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Topic: Algorithmic complexity theory

	No. 1897:
		Dijkstra's algorithm is guaranteed to find a shortest route in no more than O(n*n) steps, where n is the number of intersections.
		Bellman-Ford algorithm is guaranteed to find a shortest route in no more than O(n*m) steps, where m is the number of road segments.
		Algorithmic complexity theory goes further by showing that there are problems in NP that are NP-complete.
	www.uh.edu /engines/epi1897.htm (1605 words)

	algorithmic complexity
		A measure of complexity that has been developed by Gregory Chaitin and others and is based on Claude Shannon’s information theory and earlier work by the Russian mathematicians Andrei Kolmogorov and Ray Solomonoff.
		Algorithmic complexity quantifies how complex a system is in terms of the length of the shortest computer program, or set of algorithms, need to completely describe the system.
		At the other extreme, if system is totally random its algorithmic complexity is very high since the random patterns can't be condensed into a smaller set of algorithms: the program is effectively as large as the system itself.
	www.daviddarling.info /encyclopedia/A/algorithmic_complexity.html (254 words)

	20th WCP: Computational Complexity and Philosophical Dualism
		Of course, one could plausibly argue that the major problems arisen by Complexity Theory, i.e., the size of N (number of steps to solve a problem) and the speed of computing might depend on the type of machine on which the algorithm is to be run.
		Nonetheless, early writings on Complexity Theory by Bremermann (1977) which have been systematically overlooked by many authors (including Penrose) show that there are physical constraints for the design of computing machines and such constraints have a bearing on the time-length consumed by those machines no matter how improved their hardware may be.
		Since the growth of temporal complexity involved in the realization of transcomputable algorithms is exponential that means that the time-length required for running some transcomputable algorithms can be as long as the age of the universe.
	www.bu.edu /wcp/Papers/Cogn/CognTeix.htm (3006 words)

	Complexity theory Summary
		Sometimes it is the case that the algorithm used to solve a particular instance of a complex problem is not necessarily applicable as a solution to the general problem.
		Another complexity class is "class Co-NP." This class is similar to NP but has the added difficulty of having checking algorithms that are also not of class P. It is possible that to check a claimed solution to a Co-NP problem could require an exhaustive search of all possible solutions.
		the application of complexity theory to organizations, which has been influential in strategic management and organizational studies; this area is sometimes referred to as complexity strategy or the study of complex adaptive organizations.
	www.bookrags.com /Complexity_theory (766 words)

	Algorithmic Complexity in English Studies
		The application of mathematical algorithms in chaos theory is a similar given (with fractals as an illustration) but this is merely how the term is applied in the scientific community -- algorithms have made an important transition from mathematical computations to representation of spatial, scientific, and I argue, literary, relationships.
		The problem with using a postmodern algorithm to mediate between texts (at the lowest level) and practical theory (at the highest, most complex level) is the nature of postmodernity as open to disorder and order within the texts themselves.
		She defines algorithm as "a set of procedures a computer uses to solve a problem" (161), which is a common computer science definition.
	www.cwrl.utexas.edu /~tonya/Tonya/algorithm.html (4571 words)

	Complexity & Information Theory
		We define the algorithmic information content I(x) of an object (or bit string) x as the length of the shortest self delimiting program(s) for a Universal Turing Machine U which generates a description of that object for a given level of precision.
		The string's algorithmic information content, or aglorithmic complexity, is bounded from above by the length of the binary sequence specifying 'forty', together with the code which tells the computer to print a digit a certain number of times.
		Another of these short introductory papers explains why, despite the fact that algorithmically complex strings outnumber their simpler cousing by a wide margin, proving that given strings are random is generally impossible.
	www.mulhauser.net /research/tutorials/complexity/complexity.html (1571 words)

	[No title]
		\section{The complexity measure} The key idea in algorithmic complexity theory is that \begin{itemize} \setlength{\leftmargin}{1in} \item The complexity of some dataset D is equal (subject to certain caveats) to the size of the smallest computer program that generates D.
		Algorithmic complexity theory thus defines a dataset to be random provided only that it has its maximum complexity value, i.e., if its complexity is equal to its size.
		Thus algorithmic regularity is the key target in the data mining process: it is effectively the `material' that needs to be extracted from the `data mine'.
	www.cogs.susx.ac.uk /users/christ/crs/dm/111x17818 (11082 words)

	[No title]
		Complexity theory is the theory of determining the necessary resources for the solution of algorithmic problems and, therefore, the limits what is possible with the available resources.
		New branches of complexity theory react to all new algorithmic concepts.
		The Prosentential Theory of Truth suggests that the grammatical predicate "is true" does not function semantically or logically as a predicate.
	www.lycos.com /info/model-theory.html (400 words)

