Constraints

There are three types of constraints: Boolean, arithmetic, and global. Note that each of the new operators in the interface has a counterpart in CLP(FD). For example, the equality operator is $= in the interface, and its counterpart in CLP(FD) is #=.

A basic Boolean expression is made from constants (0 and 1), Boolean variables, and the following operators: $/\ (and), $\/ (or), $\ (not or xor), $<=> (equivalent), and $=> (implication). The operator $\ is used for two different purposes: $\ X indicates the negation of X, and X $\ Y is the exclusive or of X and Y (the same as (X $/\ ($\ Y)) $\/ (Y $/\ ($\ X))).

An arithmetic constraint takes the form $E_1\ R\ E_2$, where E₁ and E₂ are two arithmetic expressions, and R is one of the following constraint operators: $= (equal), $\= (not equal), $>=, $>, $=<, and $<. An arithmetic expression is made of numbers, domain variables, and the following arithmetic functions: + (sign or addition), - (sign or subtraction), * (multiplication), div (integer division), mod (remainder), abs, min, max, and sum.

In addition to the basic standard syntax for expressions, the following forms of extended expressions are also acceptable. Let C be a Boolean expression, let E₁ and E₂ be expressions, and let L be a list of expressions [E₁,E₂,$\ldots$,E_n]. The following are also valid expressions:

• if(C,E₁,E₂): This is the same as C*E₁+(1-C)*E₂.
• min(L): The minimum element of L, where L can be a list comprehension.
• max(L): The maximum element of L, where L can be a list comprehension.
• min(E₁,E₂): The minimum of E₁ and E₂.
• max(E₁,E₂): The maximum of E₁ and E₂.
• sum(L): The sum of the elements of L, where L can be a list comprehension.

An extended Boolean expression can also include arithmetic constraints as operands. In particular, the constraint B $<=> (E1 $= E2) is called a reification constraint, which uses a Boolean variable B to indicate the satisfiability of the arithmetic constraint E1 $= E2.

The following two global constraints are currently supported:

• $alldifferent(L): Logically, the constraint $alldifferent(L) is equivalent to the conjunction of the pair-wise inequality constraints on the variables in L.
• $element(I,L,V): The Ith element of L is V.