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Decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?"
A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers x and y, does x evenly divide y?" is decidable since there is a decision procedure called long division that gives the steps for determining whether x evenly divides y and the correct answer, YES or NO, accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem.
The field of computational complexity theory 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. On the other hand, the field of recursion theory categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.
A decision problem is the formal language of all inputs for which the output (the answer to the yes-no question on a given input) is YES.
A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient procedures of primality testing are known, the existence of any effective procedure is enough to establish decidability.
Problems that are not decidable are undecidable, which means it is not possible to create an algorithm (efficient or not) that solves them. The halting problem is an important undecidable decision problem; for more examples, see list of undecidable problems.
Decision problems can be ordered according to many-one reducibility and related to feasible reductions such as polynomial-time reductions. A decision problem P is said to be complete for a set of decision problems S if P is a member of S and every problem in S can be reduced to P. Complete decision problems are used in computational complexity theory to characterize complexity classes of decision problems. For example, the Boolean satisfiability problem is complete for the class NP of decision problems under polynomial-time reducibility.
Decision problems are closely related to function problems, which can have answers that are more complex than a simple YES or NO. A corresponding function problem is "given two numbers x and y, what is x divided by y?".
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Decision problem AI simulator
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Decision problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?"
A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers x and y, does x evenly divide y?" is decidable since there is a decision procedure called long division that gives the steps for determining whether x evenly divides y and the correct answer, YES or NO, accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem.
The field of computational complexity theory 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. On the other hand, the field of recursion theory categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.
A decision problem is the formal language of all inputs for which the output (the answer to the yes-no question on a given input) is YES.
A classic example of a decidable decision problem is the set of prime numbers. It is possible to effectively decide whether a given natural number is prime by testing every possible nontrivial factor. Although much more efficient procedures of primality testing are known, the existence of any effective procedure is enough to establish decidability.
Problems that are not decidable are undecidable, which means it is not possible to create an algorithm (efficient or not) that solves them. The halting problem is an important undecidable decision problem; for more examples, see list of undecidable problems.
Decision problems can be ordered according to many-one reducibility and related to feasible reductions such as polynomial-time reductions. A decision problem P is said to be complete for a set of decision problems S if P is a member of S and every problem in S can be reduced to P. Complete decision problems are used in computational complexity theory to characterize complexity classes of decision problems. For example, the Boolean satisfiability problem is complete for the class NP of decision problems under polynomial-time reducibility.
Decision problems are closely related to function problems, which can have answers that are more complex than a simple YES or NO. A corresponding function problem is "given two numbers x and y, what is x divided by y?".