Dictionary Definition
algorithm n : a precise rule (or set of rules)
specifying how to solve some problem [syn: algorithmic
rule, algorithmic
program]
User Contributed Dictionary
English
Etymology
From algorithme; from the algorisme ("the Arabic numeral
system"), a modification likely due to a mistaken connection with
Greek
ἀριθμός (number);
from Medieval Latin algorismus, a mangled
transliteration of the name of the Islamic mathematician
alKhwārizmī (Arabic: , "native of Khwarezm.")
Alternative spellings
 algorism (obsolete)
Pronunciation
 /ˈælgəɹɪðm/, /"
Extensive Definition
In mathematics, computing, linguistics and related
disciplines, an algorithm is a sequence of instructions, often used
for calculation,
data
processing. It is formally a type of effective
method in which a list of welldefined instructions for
completing a task will, when given an initial state, proceed
through a welldefined series of successive states, eventually
terminating in an endstate. The transition from one state to the
next is not necessarily deterministic; some
algorithms, known as probabilistic
algorithms, incorporate randomness.
A partial formalization of the
concept began with attempts to solve the Entscheidungsproblem
(the "decision problem") posed by David
Hilbert in 1928. Subsequent formalizations were framed as
attempts to define "effective
calculability" (Kleene 1943:274) or "effective method" (Rosser
1939:225); those formalizations included the GödelHerbrandKleene
recursive functions of 1930, 1934 and 1935, Alonzo
Church's lambda
calculus of 1936, Emil Post's
"Formulation I" of 1936, and Alan Turing's
Turing
machines of 19367 and 1939.
Etymology
AlKhwārizmī, Persian astronomer and mathematician, wrote a treatise in Arabic in 825 AD, On Calculation with Hindu Numerals. (See algorism). It was translated into Latin in the 12th century as Algoritmi de numero Indorum (alDaffa 1977), which title was likely intended to mean "[Book by] Algoritmus on the numbers of the Indians", where "Algoritmi" was the translator's rendition of the author's name in the genitive case; but people misunderstanding the title treated Algoritmi as a Latin plural and this led to the word "algorithm" (Latin algorismus) coming to mean "calculation method". The intrusive "th" is most likely due to a false cognate with the Greek αριθμος (arithmos) meaning "number".Why algorithms are necessary: an informal definition
No generally accepted formal
definition of "algorithm" exists yet. We can, however, derive clues
to the issues involved and an informal meaning of the word from the
following quotation from (boldface added):
No human being can write fast
enough, or long enough, or small enough to list all members of an
enumerably infinite set by writing out their names, one after
another, in some notation. But humans can do something equally
useful, in the case of certain enumerably infinite sets: They can
give explicit instructions for determining the nth member of the
set, for arbitrary finite n. Such instructions are to be given
quite explicitly, in a form in which they could be followed by a
computing machine, or by a human who is capable of carrying out
only very elementary operations on symbols
The words "enumerably
infinite" mean "countable using integers perhaps extending to
infinity". Thus Boolos and Jeffrey are saying that an algorithm
implies instructions for a process that "creates" output integers
from an arbitrary "input" integer or integers that, in theory, can
be chosen from 0 to infinity. Thus we might expect an algorithm to
be an algebraic equation such as y = m + n — two arbitrary "input
variables" m and n that produce an output y. As we see in Algorithm
characterizations — the word algorithm implies much more than
this, something on the order of (for our addition example):
 Precise instructions (in language understood by "the computer") for a "fast, efficient, good" process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internallycontained information and capabilities) to find, decode, and then munch arbitrary input integers/symbols m and n, symbols + and = ... and (reliably, correctly, "effectively") produce, in a "reasonable" time, outputinteger y at a specified place and in a specified format.
The concept of algorithm is
also used to define the notion of decidability.
That notion is central for explaining how formal
systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm
requires to complete cannot be measured, as it is not apparently
related with our customary physical dimension. From such
uncertainties, that characterize ongoing work, stems the
unavailability of a definition of algorithm that suits both
concrete (in some sense) and abstract usage of the
term.
 ''For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.''
