Computational Theory Questions Long
In computational theory, decision problems and search problems are two fundamental types of problems that are commonly studied. While both types involve solving problems using computational methods, they differ in their objectives and the nature of their solutions.
1. Decision Problems:
Decision problems are concerned with determining whether a given input satisfies a specific property or condition. The goal is to provide a yes/no answer, indicating whether the input belongs to a particular set or not. In other words, decision problems aim to classify inputs into two distinct categories: those that have the desired property and those that do not.
For example, consider the decision problem of determining whether a given number is prime. The input is a number, and the objective is to determine whether it is divisible only by 1 and itself. The answer to this decision problem would be either "yes" if the number is prime or "no" if it is not.
Decision problems are typically represented as languages, where the language consists of all inputs that satisfy the desired property. The decision problem is then to determine whether a given input belongs to the language or not.
2. Search Problems:
Search problems, on the other hand, involve finding a solution or set of solutions that satisfy a specific criterion. Unlike decision problems, search problems do not require a simple yes/no answer but rather aim to find a specific solution or a set of solutions.
For example, consider the search problem of finding the shortest path between two cities on a map. The input is a map with cities and their connections, and the objective is to find the path with the minimum distance between two specified cities. The solution to this search problem would be the actual path that satisfies the criterion.
Search problems are typically represented as functions, where the function takes an input and returns a solution that satisfies the desired criterion. The goal is to find an algorithm or method that efficiently computes the solution(s) for a given input.
In summary, the main difference between decision problems and search problems lies in their objectives and the nature of their solutions. Decision problems aim to classify inputs into two distinct categories, while search problems involve finding specific solutions that satisfy a given criterion.