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In constraint satisfaction, local consistency conditions are properties of constraint satisfaction problems related to the consistency of subsets of variables or constraints. They can be used to reduce the search space and make the problem easier to solve. Various kinds of local consistency conditions are leveraged, including node consistency, arc consistency, and path consistency.

Every local consistency condition can be enforced by a transformation that changes the problem without changing its solutions; such a transformation is called constraint propagation. Constraint propagation works by reducing domains of variables, strengthening constraints, or creating new constraints. This leads to a reduction of the search space, making the problem easier to solve by some algorithms. Constraint propagation can also be used as an unsatisfiability checker, incomplete in general but complete in some particular cases.

Local consistency conditions can be grouped into various classes. The original local consistency conditions require that every consistent partial assignment (of a particular kind) can be consistently extended to another variable. Directional consistency only requires this condition to be satisfied when the other variable is greater than the ones in the assignment, according to a given order. Relational consistency includes extensions to more than one variable, but this extension is only required to satisfy a given constraint or set of constraints.

Assumptions

In this article, a constraint satisfaction problem is defined as a set of variables, a set of domains, and a set of constraints. Variables and domains are associated: the domain of a variable contains all values the variable can take. A constraint is composed of a sequence of variables, called its scope, and a set of their evaluations, which are the evaluations satisfying the constraint.

The constraint satisfaction problems referred to in this article are assumed to be in a special form. A problem is in normalized form, respectively regular form, if every sequence of variables is the scope of at most one constraint or exactly one constraint. The assumption of regularity done only for binary constraints leads to the standardized form. These conditions can always be enforced by combining all constraints over a sequence of variables into a single one and/or adding a constraint that is satisfied by all values of a sequence of variables.

In the figures used in this article, the lack of links between two variables indicate that either no constraint or a constraint satisfied by all values exists between these two variables.^{[clarification needed]}

Local consistency

The "standard" local consistency conditions all require that all consistent partial evaluations can be extended to another variable in such a way that the resulting assignment is consistent. A partial evaluation is consistent if it satisfies all constraints whose scope is a subset of the assigned variables.^{[clarification needed]}

Node consistency

Node consistency requires that every unary constraint on a variable is satisfied by all values in the domain of the variable, and vice versa. This condition can be trivially enforced by reducing the domain of each variable to the values that satisfy all unary constraints on that variable. As a result, unary constraints can be neglected and assumed incorporated into the domains.

For example, given a variable ${\displaystyle V}$ with a domain of ${\displaystyle \left\{1,2,3,4\right\}}$ and a constraint ${\displaystyle V\leq 3}$, node consistency would restrict the domain to ${\displaystyle \left\{1,2,3\right\}}$ and the constraint could then be discarded. This pre-processing step simplifies later stages.

Arc consistency

A variable of a constraint satisfaction problem is arc consistent with another one if each of its admissible values are consistent with some admissible value of the second variable. Formally, a variable ${\displaystyle x_{i}}$ is arc consistent with another variable ${\displaystyle x_{j}}$ if, for every value ${\displaystyle a}$ in the domain of ${\displaystyle x_{i}}$ there exists a value ${\displaystyle b}$ in the domain of ${\displaystyle x_{j}}$ such that ${\displaystyle (a,b)}$ satisfies the binary constraint between ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$. A problem is arc consistent if every variable is arc consistent with every other one.

For example, consider the constraint ${\displaystyle x<y}$ where the variables range over the domain 1 to 3. Because ${\displaystyle x}$ can never be 3, there is no arc from 3 to a value in ${\displaystyle y}$ so it is safe to remove. Likewise, ${\displaystyle y}$ can never be 1, so there is no arc, therefore it can be removed.

Arc consistency can also be defined relative to a specific binary constraint: a binary constraint is arc consistent if every value of one variable has a value of the second variable such that they satisfy the constraint. This definition of arc consistency is similar to the above, but is given specific to a constraint. This difference is especially relevant for non-normalized problems, where the above definition would consider all constraints between two variables while this one considers only a specific one.

If a variable is not arc consistent with another one, it can be made so by removing some values from its domain. This is the form of constraint propagation that enforces arc consistency: it removes, from the domain of the variable, every value that does not correspond to a value of the other variable. This transformation maintains the problem solutions, as the removed values are in no solution anyway.

Constraint propagation can make the whole problem arc consistent by repeating this removal for all pairs of variables. This process might have to consider a given pair of variables more than once. Indeed, removing values from the domain of a variable may cause other variables to become no longer arc consistent with it. For example, if ${\displaystyle x_{3}}$ is arc consistent with ${\displaystyle x_{2}}$ but the algorithm reduces the domain of ${\displaystyle x_{2}}$, arc consistency of ${\displaystyle x_{3}}$ with ${\displaystyle x_{2}}$ does not hold any longer, and has to be enforced again.

A simplistic algorithm would cycle over the pairs of variables, enforcing arc consistency, repeating the cycle until no domains change for a whole cycle. The AC-3 algorithm improves over this algorithm by ignoring constraints that have not been modified since they were last analyzed. In particular, it works on a set of constraints that initially contains all constraints; at each step, it takes a constraint and enforces arc consistency; if this operation may have produced a violation of arc consistency over another constraint, it places that constraint back in the set of constraints to analyze. This way, once arc consistency is enforced on a constraint, this constraint is not considered again unless the domain of one of its variables is changed.

Path consistency (k-consistency)

Path consistency is a property similar to arc consistency, but considers pairs of variables instead of only one. A pair of variables is path-consistent with a third variable if each consistent evaluation of the pair can be extended to the other variable in such a way that all binary constraints are satisfied. Formally, ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$ are path consistent with ${\displaystyle x_{k}}$ if, for every pair of values ${\displaystyle (a,b)}$ that satisfies the binary constraint between ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$, there exists a value ${\displaystyle c}$ in the domain of ${\displaystyle x_{k}}$ such that ${\displaystyle (a,c)}$ and ${\displaystyle (b,c)}$ satisfy the constraint between ${\displaystyle x_{i}}$ and ${\displaystyle x_{k}}$ and between ${\displaystyle x_{j}}$ and ${\displaystyle x_{k}}$, respectively.

The form of constraint propagation that enforces path consistency works by removing some satisfying assignment from a constraint. Indeed, path consistency can be enforced by removing from a binary constraint all evaluations that cannot be extended to another variable. As for arc consistency, this removal might have to consider a binary constraint more than once. As for arc consistency, the resulting problem has the same solutions of the original one, as the removed values are in no solution.

The form of constraint propagation that enforces path consistency might introduce new constraints. When two variables are not related by a binary constraint, they are virtually related by the constraint allowing any pair of values. However, some pair of values might be removed by constraint propagation. The resulting constraint is no longer satisfied by all pairs of values. Therefore, it is no longer a virtual, trivial constraint.

The name "path consistency" derives from the original definition, which involved a pair of variables and a path between them, rather than a pair and a single variable. While the two definitions are different for a single pair of variables, they are equivalent when referring to the whole problem.

Generalizations

Arc and path consistency can be generalized to non-binary constraints using tuples of variables instead of a single one or a pair. A tuple of ${\displaystyle i-1}$ variables is ${\displaystyle i}$-consistent with another variable if every consistent evaluation of the ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=Local_consistency
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