Formalization of algorithms
Algorithms are essential to the way computers process information, because a computer program is essentially an algorithm that tells the computer what specific steps to perform (in what specific order) in order to carry out a specified task, such as calculating employees’ paychecks or printing students’ report cards. Thus, an algorithm can be considered to be any sequence of operations that can be performed by a Turingcomplete system. Authors who assert this thesis include Savage (1987) and Gurevich (2000):...Turing's informal argument
in favor of his thesis justifies a stronger thesis: every algorithm
can be simulated by a Turing machine (Gurevich 2000:1)...according
to Savage [1987], an algorithm is a computational process defined
by a Turing machine. (Gurevich 2000:3)
Typically, when an algorithm
is associated with processing information, data are read from an
input source or device, written to an output sink or device, and/or
stored for further processing. Stored data are regarded as part of
the internal state of the entity performing the algorithm. In
practice, the state is stored in a data
structure, but an algorithm requires the internal data only for
specific operation sets called abstract
data types.
For any such computational
process, the algorithm must be rigorously defined: specified in the
way it applies in all possible circumstances that could arise. That
is, any conditional steps must be systematically dealt with,
casebycase; the criteria for each case must be clear (and
computable).
Because an algorithm is a
precise list of precise steps, the order of computation will almost
always be critical to the functioning of the algorithm.
Instructions are usually assumed to be listed explicitly, and are
described as starting "from the top" and going "down to the
bottom", an idea that is described more formally by flow of
control.
So far, this discussion of the
formalization of an algorithm has assumed the premises of imperative
programming. This is the most common conception, and it
attempts to describe a task in discrete, "mechanical" means. Unique
to this conception of formalized algorithms is the assignment
operation, setting the value of a variable. It derives from the
intuition of "memory" as
a scratchpad. There is an example below of such an
assignment.
For some alternate conceptions
of what constitutes an algorithm see functional
programming and logic
programming .
Termination
Some writers restrict the definition of algorithm to procedures that eventually finish. In such a category Kleene places the "decision procedure or decision method or algorithm for the question" (Kleene 1952:136). Others, including Kleene, include procedures that could run forever without stopping; such a procedure has been called a "computational method" (Knuth 1997:5) or "calculation procedure or algorithm" (Kleene 1952:137); however, Kleene notes that such a method must eventually exhibit "some object" (Kleene 1952:137).Minsky makes the pertinent
observation, in regards to determining whether an algorithm will
eventually terminate (from a particular starting state): But if the
length of the process is not known in advance, then "trying" it may
not be decisive, because if the process does go on forever — then
at no time will we ever be sure of the answer (Minsky
1967:105).
As it happens, no other method
can do any better, as was shown by Alan Turing
with his celebrated result on the undecidability of the socalled
halting
problem. There is no algorithmic procedure for determining of
arbitrary algorithms whether or not they terminate from given
starting states. The analysis of algorithms for their likelihood of
termination is called termination
analysis.
See the examples of
(im)"proper" subtraction at partial
function for more about what can happen when an algorithm fails
for certain of its input numbers — e.g., (i) nontermination, (ii)
production of "junk" (output in the wrong format to be considered a
number) or no number(s) at all (halt ends the computation with no
output), (iii) wrong number(s), or (iv) a combination of these.
Kleene proposed that the production of "junk" or failure to produce
a number is solved by having the algorithm detect these instances
and produce e.g., an error message (he suggested "0"), or
preferably, force the algorithm into an endless loop (Kleene
1952:322). Davis does this to his subtraction algorithm — he fixes
his algorithm in a second example so that it is proper subtraction
(Davis 1958:1215). Along with the logical outcomes "true" and
"false" Kleene also proposes the use of a third logical symbol "u"
— undecided (Kleene 1952:326) — thus an algorithm will always
produce something when confronted with a "proposition". The problem
of wrong answers must be solved with an independent "proof" of the
algorithm e.g., using induction: We normally require auxiliary
evidence for this (that the algorithm correctly defines a mu
recursive function), e.g., in the form of an inductive proof
that, for each argument value, the computation terminates with a
unique value (Minsky 1967:186).
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, and programming languages. Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode and flowcharts are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements, while remaining independent of a particular implementation language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.There is a wide variety of
representations possible and one can express a given Turing
machine program as a sequence of machine tables (see more at
finite
state machine and state
transition table), as flowcharts (see more at state
diagram), or as a form of rudimentary machine code
or assembly
code called "sets of quadruples" (see more at Turing
machine).
Sometimes it is helpful in the
description of an algorithm to supplement small "flow charts"
(state diagrams) with naturallanguage and/or arithmetic
expressions written inside "block
diagrams" to summarize what the "flow charts" are
accomplishing.
