The Cambridge mathematician Charles Babbage — conceived and designed the first machine that could in principle resolve all well-defined arithmetic problems. Nevertheless, he was unable to build a working prototype. Over a century later another Cambridge mathematician, Alan Turing — , laid the theoretical foundations for effectively implementing a universal algorithm. Turing proposed a very simple conceptual device involving a tape with a movable reader that could mark and erase letters on the tape.

Turing showed that all algorithms could be mapped onto the tape as data and then run by a universal algorithm already inscribed on the tape. This machine, known as a universal Turing machine , became the basis for the modern theory of computation known as recursion theory and inspired the modern digital computer. Turing's universal algorithm fell short of Lully's vision of an algorithm that could resolve all problems.

Turing's universal algorithm is not so much a universal problem solver as an empty box capable of housing and implementing the algorithms placed into it. Thus Turing invited into the theory of computing the very Cartesian distinction between hardware and software. Hardware is the mechanical device i.

Turing himself was fascinated with how the distinction between software and hardware illuminated immortality and the soul. Identifying personal identity with computer software ensured that humans were immortal, since even though hardware could be destroyed, software resided in a realm of mathematical abstraction and was thus immune to destruction.

It is a deep and much disputed question whether the essence of what constitutes the human person is at base computational and therefore an emergent property of algorithms, or whether it fundamentally transcends the capacity of algorithms. Cite this article Pick a style below, and copy the text for your bibliography.

September 22, Retrieved September 22, from Encyclopedia. Then, copy and paste the text into your bibliography or works cited list. Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia. Expressing an algorithm in a formal notation is one of the main parts of a program ; much that is said about programs applies to algorithms, and vice versa.

An effective algorithm is one that is effectively computable see effective computability. The study of whether effective algorithms exist to compute particular quantities forms the basis of the theory of algorithms.

### Properties of Algorithms

Save for the simplest of algorithms it is difficult to prove that an algorithm is correct see program correctness proof , or even to specify the effect it is intended to achieve. In practice it is usually necessary to be content with algorithm validation. This process certifies, or verifies, that an algorithm will perform the calculation required of it. It involves testing the routine against a variety of instances of the problem and ensuring that it performs satisfactorily for these test cases.

If the test set is chosen sufficiently well there can then be confidence in the algorithm. Algorithm analysis is the study of the performance characteristics of a given algorithm. One branch of this study, average-case analysis , examines the average behavior of the algorithm. Worst-case analysis studies the behavior when all circumstances are as unfavorable as possible.

Algorithms can be analyzed in terms of their complexity and efficiency, where algorithm efficiency is characterized by its order. Much of ordinary arithmetic as traditionally taught consists of algorithms involving the fundamental operations of addition, subtraction, multiplication, and division.

An example of an algorithm is the common procedure for division, e. The software that instructs modern computers embodies algorithms, often of great sophistication. In sociology, the term is generally used more loosely than this, to describe the steps which must be followed to construct a new variable from a set of other variables. A good example would be the algorithm used by Erik Olin Wright to arrive at his social class variable, by combining particular combinations of ownership and decision-making responsibilities, said to characterize the various class locations see his Classes , The term is also used in computer science for the method of a computer in following an established series of steps in the solution of a problem.

An algorithm is a set of instructions for accomplishing a task that can be couched in mathematical terms. If followed correctly, an algorithm guarantees successful completion of the task.

The term algorithm is derived from the name al-Khowarizmi, a ninth- century Arabian mathematician credited with discovering algebra. The latter studies algorithms from the point of view of the complexity both of the algorithms themselves cf. Algorithm, complexity of description of an and of the "computations" defined by the algorithms, i.

Algorithm, computational complexity of an. It is important to note that both the computational complexity of the algorithm and the complexity of description may be defined in various ways, and it may well be that will turn out to be more complex than when one definition is adopted, while will turn out to be more complex than according to another definition.

