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Computational problem
In theoretical computer science, a problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring
is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every instance/case. The question then is, whether there exists an algorithm that maps instances to solutions. For example, in the factoring problem, the instances are the integers n, and solutions are prime numbers p that are the nontrivial prime factors of n. An example of a computational problem without a solution is the Halting problem. Computational problems are one of the main objects of study in theoretical computer science.
One is often interested not only in mere existence of an algorithm, but also how efficient the algorithm can be. The field of computational complexity theory addresses such questions by determining the amount of resources (computational complexity) solving a given problem will require, and explain why some problems are intractable or undecidable. Solvable computational problems belong to complexity classes that define broadly the resources (e.g. time, space/memory, energy, circuit depth) it takes to compute (solve) them with various abstract machines. For example, the complexity classes
Both instances and solutions are represented by binary strings, namely elements of {0, 1}*. For example, natural numbers are usually represented as binary strings using binary encoding. This is important since the complexity is expressed as a function of the length of the input representation.
A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing:
A decision problem is typically represented as the set of all instances for which the answer is yes. For example, primality testing can be represented as the infinite set
In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.
A search problem is represented as a relation consisting of all the instance-solution pairs, called a search relation. For example, factoring can be represented as the relation
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Computational problem
In theoretical computer science, a problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring
is a computational problem that has a solution, as there are many known integer factorization algorithms. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every instance/case. The question then is, whether there exists an algorithm that maps instances to solutions. For example, in the factoring problem, the instances are the integers n, and solutions are prime numbers p that are the nontrivial prime factors of n. An example of a computational problem without a solution is the Halting problem. Computational problems are one of the main objects of study in theoretical computer science.
One is often interested not only in mere existence of an algorithm, but also how efficient the algorithm can be. The field of computational complexity theory addresses such questions by determining the amount of resources (computational complexity) solving a given problem will require, and explain why some problems are intractable or undecidable. Solvable computational problems belong to complexity classes that define broadly the resources (e.g. time, space/memory, energy, circuit depth) it takes to compute (solve) them with various abstract machines. For example, the complexity classes
Both instances and solutions are represented by binary strings, namely elements of {0, 1}*. For example, natural numbers are usually represented as binary strings using binary encoding. This is important since the complexity is expressed as a function of the length of the input representation.
A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing:
A decision problem is typically represented as the set of all instances for which the answer is yes. For example, primality testing can be represented as the infinite set
In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.
A search problem is represented as a relation consisting of all the instance-solution pairs, called a search relation. For example, factoring can be represented as the relation