Representations of algorithms
are generally classed into three accepted levels of Turing machine
description (Sipser 2006:157):
 1 Highlevel description:

 "...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head"
 2 Implementation description:

 "...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function"
 3 Formal description:

 Most detailed, "lowest level", gives the Turing machine's "state table".
 For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.
Implementation
Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.Example
One of the simplest algorithms is to find the largest number in an (unsorted) list of numbers. The solution necessarily requires looking at every number in the list, but only once at each. From this follows a simple algorithm, which can be stated in a highlevel description English prose, as:Highlevel description:
 Assume the first item is largest.
 Look at each of the remaining items in the list and if it is larger than the largest item so far, make a note of it.
 The last noted item is the largest in the list when the process is complete.
(Quasi)formal description:
Written in prose but much closer to the highlevel language of a
computer program, the following is the more formal coding of the
algorithm in pseudocode or pidgin
code:
Input: A nonempty list of
numbers L. Output: The largest number in the list L. largest ← L0
for each item in the list L≥1, do if the item > largest, then
largest ← the item return largest For a more complex example of an
algorithm, see Euclid's
algorithm for the greatest
common divisor, one of the earliest algorithms
known.
Algorithm analysis
As it happens, it is important to know how much of a particular resource (such as time or storage) is required for a given algorithm. Methods have been developed for the analysis of algorithms to obtain such quantitative answers; for example, the algorithm above has a time requirement of O(n), using the big O notation with n as the length of the list. At all times the algorithm only needs to remember two values: the largest number found so far, and its current position in the input list. Therefore it is said to have a space requirement of O(1), if the space required to store the input numbers is not counted, or O (log n) if it is counted.Different algorithms may
complete the same task with a different set of instructions in less
or more time, space, or effort than others. For example, given two
different recipes for making potato salad, one may have peel the
potato before boil the potato while the other presents the steps in
the reverse order, yet they both call for these steps to be
repeated for all potatoes and end when the potato salad is ready to
be eaten.
The analysis
and study of algorithms is a discipline of computer
science, and is often practiced abstractly without the use of a
specific programming
language or implementation. In this sense, algorithm analysis
resembles other mathematical disciplines in that it focuses on the
underlying properties of the algorithm and not on the specifics of
any particular implementation. Usually pseudocode is used for
analysis as it is the simplest and most general
representation.
Classes
There are various ways to classify algorithms, each with its own merits.Classification by implementation
One way to classify algorithms is by implementation means. Recursion or iteration: A recursive algorithm is one that invokes (makes reference to) itself repeatedly until a certain condition matches, which is a method common to functional programming. Iterative algorithms use repetitive constructs like loops and sometimes additional data structures like stacks to solve the given problems. Some problems are naturally suited for one implementation or the other. For example, towers of hanoi is well understood in recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.
 Logical: An algorithm may be viewed as controlled logical deduction. This notion may be expressed as: Algorithm = logic + control (Kowalski 1979). The logic component expresses the axioms that may be used in the computation and the control component determines the way in which deduction is applied to the axioms. This is the basis for the logic programming paradigm. In pure logic programming languages the control component is fixed and algorithms are specified by supplying only the logic component. The appeal of this approach is the elegant semantics: a change in the axioms has a well defined change in the algorithm.
 Serial or parallel or distributed: Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time. Those computers are sometimes called serial computers. An algorithm designed for such an environment is called a serial algorithm, as opposed to parallel algorithms or distributed algorithms. Parallel algorithms take advantage of computer architectures where several processors can work on a problem at the same time, whereas distributed algorithms utilize multiple machines connected with a network. Parallel or distributed algorithms divide the problem into more symmetrical or asymmetrical subproblems and collect the results back together. The resource consumption in such algorithms is not only processor cycles on each processor but also the communication overhead between the processors. Sorting algorithms can be parallelized efficiently, but their communication overhead is expensive. Iterative algorithms are generally parallelizable. Some problems have no parallel algorithms, and are called inherently serial problems.
 Deterministic or nondeterministic: Deterministic algorithms solve the problem with exact decision at every step of the algorithm whereas nondeterministic algorithm solve problems via guessing although typical guesses are made more accurate through the use of heuristics.
 Exact or approximate: While many algorithms reach an exact solution, approximation algorithms seek an approximation that is close to the true solution. Approximation may use either a deterministic or a random strategy. Such algorithms have practical value for many hard problems.
