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Dragonfall
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What are complexity classes (P, NP, etc) in terms of pure sets? ZFC, I mean.
AUMathTutor said:I thought they were just sets. Like
P = {problem q | there is a polynomial time algorithm for solving q}
etc.
Complexity classes are sets of problems that share similar characteristics in terms of their computational complexity. They are used to classify problems according to the amount of time or resources required to solve them.
Pure sets, also known as decision problems, are a subset of complexity classes that contain problems with a yes/no answer. They are important in complexity classes as they are used to define the hardness of problems and determine which problems are solvable in polynomial time.
Complexity classes are defined based on the complexity of pure sets. This means that a problem belongs to a particular complexity class if it can be reduced to a pure set within that class. In other words, the complexity of a problem is determined by its relationship to pure sets in a particular complexity class.
Examples of complexity classes include P, NP, and EXPTIME. Some examples of pure sets within these classes are the Boolean satisfiability problem, the traveling salesman problem, and the subset sum problem, respectively.
Complexity classes and pure sets are used in practical applications to analyze the time and space complexity of algorithms, to determine the feasibility of solving a problem, and to classify problems based on their difficulty. They also provide a framework for understanding the relationship between different types of problems and the resources required to solve them.