Van Emde Boas observes " even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains.

In algorithmic information theory ( a subfield of computer science ), the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object.

Kolmogorov complexity is also known as " descriptive complexity " ( not to be confused with descriptive complexity theory ), Kolmogorov – Chaitin complexity, algorithmic entropy, or program-size complexity.

Algorithmic information theory is the area of computer science that studies Kolmogorov complexity and other complexity measures on strings ( or other data structures ).

Algorithm analysis is an important part of a broader computational complexity theory, which provides theoretical estimates for the resources needed by any algorithm which solves a given computational problem.

The term was coined by Fanya Montalvo by analogy with NP-complete and NP-hard in complexity theory, which formally describes the most famous class of difficult problems.

Computational complexity theory deals with the relative computational difficulty of computable functions.

Since many AI problems have no formalisation yet, conventional complexity theory does not allow the definition of AI-completeness.

To address this problem, a complexity theory for AI has been proposed.

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

Category: Quantum complexity theory

In complexity theory, the satisfiability problem ( SAT ) is a decision problem, whose instance is a Boolean expression written using only AND, OR, NOT, variables, and parentheses.

This property is used in several theorems in complexity theory:

Some, such as computational complexity theory, which studies fundamental properties of computational problems, are highly abstract, while others, such as computer graphics, emphasize real-world applications.

Some fields, such as computational complexity theory ( which explores the fundamental properties of computational problems ), are highly abstract, whilst fields such as computer graphics emphasise real-world applications.

:* Complexity theory and organizations, the application of complexity theory to strategy

:* Complexity economics, the application of complexity theory to economics

* Computational complexity theory, a field in theoretical computer science and mathematics

The following relations are known between PSPACE and the complexity classes NL, P, NP, PH, EXPTIME and EXPSPACE ( note that is not ):

Moral dread is seen as the other face of desire, and here psychoanalysis delivers to the writer a magnificent irony and a moral problem of great complexity.

With the growing complexity of markets and intensity of competition, sales management, whether at the district, region or headquarters level, is a tough job today -- and it will be tougher in the future.

In the `` typical tone language '', tonal morphophonemics is of the same order of complexity as consonantal morphophonemics.

The difficulty of analysis of any subsystem in the phonology is an inverse function of the size -- smaller systems are more troublesome -- for any given degree of morphophonemic complexity.

Improved bandwidth efficiency is achieved at the expense of increased transmitter and receiver complexity by completely suppressing both the carrier and one of the sidebands.

Thus, the Kolmogorov complexity of the raw file encoding this bitmap is much less than 1. 62 million.

More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language ( the sensitivity of complexity relative to the choice of description language is discussed below ).

Strings whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.

If a description of s, d ( s ), is of minimal length ( i. e. it uses the fewest number of characters ), it is called a minimal description of s. Thus, the length of d ( s ) ( i. e. the number of characters in the description ) is the Kolmogorov complexity of s, written K ( s ).

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

Some consider that naming the concept " Kolmogorov complexity " is an example of the Matthew effect.

The program tries every string, starting with the shortest, until it finds a string with complexity at least n ( if there is one ), then returns that string ( or goes into an infinite loop if there is no such string ).

* Complexity class, a set of problems of related complexity in computational complexity theory

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.

The data compression ratio can serve as an measure of the complexity of a data set or signal, in particular it is used to approximate the algorithmic complexity.

Biological evolution proceeds by random variation in ensemble averages of organisms combined with culling of the less-successful variants and reproduction of the more-successful variants, and macroscale engineering design also proceeds by a process of design evolution from simplicity to complexity as set forth somewhat satirically by John Gall: " A complex system that works is invariably found to have evolved from a simple system that worked.

In order to restrict computational complexity, one or more ' Levels ' are set for each Profile.

Each subroutine call required its own set of registers, which in turn required more real estate on the CPU and more complexity in its design.

Euler diagram for P ( complexity ) | P, NP, NP-complete, and NP-hard set of problems.

In complexity theory, the class NC ( for " Nick's Class ") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors.

The notation A < sup > B </ sup > can be extended to a set of languages B ( or a complexity class B ), by using the following definition:

The Confluence trilogy, set in an even more distant future ( about ten million years from now ), is one of a number of novels to use Frank J. Tipler's Omega Point Theory ( that the universe seems to be evolving toward a maximum degree of complexity and consciousness ) as one of its themes.

The set of primitive recursive functions is known as PR in complexity theory.

* Reduced instruction set computing, a CPU design philosophy that favors an instruction set reduced both in size and complexity of addressing modes, in order to enable easier implementation, greater instruction level parallelism, and more efficient compilers

The set of all recursive functions is known as R in computational complexity theory.

In computational complexity theory, the complexity class # P ( pronounced " number P " or, sometimes " sharp P " or " hash P ") is the set of the counting problems associated with the decision problems in the set NP.

In computational complexity theory, PSPACE is the set of all decision problems which can be solved by a Turing machine using a polynomial amount of space.

A logical characterization of PSPACE from descriptive complexity theory is that it is the set of problems expressible in second-order logic with the addition of a transitive closure operator.

Euler diagram for P ( complexity ) | P, NP ( complexity ) | NP, NP-complete, and NP-hard set of problems

In complexity theory, the complexity class NP-easy is the set of function problems that are solvable in polynomial time by a deterministic Turing machine with an oracle for some decision problem in NP.