Classification by design paradigm
Another way of classifying algorithms is by their design methodology or paradigm. There is a certain number of paradigms, each different from the other. Furthermore, each of these categories will include many different types of algorithms. Some commonly found paradigms include: Divide and conquer. A divide and conquer algorithm repeatedly reduces an instance of a problem to one or more smaller instances of the same problem (usually recursively), until the instances are small enough to solve easily. One such example of divide and conquer is merge sorting. Sorting can be done on each segment of data after dividing data into segments and sorting of entire data can be obtained in conquer phase by merging them. A simpler variant of divide and conquer is called decrease and conquer algorithm, that solves an identical subproblem and uses the solution of this subproblem to solve the bigger problem. Divide and conquer divides the problem into multiple subproblems and so conquer stage will be more complex than decrease and conquer algorithms. An example of decrease and conquer algorithm is binary search algorithm.
 Dynamic programming. When a problem shows optimal substructure, meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems, and overlapping subproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called dynamic programming avoids recomputing solutions that have already been computed. For example, the shortest path to a goal from a vertex in a weighted graph can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and memoization go together. The main difference between dynamic programming and divide and conquer is that subproblems are more or less independent in divide and conquer, whereas subproblems overlap in dynamic programming. The difference between dynamic programming and straightforward recursion is in caching or memoization of recursive calls. When subproblems are independent and there is no repetition, memoization does not help; hence dynamic programming is not a solution for all complex problems. By using memoization or maintaining a table of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
 The greedy method. A greedy algorithm is similar to a dynamic programming algorithm, but the difference is that solutions to the subproblems do not have to be known at each stage; instead a "greedy" choice can be made of what looks best for the moment. The greedy method extends the solution with the best possible decision (not all feasible decisions) at an algorithmic stage based on the current local optimum and the best decision (not all possible decisions) made in previous stage. It is not exhaustive, and does not give accurate answer to many problems. But when it works, it will be the fastest method. The most popular greedy algorithm is finding the minimal spanning tree as given by Kruskal.
 Linear programming. When solving a problem using linear programming, specific inequalities involving the inputs are found and then an attempt is made to maximize (or minimize) some linear function of the inputs. Many problems (such as the maximum flow for directed graphs) can be stated in a linear programming way, and then be solved by a 'generic' algorithm such as the simplex algorithm. A more complex variant of linear programming is called integer programming, where the solution space is restricted to the integers.
 Reduction. This technique involves solving a difficult problem by transforming it into a better known problem for which we have (hopefully) asymptotically optimal algorithms. The goal is to find a reducing algorithm whose complexity is not dominated by the resulting reduced algorithm's. For example, one selection algorithm for finding the median in an unsorted list involves first sorting the list (the expensive portion) and then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as transform and conquer.
 Search and enumeration. Many problems (such as playing chess) can be modeled as problems on graphs. A graph exploration algorithm specifies rules for moving around a graph and is useful for such problems. This category also includes search algorithms, branch and bound enumeration and backtracking.
 The probabilistic and heuristic paradigm. Algorithms belonging to this class fit the definition of an algorithm more loosely.
 Probabilistic algorithms are those that make some choices randomly (or pseudorandomly); for some problems, it can in fact be proven that the fastest solutions must involve some randomness.
 Genetic algorithms attempt to find solutions to problems by mimicking biological evolutionary processes, with a cycle of random mutations yielding successive generations of "solutions". Thus, they emulate reproduction and "survival of the fittest". In genetic programming, this approach is extended to algorithms, by regarding the algorithm itself as a "solution" to a problem.
 Heuristic algorithms, whose general purpose is not to find an optimal solution, but an approximate solution where the time or resources are limited. They are not practical to find perfect solutions. An example of this would be local search, tabu search, or simulated annealing algorithms, a class of heuristic probabilistic algorithms that vary the solution of a problem by a random amount. The name "simulated annealing" alludes to the metallurgic term meaning the heating and cooling of metal to achieve freedom from defects. The purpose of the random variance is to find close to globally optimal solutions rather than simply locally optimal ones, the idea being that the random element will be decreased as the algorithm settles down to a solution.
