knowledge, combined with the following formula, to obtain the product of any two numbers: We also have a reduction in the other direction; obviously, if we can multiply two numbers, we can square a number. This seems to imply that these two problems are equally hard. This kind of reduction corresponds to Turing reduction. However, the reduction becomes much harder if we add the restriction that we can only use the squaring function one time, and only at the end. In this case, even if we're allowed to use all the basic arithmetic operations, including multiplication, no reduction exists in general, because in order to get the desired result as a square we have to compute its square root first, and this square root could be an irrational number like that cannot be constructed by arithmetic operations on rational numbers. Going in the other direction, however, we can certainly square a number with just one multiplication, only at the end. Using this limited form of reduction, we have shown the unsurprising result that multiplication is harder in general than squaring. This corresponds to many-one reduction. Properties A reduction is a preordering, that is a reflexive and transitive relation, on P(N)×P(N), where P(N) is the power set of the natural numbers. Types and applications of reductions As described in the example above, there are two main types of reductions used in computational complexity, the many-one reduction and the Turing reduction. Many-one reductions map instances of one problem to instances of another; Turing reductions compute the solution to one problem, assuming the other problem is easy to solve. The many-one reduction is a stronger type of Turing reduction, and is more effective at separating problems into distinct complexity classes. However, the increased restrictions on many-one reductions make them more difficult to find. A problem is complete for a complexity class if every problem in the class reduces to that problem, and it is also in the class itself. In this sense the problem represents the class, since any solution to it can, in combination with the reductions, be used to solve every problem in the class. However, in order to be useful, reductions must be easy. For example, it's quite possible to reduce a difficult-to-solve NP-complete problem like the boolean satisfiability problem to a trivial problem, like determining if a number equals zero, by having the reduction machine solve the problem in