	Algorithmic information theory - Scholarpedia
		Algorithmic Information Theory (AIT) is the information theory of individual objects, using computer science, and concerns itself with the relationship between computation, information, and randomness.
		Algorithmic complexity formalizes the notions of simplicity and complexity.
		Zvonkin and L. Levin The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms.
	www.scholarpedia.org /article/Algorithmic_information_theory (3086 words)

	A SYSTEM FOR INCREMENTAL LEARNING BASED ON ALGORITHMIC PROBABILITY (Site not responding. Last check: 2007-10-25)
		The algorithm we use is able to solve inversion problems (such as the P and NP problems of computational complexity theory) as well as time limited optimization problems (See Section 2 for definitions and discussion of these problem types).
		Algorithmic probability defines the apriori probability of a string of symbols, x, as the probability that x will be produced as the output of a reference universal Turing machine having random input.
		Algorithmic probability was able to refine these ideas by giving an optimum "goodness of fit criterion" for the fitting of probabilistic grammars to data (Sol 62, 64b, 75).
	world.std.com /~rjs/IncLrn89.html (7066 words)

	What is Algorithmic Complexity?
		Algorithmic complexity, (computational complexity, or Kolmogorov complexity), is a foundational idea in both computational complexity theory and algorithmic information theory, and plays an important role in formal induction.
		The algorithmic complexity of a binary string is defined as the shortest and most efficient program that can produce the string.
		Complexity classes include problems far more difficult than anything one might confront in mathematics up to calculus.
	www.wisegeek.com /what-is-algorithmic-complexity.htm (398 words)

	BASIC CONCEPTS RELEVANT TO LEARNING
		Algorithmic complexity theory provides various different but closely related measures for the complexity (or simplicity) of objects.
		Informally speaking, the complexity of a computable object is the length of the shortest program that computes it and halts, where the set of possible programs forms a prefix code.
		The Levin complexity of the string is the minimal possible value of this.
	www.idsia.ch /~juergen/loconet/node2.html (1388 words)

	hetland.org: Algorithmic Complexity (Site not responding. Last check: 2007-10-25)
		This is a very short introduction to algorithmic complexity, with focus on topics that seem to confuse students when they first meet the subject.
		In algorithmic complexity theory the choice is the latter, that is, the number of characters — or digits.
		While these algorithms technically have exponential running times, they often work well in practice, because the number in question (the one giving rise to the exponential time) can be bounded, and the rest of the running time is a polynomial.
	hetland.org /algorithms/complexity.php (3122 words)

	Algorithmic information theory - Wikipedia, the free encyclopedia
		Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information.
		There are several variants of Kolmogorov complexity or algorithmic information; the most widely used one is based on self-delimiting programs and is mainly due to Leonid Levin (1974).
		Unlike classical information theory, algorithmic information theory gives formal, rigorous definitions of a random string and a random infinite sequence that do not depend on physical or philosophical intuitions about nondeterminism or likelihood.
	en.wikipedia.org /wiki/Algorithmic_information_theory (390 words)

	The Observer as a Classical Computer in a Quantum Universe
		The Kolmogorov complexity of a symbol string is defined as the length of the shortest program, for a particular Turing machine (TM), which will output that string.
		This ambiguity in the notion of complexity is not very important in most practical applications, because for two fixed Turing machines the difference in complexity does not grow to infinity as the complexity of the string goes to infinity, but approaches a (usually small) constant.
		An absolute algorithmic complexity must be defined if it is to play any role in solving the implementation problem.
	www.finney.org /~hal/mallah1.html (4315 words)

	Algorithmic coding theory
		In addition to their obvious applications in communications, codes have found numerous applications in complexity theory and cryptography, and are by now indispensable items in the toolkit of theoretical computer scientists.
		Traditionally, decoding algorithms were required to always output a unique answer which limited them to correcting a number of errors which is at most half the distance, say d/2, of the code.
		In addition to serving as the core algorithmic results, these in turn renewed interest in questions on the exact power of list decoding, and in particular on the trade-off between number of errors that can be list decoded and the amount of redundancy needed in the code.
	www.cs.washington.edu /homes/venkat/RD-coding.html (604 words)

	Kolmogorov complexity - Wikipedia, the free encyclopedia
		In computer science, the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity) of an object such as a piece of text is a measure of the computational resources needed to specify the object.
		The notion of Kolmogorov complexity is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.
		Algorithmic information theory is the area of computer science that studies Kolmogorov complexity and other complexity measures on strings (or other data structures).
	en.wikipedia.org /wiki/Algorithmic_complexity_theory (2081 words)

	Algorithmic Complexity
		String maps of highly complex structures can be computed, in general, with the same computational overhead as those of simple structures (the computational overhead is nearly constant), so for complex structures (large I) the negative component of informational complexity is negligible.
		Furthermore, in comparisons of algorithmic complexity, the overhead drops out except for a very small part required to make comparisons of complexity (even this drops out in comparisons of comparisons of complexity), so the relative algorithmic complexity is almost a direct measure of the relative informational complexity.
		In a formal system of complexity n it is impossible to prove that a particular series of binary digits is of complexity greater than n+c, where c is a constant that is independent of the particular system employed, but depends only on the implementation of the system.
	www.kli.ac.at /theorylab/jdc/information/algorith.html (1134 words)

