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- W3100189114 abstract "In this chapter we study two related areas of theoretical computer science: computability theory and computational complexity. Each of these subjects take mathematical problems as objects of study. The aim is not to solve these problems, but rather to classify them by level of difficulty. Time complexity classifies a given problem according to the length of time required for a computer to solve the problem. The polynomial-time problems P and the nondeterministic polynomial-time problems NP are the two most prominent classes of time complexity. Some problems cannot be solved by the algorithmic process of a computer. We refer to problems as decidable or undecidable according to whether or not there exists an algorithm that solves the problem. Computability theory considers undecidable problems and the brink between the undecidable and the decidable. There are only countably many algorithms and uncountably many problems to solve. From this fact we deduce that most problems are not decidable. To proceed beyond this fact, we must state precisely what we mean by an “algorithm” and a “problem.” One of the aims of this chapter is to provide a formal definition for the notion of an algorithm. The types of problems we shall consider are represented by the following examples. • The even problem: Given an n ∈ ℕ, determine whether or not n is even. • The 10-clique problem: Given finite graph, determine whether or not there exists a subgraph that is isomorphic to the 10-clique. • The satisfiability problem for first-order logic: Given a sentence of first-order logic, determine whether or not it is satisfiable. The first problem is quite easy. To determine whether a given number is even, we simply check whether the last digit of the number is 0, 2, 4, 6 or 8. The second problem is harder. If the given graph is large and does contain a 10-clique as a subgraph, then we may have to check many subsets of the graph before we find it. Time complexity gives precise meaning to the ostensibly subjective idea of one problem being “harder” than another. The third problem is the most difficult of the three problems." @default.
- W3100189114 created "2020-11-23" @default.
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- W3100189114 date "2004-07-08" @default.
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- W3100189114 title "Computability and complexity" @default.
- W3100189114 doi "https://doi.org/10.1093/oso/9780198529804.003.0011" @default.
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