Computational Complexity Theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. In this context, a computational problem is understood to be a task that is in principle amenable to being solved by a computer (i.e. the problem can be stated by a set of mathematical instructions). Informally, a computational problem consists of problem instances and solutions to these problem instances. For example, primality testing is the problem of determining whether a given number is prime or not. The instances of this problem are natural numbers, and the solution to an instance is yes or no based on whether the number is prime or not.

A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Other complexity measures are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computational complexity theory is to determine the practical limits on what computers can and cannot do.

Closely related fields in theoretical computer science are analysis of algorithms and computability theory. A key distinction between analysis of algorithms and computational complexity theory is that the former is devoted to analyzing the amount of resources needed by a particular algorithm to solve a problem, whereas the latter asks a more general question about all possible algorithms that could be used to solve the same problem. More precisely, it tries to classify problems that can or cannot be solved with appropriately restricted resources. In turn, imposing restrictions on the available resources is what distinguishes computational complexity from computability theory: the latter theory asks what kind of problems can, in principle, be solved algorithmically.

Read more about Computational Complexity TheoryIntractability, Continuous Complexity Theory, History

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... the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers ... Fortnow Homer (2003) date the beginning of systematic studies in computational complexity to the seminal paper "On the Computational Complexity of Algorithms" by Juris Hartmanis and Richard Stearns (1965 ... from the USSR, studied another specific complexity measure ...
Complexity, Problem Solving, And Sustainable Societies - Focus - Requirement of Knowledge
... Lowering the costs of complexity in one sphere causes them to rise in another." "Industrialism illustrates this point ... It generated its own problems of complexity and costliness ... While such elements of complexity are usually thought to facilitate economic growth, in fact they can do so only when subsidized by energy." (italics ours) ...
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... differential geometry) Sarkovskii's theorem (dynamical systems) Savitch's theorem (computational complexity theory) Sazonov's theorem (functional analysis) Schaefer's dichotomy ...

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