Method

Computational complexity theory

Overview

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer and is solvable by mechanical application of mathematical steps, such as an algorithm.

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 their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage.

Other measures of complexity 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. The P versus NP problem, one of the seven Millennium Prize Problems, is part of the field of computational complexity.

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, computational complexity theory 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 kinds of problems can, in principle, be solved algorithmically.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

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Then _R(A, B, C)_ is another eternal object of the lowest complex grade. Analogously there are eternal objects of successively higher grades. In respect to any complex eternal object, _S(D1, D2, ... Dn)_, the eternal objects _D1, ... Dn_, whose individual essences are constitutive of the individual essence of _S(D1, ... Dn)_, are called the components of _S(D1, ... Dn)_. It is obvious that the grade of complexity to…
Alfred North Whitehead, Science and the modern world
States what Computational complexity theory is.
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227.] Finite Cognition, when it pre-supposes what is distinguished from it to be something already existing and confronting it,--to be the various facts of external nature or of consciousness--has, in the first place, (1) Formal identity or the abstraction of universality for the form of its action. Its activity therefore consists in analysing the given concrete object, isolating its differences, and giving them the…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Computational complexity theory is.
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It doubtless makes little difference whether a disease be attributed to big or little, black or red, demons or whether the cause be represented by a, b, or c, etc. But where data and hypotheses are such as are capable of verification, i.e., of mutually checking up each other, a change in one without a corresponding modification of the other is the principle of all formal fallacies.[18]
John Dewey, Creative Intelligence
States what Computational complexity theory is.
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Related ideas

Place in the tradition

Part of First-order logic.

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