Classification by field of study
Every field of science has its own problems and needs efficient algorithms. Related problems in one field are often studied together. Some example classes are search algorithms, sorting algorithms, merge algorithms, numerical algorithms, graph algorithms, string algorithms, computational geometric algorithms, combinatorial algorithms, machine learning, cryptography, data compression algorithms and parsing techniques.Fields tend to overlap with
each other, and algorithm advances in one field may improve those
of other, sometimes completely unrelated, fields. For example,
dynamic programming was originally invented for optimization of
resource consumption in industry, but is now used in solving a
broad range of problems in many fields.
Classification by complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size. There is a wide variety: some algorithms complete in linear time relative to input size, some do so in an exponential amount of time or even worse, and some never halt. Additionally, some problems may have multiple algorithms of differing complexity, while other problems might have no algorithms or no known efficient algorithms. There are also mappings from some problems to other problems. Owing to this, it was found to be more suitable to classify the problems themselves instead of the algorithms into equivalence classes based on the complexity of the best possible algorithms for them.Classification by computing power
Another way to classify
algorithms is by computing power. This is typically done by
considering some collection (class) of algorithms. A recursive
class of algorithms is one that includes algorithms for all Turing
computable functions. Looking at classes of algorithms allows for
the possibility of restricting the available computational
resources (time and memory) used in a computation. A subrecursive
class of algorithms is one in which not all Turing computable
functions can be obtained. For example, the algorithms that run in
polynomial
time suffice for many important types of computation but do not
exhaust all Turing computable functions. The class algorithms
implemented by
primitive recursive functions is another subrecursive
class.
Burgin (2005, p. 24) uses a
generalized definition of algorithms that relaxes the common
requirement that the output of the algorithm that computes a
function must be determined after a finite number of steps. He
defines a superrecursive class of algorithms as "a class of
algorithms in which it is possible to compute functions not
computable by any Turing machine" (Burgin 2005, p. 107). This is
closely related to the study of methods of hypercomputation.
Legal issues
 See also: Software patents for a general overview of the patentability of software, including computerimplemented algorithms.
Algorithms, by themselves, are
not usually patentable. In the United
States, a claim consisting solely of simple manipulations of
abstract concepts, numbers, or signals do not constitute
"processes" (USPTO 2006) and hence algorithms are not patentable
(as in Gottschalk
v. Benson). However, practical applications of algorithms are
sometimes patentable. For example, in Diamond v.
Diehr, the application of a simple feedback algorithm to aid in
the curing of synthetic
rubber was deemed patentable. The patenting
of software is highly controversial, and there are highly
criticized patents involving algorithms, especially data
compression algorithms, such as Unisys'
LZW patent.
Additionally, some
cryptographic algorithms have export restrictions (see export
of cryptography).
History: Development of the notion of "algorithm"
Origin of the word
The word algorithm comes from
the name of the 9th century Persian
mathematician Abu Abdullah
Muhammad ibn Musa alKhwarizmi whose works introduced Indian
numerals and algebraic concepts. He worked in Baghdad at the time
when it was the centre of scientific studies and trade. The word
algorism originally
referred only to the rules of performing arithmetic using
Arabic numerals but evolved via European Latin translation of
alKhwarizmi's name into algorithm by the 18th century. The word
evolved to include all definite procedures for solving problems or
performing tasks.
Discrete and distinguishable symbols
Tallymarks: To keep track of
their flocks, their sacks of grain and their money the ancients
used tallying: accumulating stones or marks scratched on sticks, or
making discrete symbols in clay. Through the Babylonian and
Egyptian use of marks and symbols, eventually Roman
numerals and the abacus evolved (Dilson, p.16–41).
Tally marks appear prominently in unary
numeral system arithmetic used in Turing
machine and PostTuring
machine computations.
Manipulation of symbols as "place holders" for numbers: algebra
The work of the ancient Greek geometers, Persian mathematician AlKhwarizmi (often considered as the "father of algebra"), and Western European mathematicians culminated in Leibniz's notion of the calculus ratiocinator (ca 1680): "A good century and a half ahead of his time, Leibniz proposed an algebra of logic, an algebra that would specify the rules for manipulating logical concepts in the manner that ordinary algebra specifies the rules for manipulating numbers" (Davis 2000:1)
Mechanical contrivances with discrete states
The clock: Bolter credits the
invention of the weightdriven clock as “The key invention [of
Europe in the Middle Ages]", in particular the verge
escapement< (Bolter 1984:24) that provides us with the tick
and tock of a mechanical clock. “The accurate automatic machine”
(Bolter 1984:26) led immediately to "mechanical automata" beginning in the
thirteenth century and finally to “computational machines" – the
difference
engine and analytical
engines of Charles
Babbage and Countess Ada Lovelace
(Bolter p.33–34, p.204–206).
