Word problem (computability)

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values. Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set; some of the most important problems in mathematics are undecidable.

For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and depends upon the values of x and y. A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem. A decision procedure for the decision problem "given two numbers x and y, does x evenly divide y?" would give the steps for determining whether x evenly divides y, given x and y. One such algorithm is long division, taught to many school children. If the remainder is zero the answer produced is 'yes', otherwise it is 'no'. A decision problem which can be solved by an algorithm, such as this example, is called decidable.

The field of computational complexity categorizes decidable decision problems by how difficult they are to solve. "Difficult", in this sense, is described in terms of the computational resources needed by the most efficient algorithm for a certain problem. The field of recursion theory, meanwhile, categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.

A decision problem is any arbitrary yes-or-no question on an infinite set of inputs. Because of this, it is traditional to define the decision problem equivalently as: the set of possible inputs together with the set of inputs for which the problem returns yes.

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