Editing Constraint satisfaction problem (section)

==Formal definition==
Formally, a constraint satisfaction problem is defined as a triple <math>\langle X,D,C \rangle</math>, where<ref name=Russell2010>{{cite book|author1=Stuart Jonathan Russell |author2=Peter Norvig |title=Artificial Intelligence: A Modern Approach|date=2010|publisher=Prentice Hall|isbn=9780136042594|page=Chapter 6}}</ref>
* <math>X = \{X_1, \ldots,X_n\}</math> is a set of variables, 
* <math>D = \{D_1, \ldots, D_n\}</math> is a set of their respective domains of values, and
* <math>C = \{C_1, \ldots, C_m\}</math> is a set of constraints. 
Each variable <math>X_i</math> can take on the values in the nonempty domain <math>D_i</math>.
Every constraint <math>C_j \in C</math> is in turn a pair <math>\langle t_j,R_j \rangle</math>, where <math>t_j \subseteq \{1, 2, \ldots, n\}</math> is a set of <math>k</math> indices and <math>R_j</math> is a <math>k</math>-ary [[relation (mathematics)|relation]] on the corresponding product of domains <math>\times_{i \in t_j} D_i</math> where the product is taken with indices in ascending order. An ''evaluation'' of the variables is a function from a subset of variables to a particular set of values in the corresponding subset of domains. An evaluation <math>v</math> satisfies a constraint <math>\langle t_j, R_j \rangle</math> if the values assigned to the variables <math>t_j</math> satisfy the relation <math>R_j</math>.

An evaluation is ''consistent'' if it does not violate any of the constraints. An evaluation is ''complete'' if it includes all variables. An evaluation is a ''solution'' if it is consistent and complete; such an evaluation is said to ''solve'' the constraint satisfaction problem.