Jacquard loom, Hollerith punch
cards, telegraphy and telephony — the electromechanical relay: Bell
and Newell (1971) indicate that the Jacquard
loom (1801), precursor to Hollerith
cards (punch cards, 1887), and “telephone switching
technologies” were the roots of a tree leading to the development
of the first computers (Bell and Newell diagram p. 39, cf Davis
2000). By the mid1800s the telegraph, the precursor of
the telephone, was in use throughout the world, its discrete and
distinguishable encoding of letters as “dots and dashes” a common
sound. By the late 1800s the ticker tape
(ca 1870s) was in use, as was the use of Hollerith
cards in the 1890 U.S. census. Then came the Teletype (ca 1910)
with its punchedpaper use of Baudot code
on tape.
Telephoneswitching networks
of electromechanical relays (invented 1835) was behind
the work of George
Stibitz (1937), the inventor of the digital adding device. As
he worked in Bell Laboratories, he observed the “burdensome’ use of
mechanical calculators with gears. "He went home one evening in
1937 intending to test his idea.... When the tinkering was over,
Stibitz had constructed a binary adding device". (Valley News, p.
13).
Davis (2000) observes the
particular importance of the electromechanical relay (with its two
"binary states" open and closed):
 It was only with the development, beginning in the 1930s, of electromechanical calculators using electrical relays, that machines were built having the scope Babbage had envisioned." (Davis, p. 14).
Mathematics during the 1800s up to the mid1900s
Symbols and rules: In rapid succession the mathematics of George Boole (1847, 1854), Gottlob Frege (1879), and Giuseppe Peano (1888–1889) reduced arithmetic to a sequence of symbols manipulated by rules. Peano's The principles of arithmetic, presented by a new method (1888) was "the first attempt at an axiomatization of mathematics in a symbolic language" (van Heijenoort:81ff).But Heijenoort gives Frege
(1879) this kudos: Frege’s is "perhaps the most important single
work ever written in logic. ... in which we see a " 'formula
language', that is a lingua characterica, a language written with
special symbols, "for pure thought", that is, free from rhetorical
embellishments ... constructed from specific symbols that are
manipulated according to definite rules" (van Heijenoort:1). The
work of Frege was further simplified and amplified by Alfred
North Whitehead and Bertrand
Russell in their Principia
Mathematica (1910–1913).
The paradoxes: At the same
time a number of disturbing paradoxes appeared in the literature,
in particular the BuraliForti
paradox (1897), the Russell
paradox (1902–03), and the Richard
Paradox (Dixon 1906, cf Kleene 1952:36–40). The resultant
considerations led to Kurt
Gödel’s paper (1931) — he specifically cites the paradox of the
liar — that completely reduces rules of recursion to
numbers.
Effective calculability: In an
effort to solve the Entscheidungsproblem
defined precisely by Hilbert in 1928, mathematicians first set
about to define what was meant by an "effective method" or
"effective calculation" or "effective calculability" (i.e., a
calculation that would succeed). In rapid succession the following
appeared: Alonzo
Church, Stephen
Kleene and J.B. Rosser's
λcalculus,
(cf footnote in Alonzo
Church 1936a:90, 1936b:110) a finelyhoned definition of
"general recursion" from the work of Gödel acting on suggestions of
Jacques
Herbrand (cf Gödel's Princeton lectures of 1934) and subsequent
simplifications by Kleene (19356:237ff, 1943:255ff). Church's
proof (1936:88ff) that the Entscheidungsproblem
was unsolvable, Emil Post's
definition of effective calculability as a worker mindlessly
following a list of instructions to move left or right through a
sequence of rooms and while there either mark or erase a paper or
observe the paper and make a yesno decision about the next
instruction (cf "Formulation I", Post 1936:289290). Alan Turing's
proof of that the Entscheidungsproblem was unsolvable by use of his
"a [automatic] machine"(Turing 19367:116ff)  in effect almost
identical to Post's "formulation", J.
Barkley Rosser's definition of "effective method" in terms of
"a machine" (Rosser 1939:226). S. C.
Kleene's proposal of a precursor to "Church
thesis" that he called "Thesis I" (Kleene 1943:273–274), and a
few years later Kleene's renaming his Thesis "Church's Thesis"
(Kleene 1952:300, 317) and proposing "Turing's Thesis" (Kleene
1952:376).
