relational operator Op. i.e., if Σ

e∈L

Op V .

• Example at the TOY(FD) command level:

Next goal returns true and constrains X, Y and Z to be 1.

TOY(FD)> L == [X,Y,Z], domain L 1 3, sum L (#<) 4

yes

L == [ 1, 1, 1 ]

Z == 1

Y == 1

X == 1

Elapsed time: 0 ms.

scalar product/4

• Type declaration:

scalar product :: [int] → [int] → (int → int → bool) → int → bool

• Deﬁnition: ’scalar product L1 L2 Op V’ is true if the scalar product (in the

sense of FD constraint solving) of the integers or FD variables in L1 and L2 is

related with the value V via the operator Op, i.e., if ’(L1 ∗

L2) Op V’ is satisﬁed

is satisﬁed with ∗

deﬁned as the usual scalar product of integer vectors.

• Example at the TOY(FD) command level:

TOY(FD)> domain [X,Y,Z] 1 10, scalar_product [1,2,3] [X,Y,Z] (#<) 10

yes

Z in 1..2

Y in 1..2

X in 1..4

Elapsed time: 0 ms.

As expected, the expressions constructed from both arithmetic and relational con-

straints may be non-linear.

1.4.5 Combinatorial Constraints

Combinatorial Constraints include well-known global constraints that are useful to

solve problems deﬁned on discrete domains. Often, these constraints are also called

symbolic constraints Beldiceanu, 2000.

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