<P> While usually not included in the above definition of a constraint satisfaction problem, arithmetic equations and inequalities bound the values of the variables they contain and can therefore be considered a form of constraints . Their domain is the set of numbers (either integer, rational, or real), which is infinite: therefore, the relations of these constraints may be infinite as well; for example, X = Y + 1 (\ displaystyle X = Y + 1) has an infinite number of pairs of satisfying values . Arithmetic equations and inequalities are often not considered within the definition of a "constraint satisfaction problem", which is limited to finite domains . They are however used often in constraint programming . </P> <P> It can be shown that the arithmetic inequalities or equations present in some types of finite logic puzzles such as Futoshiki or Kakuro (also known as Cross Sums) can be dealt with as non-arithmetic constraints (see Pattern - Based Constraint Satisfaction and Logic Puzzles). </P> <P> Constraint satisfaction problems on finite domains are typically solved using a form of search . The most used techniques are variants of backtracking, constraint propagation, and local search . These techniques are used on problems with nonlinear constraints . </P> <P> Variable elimination and the simplex algorithm are used for solving linear and polynomial equations and inequalities, and problems containing variables with infinite domain . These are typically solved as optimization problems in which the optimized function is the number of violated constraints . </P>

Constraints or problems in a study are known as