Emil Post (1936) and Alan Turing (19367, 1939)
Here is a remarkable
coincidence of two men not knowing each other but describing a
process of menascomputers working on computations — and they
yield virtually identical definitions.
Emil Post
(1936) described the actions of a "computer" (human being) as
follows:
 "...two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions.
His symbol space would
be
 "a two way infinite sequence of spaces or boxes... The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time.... a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.
 "One box is to be singled out and called the starting point. ...a specific problem is to be given in symbolic form by a finite number of boxes [i.e., INPUT] being marked with a stroke. Likewise the answer [i.e., OUTPUT] is to be given in symbolic form by such a configuration of marked boxes....
 "A set of directions applicable to a general problem sets up a deterministic process when applied to each specific problem. This process will terminate only when it comes to the direction of type (C ) [i.e., STOP]." (U p. 289–290) See more at PostTuring machine
Alan Turing’s
work (1936, 1939:160) preceded that of Stibitz (1937); it is
unknown whether Stibitz knew of the work of Turing. Turing’s
biographer believed that Turing’s use of a typewriterlike model
derived from a youthful interest: “Alan had dreamt of inventing
typewriters as a boy; Mrs. Turing had a typewriter; and he could
well have begun by asking himself what was meant by calling a
typewriter 'mechanical'" (Hodges, p. 96). Given the prevalence of
Morse code and telegraphy, ticker tape machines, and Teletypes we
might conjecture that all were influences.
Turing — his model of
computation is now called a Turing
machine — begins, as did Post, with an analysis of a human
computer that he whittles down to a simple set of basic motions and
"states of mind". But he continues a step further and creates a
machine as a model of computation of numbers (Turing
19367:116).
 "Computing is normally done by writing certain symbols on paper. We may suppose this paper is divided into squares like a child's arithmetic book....I assume then that the computation is carried out on onedimensional paper, i.e., on a tape divided into squares. I shall also suppose that the number of symbols which may be printed is finite....
 "The behavior of the computer at any moment is determined by the symbols which he is observing, and his "state of mind" at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at one moment. If he wishes to observe more, he must use successive observations. We will also suppose that the number of states of mind which need be taken into account is finite...
 "Let us imagine that the operations performed by the computer to be split up into 'simple operations' which are so elementary that it is not easy to imagine them further divided" (Turing 19367:136).
Turing's reduction yields the
following:
 "The simple operations must therefore include:

 "(a) Changes of the symbol on one of the observed squares
 "(b) Changes of one of the squares observed to another square within L squares of one of the previously observed squares.

 "(A) A possible change (a) of symbol together with a possible change of state of mind.
 "(B) A possible change (b) of observed squares, together with a possible change of state of mind"
 "We may now construct a machine to do the work of this computer." (Turing 19367:136)
A few years later, Turing
expanded his analysis (thesis, definition) with this forceful
expression of it:
 "A function is said to be "effectively calculable" if its values can be found by some purely mechanical process. Although it is fairly easy to get an intuitive grasp of this idea, it is neverthessless desirable to have some more definite, mathematical expressible definition . . . [he discusses the history of the definition pretty much as presented above with respect to Gödel, Herbrand, Kleene, Church, Turing and Post] . . . We may take this statement literally, understanding by a purely mechanical process one which could be carried out by a machine. It is possible to give a mathematical description, in a certain normal form, of the structures of these machines. The development of these ideas leads to the author's definition of a computable function, and to an identification of computability † with effective calculability . . . .

 "† We shall use the expression "computable function" to mean a function calculable by a machine, and we let "effectively calculabile" refer to the intuitive idea without particular identification with any one of these definitions."(Turing 1939:160)
J. B. Rosser (1939) and S. C. Kleene (1943)
J.
