Constraint logic programming: Difference between revisions

Revision as of 13:02, 9 March 2006

Revision as of 13:19, 9 March 2006

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'''Constraint logic programming''' is a variant of [[logic programming]] including concepts from [[constraint satisfaction]]. A constraint logic program is a logic program that contains constraints in the body of clauses. An example of a clause including a constraint is <code>A(X,Y) :- X+Y>0, B(X), C(Y)</code>. In this clause, <code>X+Y>0</code> is a constraint; <code>A(X,Y)</code>, <code>B(X)</code>, and <code>C(Y)</code> are literals like in regular logic programming. Intuitively, this clause tells one condition under which the statement <code>A(X,Y)</code> holds: this is the case if <code>X+Y</code> is greater than zero and both <code>B(X)</code> and <code>C(Y)</code> are true.

Like in regular logic programming, programs are queried about the provability of a goal. A proof for a goal is composed of clauses whose constraints are satisfied and whose literals can in turn be proved using other clauses. Execution is done by an interpreter, which starts from the goal and [[Recursion|recursively]] scans the clauses trying to prove the goal. Constraints encountered during this scan are placed in a set called ''constraint store''. If this set is found out to be unsatisfiable, the interpreter [[Backtracking|backtrack]]s, trying to use other clauses for proving the goal. In practice, satisfiability of the constraint store ismay be checked using aan formincomplete of [[local consistency]]algorithm, which is one that does not always detect inconsistency. A complete check for satisfiability can be explicitely forced by using a specific literal.

==Overview==

Formally, constraint logic programs are like regular logic programs, but the body of clause can contain constraints and labeling literals (described below), in addition to the regular logic programming literals.

The semantics of constraint logic programs can be defined in terms of a virtual interpreter that maintains a pair <math>\langle G,S \rangle</math> during execution. The first element of this pair is called current goal; the second element is called constraint store. The ''current goal'' contains the literals the interpreter is trying to prove; constraints and equality of terms are considered literals, so they can occur in the goal; the ''constraint store'' contains all constraints the interpreter has assumed satisfiable so far.

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Each step of the algorithm is as follows. The first literal of the goal is considered and removed from the current goal. If it is a constraint, it is added to the constraint store. If it is a literal, a clause whose head has the same predicate of the literal is chosen; the clause is rewritten by replacing its variables with new variables (what is important is that these new variables do not already occur in the goal); the body of the clause is then placed in front of the goal; the equality of each argument of the literal with the corresponding one of the rewritten clause head is placed in front of the goal as well.

Some checks are done during these operations. In particular, the constraint store is checked for consistency every time a new constraint is added to it. In principle, whenever the constraint store is unsatisfiable the algorithm shouldcould backtrack. However, checking unsatisfiability at each step would be inefficient. For this reason, aan formincomplete ofsatisfiability localchecker consistencymay isbe checkedused instead.

The interpreter stops when the current goal is empty and the constraint store is not detected unsatisfiable.

When the current goal is empty, a regular logic program interpreter would stop and output the current substitution. In the same conditions, a constraint logic program also stops, and its output may be the current domains as reduced via the local consistency conditions on the constraint store. Actual satisfiability and finding a solution is enforced via labeling literals. In particular, whenever the interpreter encounters the literal <math>labeling(variables)</math> during the evaluation of a clause, it runs a satisfiability checker on the current constraint store to try and find a satisfying assignment.

The constraints of constraints logic programming are typically of three classes: constraints over terms, constraints over reals, and constraints over finite domains, which are usually interpreted as sets of integers.

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==Labeling==

The labeling literals are used to check satisfiability or partial satisfiability of the constraint store and to find a satisfying assignment. Wheneverof thevariables interpreteron evaluatesfinite adomains. A labeling literal is of the form <code>labeling([variables])</code>, where the argument is a list of variables over finite domains. Whenever the interpreter evaluates such a literal, it performs a search over the domains of variables to find an assignment that satisfies all relevant constraints. Typically, this is done by a form of [[backtracking]]: variables are evaluated in order, trying all possible values for each of them, and backtracking when inconsistency is detected.

The first use of the labeling literal is to actual check the satisfiability of the constraint store. Indeed, when the interpreter adds a constraint to the constraint store, it only checks or enforces a form of local consistency on it. This operation may not detect inconsistency even if the constraint store is unsatisfiable. A labeling literal over all variables of the constraint store enforces a complete satisfiability test to be performed.