In constraint satisfaction, the AC-3 algorithm (short for Arc Consistency Algorithm #3) is one of a series of algorithms used for the solution of constraint satisfaction problems (or CSP's). It was developed by Alan Mackworth in 1977. The earlier AC algorithms are often considered too inefficient, and many of the later ones are difficult to implement, and so AC-3 is the one most often taught and used in very simple constraint solvers. The algorithm AC-3 operates on constraints, variables, and the variables' domains (scopes). A variable can take any of several discrete values; the set of values for a particular variable is known as its domain. A constraint is a relation that limits or constrains the values a variable may have. The constraint may involve the values of other variables. The current status of the CSP during the algorithm can be viewed as a directed graph, where the nodes are the variables of the problem, with edges or arcs between variables that are related by symmetric constraints, where each arc in the worklist represents a constraint that needs to be checked for consistency. AC-3 proceeds by examining the arcs between pairs of variables (x, y). It removes those values from the domain of x which aren't consistent with the constraints between x and y. The algorithm keeps a collection of arcs that are yet to be checked; when the domain of a variable has any values removed, all the arcs of constraints pointing to that pruned variable (except the arc of the current constraint) are added to the collection. Since the domains of the variables are finite and either one arc or at least one value are removed at each step, this algorithm is guaranteed to terminate. For illustration, here is an example of a very simple constraint problem: X (a variable) has the possible values {0, 1, 2, 3, 4, 5} -- the set of these values are the domain of X, or D(X). The variable Y has the domain D(Y) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Together with the constraints C1 = "X must be even" and C2 = "X + Y must equal 4" we have a CSP which AC-3 can solve. Notice that the actual constraint graph representing this problem must contain two edges between X and Y since C2 is undirected but the graph representation being used by AC-3 is