In order to be able to speak about the complexity of algorithms, one must first specify some exact language in which the algorithms are to be written, and define the complexity of an algorithm as the complexity of its notation; the complexity of notation can in turn be determined in various ways e. In order to speak about computational complexity, one must specify the exact form of the computation as a chain of constructive objects successively replacing each other, as well as some criterion of the complexity of such a chain — the number of "links" or steps in the chain alone or in combination with the "dimension" of the links, etc.

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## Encyclopedia of Algorithms

In any event, the computational complexity will depend on the input with which the computation is begun; for this reason, the computational complexity is a function which assigns to each object within the domain of the algorithm the complexity of the corresponding chain. The development of the methods of evaluation of the complexity of algorithms and of computations has a high theoretical and practical importance; however, as distinct from the descriptive theory of algorithms, which has now crystallized into an integral mathematical discipline [11] , [15] , [16] , the metric theory of algorithms is only in the process of being created [17] , [18] , [19] , [20].

These are encountered in all branches of mathematics in which algorithmic problems are encountered. Such problems may arise in mathematical logic and in model theory ; for each theory, the problem which arises is to solve the set of all true or demonstrable statements of this theory with respect to the set of all its statements the theories divide into solvable or unsolvable theories, depending on the solvability or unsolvability of the problem in question. Church [21] , [22] proved in that the problem of solving the set of all true assumptions of predicate logic was unsolvable; other important results on the subject are due to A.

Tarski, A. Mal'tsev and to others [23]. Unsolvable algorithmic problems are encountered in algebra the word identity problem for semi-groups and, in particular, for groups. The first examples of semi-groups with unsolvable word problem were found, independently of each other, by Markov [9] and by Post [8] in , while an example of a group with unsolvable word problem was found by P. Novikov [24] , [25] in Markov [26] in proved that the problem of homeomorphy in topology was unsolvable for an important class of cases.

In number theory, Yu. Matiyasevich proved in that the problem of solvability of Diophantine equations was unsolvable [27] , [28]. Similar examples can also be quoted from many other branches of mathematics. Finally, the theory of algorithms is closely connected with the foundations of mathematics, where one of the key problems is the relation between the constructive and the non-constructive.

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In particular, the theory of algorithms provides the apparatus for the development of the constructive direction in mathematics. It was suggested by Kolmogorov [30] in that the theory of algorithms be used as the foundation of information theory cf. Algorithmic information theory. The theory of algorithms is the theoretical foundation for a number of problems in computational mathematics, and is closely related to cybernetics, in which an important subject is the study of control algorithms. Since the above text was prepared also the theory of complexity of algorithms has matured into a well-developed branch of mathematics.

See also the comments to Algorithm, computational complexity of an. In addition to [27] , [28] one may consult [a1] , [a2]. A thorough treatise on computational complexity is [a3]. Log in.

## Encyclopedia of Algorithms - Google книги

Namespaces Page Discussion. Views View View source History. Jump to: navigation , search. How to Cite This Entry: Algorithms, theory of. Uspenskii originator , Encyclopedia of Mathematics. This page was last modified on 7 February , at Church, "An unsolvable problem of elementary number theory" Amer. Turing, "On computable numbers, with an application to the Entscheidungsproblem" Proc.

London Math. Turing, "On computable numbers with an application to the Entscheidungsproblem, a correction" Proc. Post, "Finite combinatory processes - formulation 1" J. Post, "Formal reductions of the general combinatorial decision problem" Amer. Post, "Recursively enumerable sets of positive integers and their decision problems" Bull. Post, "Recursive unsolvability of a problem of Thue" J. Markov, Dokl. Markov, "Theory of algorithms" Trudy Mat. Markov, "Theory of algorithms" , Israel Program Sci.

Kolmogorov, "On the concept of an algorithm" Uspekhi Mat. Kolmogorov, V. Uspenskii, "On the definition of an algorithm" Uspekhi Mat. Verlag Wissenschaft. Rogers jr. Markov, "Normal algorithms connected with the computation of Boolean functions" Izv.