Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved 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 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.

Computational complexity

Computational complexity may refer to:

Complexity theory

Complexity theory may refer to:

See also

Complexity

Complexity is generally used to characterize something with many parts where those parts interact with each other in multiple ways. There is no absolute definition of what complexity means; the only consensus among researchers is that there is no agreement about the specific definition of complexity. However, a characterization of what is complex is possible. The study of these complex linkages at various scales is the main goal of complex systems theory.

In science, there are as of 2010 a number of approaches to characterizing complexity; this article reflects many of these. Neil Johnson states that "even among scientists, there is no unique definition of complexity - and the scientific notion has traditionally been conveyed using particular examples..." Ultimately he adopts the definition of 'complexity science' as "the study of the phenomena which emerge from a collection of interacting objects."

Overview

Definitions of complexity often depend on the concept of a "system"—a set of parts or elements that have relationships among them differentiated from relationships with other elements outside the relational regime. Many definitions tend to postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of relationships among the elements. However, what one sees as complex and what one sees as simple is relative and changes with time.

Scott Aaronson

Scott Joel Aaronson (born May 21, 1981) is a theoretical computer scientist and faculty member in the Electrical Engineering and Computer Science department at the Massachusetts Institute of Technology.

Early life and education

Aaronson grew up in the United States, though he spent a year in Asia when his father—a science writer turned public-relations executive—was posted to Hong Kong. He enrolled in a school there that permitted him to skip ahead several years in math, but upon returning to the US, he found his education restrictive, getting bad grades and having run-ins with teachers. He enrolled in a program for gifted youngsters run by Clarkson University, which enabled Aaronson to apply for colleges while only in his freshman year of high school. He was accepted into Cornell University, where he obtained his BSc in computer science in 2000, then attended the University of California, Berkeley, for his PhD, which he got in 2004 under the supervision of Umesh Vazirani.

Aaronson had shown ability in mathematics from an early age, teaching himself calculus at the age of 11, provoked by symbols in a babysitter's textbook. He discovered computer programming at age 11, and felt he lagged behind peers, who had already been coding for years. Partly for this reason, he felt drawn to theoretical computing, particularly computational complexity. At Cornell, he became interested in quantum computing, and devoted himself to computational complexity and quantum computing.