	Information Theory and Creationism: Algorithmic Information Theory (Chaitin, Solomonoff & Kolmogorov)
		An algorithm producing the digits of π is an example of a non-halting algorithm π is an irrational and transcendental number; it cannot be expressed as a polynomial with rational coefficients and therefore has an infinite number of digits).
		Algorithmically random has a different definition and is a different concept, even though very long algorithmically random strings have symbol distributions with statistical properties.
		Computational complexity theory deals with the amount of computing resources (time and memory) needed to solve a problem.
	www.talkorigins.org /faqs/information/algorithmic.html (3567 words)

	Theory of Computing
		The theory of computing is the study of efficient computation, models of computational processes, and their limits.
		Research at Cornell spans all areas of the theory of computing and is responsible for the development of modern computational complexity theory, the foundations of efficient graph algorithms, and the use of applied logic and formal verification for building reliable systems.
		In addition to its depth in the central areas of theory, Cornell is unique among top research departments in the fluency with which students can interact with faculty in both theoretical and applied areas, and work on problems at the critical juncture of theory and applications.
	www.cs.cornell.edu /Research/theory/index.htm (873 words)

	abstracts
		The modern objective prior theories that have received the most attention are the reference prior approach and the probability matching approach.
		The theory of algorithmic complexity is almost 30 years old.
		The theory, also known as Kolmogorov complexity, gave rise to several related theories and paradigms such as: Theory of Martin-L{\"o}f's tests, Algorithmic Theory of Information, Minimum Description Length Principle (MDLP), etc. In the talk we first overview some basic notions, definitions, and properties of algorithmic complexity.
	www.isds.duke.edu /courses/Spring99/sta395/abstracts.html (1471 words)

	A prime solution
		There are several algorithms in vogue for testing primality, but they are either probabilistic (with small probabilities for returning a composite number as a prime or failing on a prime number) or conditional (like using some unproven hypothesis), or deterministic and unconditional but working in non-polynomial time.
		However, the Miller-Rabin algorithm is a very efficient test for practical real-life implementation in areas such as cryptography because the level of error is extremely small and the probability of getting a correct answer far outweighs the error probability.
		Algorithmic complexity theory classifies problems depending on how difficult they are to solve.
	www.flonnet.com /fl1917/19171290.htm (2831 words)

	Are chaotic systems dynamically random ?
		A notion of ``self-generating'' complexity was proposed by Grassberger in ref. [17].
		The (incompressible) dynamic complexity of TR sequences is generated by the unfolding of a computable, but TR initial value (the associated evolution law must have positive Lyapunov exponent(s)), or by a dynamically incompressible algorithm and arbitrary initial values.
		Heuristically speaking, algorithmic complexity may be ``created'' by an investment of dynamic complexity, for instance by an (infinite) amount of time.
	tph.tuwien.ac.at /~svozil/publ/dyn.htm (1948 words)

	Scientific & Mathematical Roots of Complexity Science
		Specifically, algorithmic complexity is a measure of complexity developed by the mathematician Gregory Chaitin based on earlier work in Information Theory founded by Claude Shannon and work on probability and information conducted by the by the Russian mathematicians Kolmogorov and Solomonoff.
		Algorithm complexity theory defines and measures complexity in terms of a computer algorithm (or computer program) which could generate the data coming from a particular complex system.
		In other words, the degree of a system's complexity is a matter of how large a computer program would be needed to generate a bit string derived from the system under question (sequence of 0's and 1's, or the binary code at the core of computer languages).
	www.plexusinstitute.org /edgeware/archive/think/main_filing3.html (3531 words)

	Nick Szabo -- Introduction to Algorithmic Information Theory
		Recent discoveries have unified the fields of computer science and information theory into the field of algorithmic information theory.
		If the algorithm does not generate the exact string, we can include error (called "distortion" in information theory) as part of the description of the data:
		An example of where this makes a difference is where the object to be described is a series of events in the environment, and we are observing these events with scientific observations with instruments.
	szabo.best.vwh.net /kolmogorov.html (1710 words)

	Complexity and Randomness
		Algorithmic complexity theory allows as rigorous definition of randomness as is possible.
		All non-computable strings are algorithmically random (Li and Vitànyi, 1990).
		According to the central theorem of algorithmic complexity (see here), for a formal system with complexity n, it is impossible to prove that the complexity of a sequence is higher than n+c, c being a constant depending on the implementation of the system.
	www.kli.ac.at /theorylab/jdc/information/randomness.html (693 words)

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