containment Since the homomorphism problem is equivalent to conjunctive query evaluation and conjunctive query containment, these two problems are equivalent to constraint satisfaction as well. Join evaluation Every constraint can be viewed as a table in a database, where the variables are interpreted as attributes names and the relation is the set of records in the table. The solutions of a constraint satisfaction problem are the result of an inner join of the tables representing its constraints; therefore, the problem of existence of solutions can be reformulated as the problem of checking whether the result of an inner join of a number of tables is empty. Dichotomy theorems Some constraint languages (or non-uniform problems) are known to correspond to problems solvable in polynomial time, and some others are known to express NP-complete problems. However, it is possible that some constraint languages are neither. It is known by Ladner's theorem that if P is not equal to NP, then there exist problems in NP that are neither polynomial-time nor NP-hard. , it is not known if such problems can be expressed as constraint satisfaction problems with a fixed constraint language. If Ladner languages were not expressible in this way, the set of all constraint languages could be divided exactly into those defining polynomial-time and those defining NP-complete problems; that is, this set would exhibit a dichotomy. Partial results are known for some sets of constraint languages. The best known such result is Schaefer's dichotomy theorem, which proves the existence of a dichotomy in the set of constraint languages on a binary domain. More precisely, it proves that a relation restriction on a binary domain is tractable if all its relations belong to one of six classes, and is NP-complete otherwise. Bulatov proved a dichotomy theorem for domains of three elements. Another dichotomy theorem for constraint languages is the Hell-Nesetril theorem, which shows a dichotomy for problems on binary constraints with a single fixed symmetric relation. In terms of the homomorphism problem, every such problem is equivalent to the existence of a homomorphism from a relational structure to a given fixed undirected graph (an undirected graph can be regarded as a relational structure with a single binary symmetric relation). The Hell-Nesetril theorem proves that every such problem is either polynomial-time or NP-complete. More precisely, the problem is polynomial-time if the graph is 2-colorable, that is, it is bipartite, and is NP-complete otherwise.