Path consistency

Path consistency is a local consistency condition in constraint satisfaction that extends arc consistency from pairs of variables to triples. Where arc consistency ensures that every value in one variable's domain has a compatible value in a neighboring domain, path consistency ensures that every pair of compatible values between two variables can be extended to a third variable connected by a path. It is stronger than arc consistency but exponentially more expensive to enforce, and it is rarely used in practice except for small networks or as a theoretical benchmark in the hierarchy of local consistencies that extends to k-consistency.

The systems-theoretic significance of path consistency is that it represents the threshold where local reasoning transitions from pairwise to network-level: once path consistency is enforced, the constraint network begins to exhibit global structural properties that pairwise reasoning cannot detect. This is why path consistency is the natural next step after arc consistency in the study of how local rules generate global order.