Barkley Rosser boldly defined an ‘effective [mathematical]
method’ in the following manner (boldface added):
 "'Effective method' is used here in the rather special sense of a method each step of which is precisely determined and which is certain to produce the answer in a finite number of steps. With this special meaning, three different precise definitions have been given to date. [his footnote #5; see discussion immediately below]. The simplest of these to state (due to Post and Turing) says essentially that an effective method of solving certain sets of problems exists if one can build a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer. All three definitions are equivalent, so it doesn't matter which one is used. Moreover, the fact that all three are equivalent is a very strong argument for the correctness of any one." (Rosser 1939:225–6)
Rosser's footnote #5
references the work of (1) Church and Kleene and their definition
of λdefinability, in particular Church's use of it in his An
Unsolvable Problem of Elementary Number Theory (1936); (2) Herbrand
and Gödel and their use of recursion in particular Gödel's use in
his famous paper On Formally Undecidable Propositions of Principia
Mathematica and Related Systems I (1931); and (3) Post (1936) and
Turing (19367) in their mechanismmodels of
computation.
Stephen
C. Kleene defined as his nowfamous "Thesis I" known as "the
ChurchTuring
Thesis". But he did this in the following context (boldface in
original):
 "12. Algorithmic theories... In setting up a complete algorithmic theory, what we do is to describe a procedure, performable for each set of values of the independent variables, which procedure necessarily terminates and in such manner that from the outcome we can read a definite answer, "yes" or "no," to the question, "is the predicate value true?”"(Kleene 1943:273)
History after 1950
A number of efforts have been directed toward further refinement of the definition of "algorithm", and activity is ongoing because of issues surrounding, in particular, foundations of mathematics (especially the ChurchTuring Thesis) and philosophy of mind (especially arguments around artificial intelligence). For more, see Algorithm characterizations.See also
wikibooks Algorithms Abstract machine
 Algorithm characterizations
 Algorithm examples
 Algorithmic music
 Algorithmic trading
 Computability theory (computer science)
 Computational complexity theory
 Data structure
 Heuristics
 Introduction to Algorithms
 Important algorithmrelated publications
 List of algorithms
 List of algorithm general topics
 List of terms relating to algorithms and data structures
 Partial function
 Runtime analysis
 Theory of computation
References
 Axt, P. (1959) On a Subrecursive Hierarchy and Primitive Recursive Degrees, Transactions of the American Mathematical Society 92, pp. 85105
 Algorithms:">http://research.microsoft.com/~gurevich/Opera/164.pdf. Includes an excellent bibliography of 56 references.
 Computability and Logic: cf. Chapter 3 Turing machines where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
 Burgin, M. Superrecursive algorithms, Monographs in computer science, Springer, 2005. ISBN 0387955690
 Campagnolo, M.L., Moore, C., and Costa, J.F. (2000) An analog characterization of the subrecursive functions. In Proc. of the 4th Conference on Real Numbers and Computers, Odense University, pp. 91109
 Reprinted in The Undecidable, p. 89ff. The first expression of "Church's Thesis". See in particular page 100 (The Undecidable) where he defines the notion of "effective calculability" in terms of "an algorithm", and he uses the word "terminates", etc.
 Reprinted in The Undecidable, p. 110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.
 The Muslim contribution to mathematics
 The Undecidable: Basic Papers On Undecidable Propostions, Unsolvable Problems and Computable Functions Davis gives commentary before each article. Papers of Gödel, Alonzo Church, Turing, Rosser, Kleene, and Emil Post are included; those cited in the article are listed here by author's name.
 Engines of Logic: Mathematicians and the Origin of the Computer Davis offers concise biographies of Leibniz, Boole, Frege, Cantor, Hilbert, Gödel and Turing with von Neumann as the showstealing villain. Very brief bios of JosephMarie Jacquard, Babbage, Ada Lovelace, Claude Shannon, Howard Aiken, etc.
 Darwin's Dangerous Idea
 Yuri Gurevich, Sequential Abstract State Machines Capture Sequential Algorithms, ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pages 77–111. Includes bibliography of 33 sources.
 Presented to the American Mathematical Society, September 1935. Reprinted in The Undecidable, p. 237ff. Kleene's definition of "general recursion" (known now as murecursion) was used by Church in his 1935 paper An Unsolvable Problem of Elementary Number Theory that proved the "decision problem" to be "undecidable" (i.e., a negative result).
 Reprinted in The Undecidable, p. 255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p. 274); he would later repeat this thesis (in Kleene 1952:300) and name it "Church's Thesis"(Kleene 1952:317) (i.e., the Church Thesis).
 Introduction to Metamathematics Excellent — accessible, readable — reference source for mathematical "foundations".
 Fundamental Algorithms, Third Edition
 Kosovsky, N. K. Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms, LSU Publ., Leningrad, 1981
 A. A. Markov (1954) Theory of algorithms. [Translated by Jacques J. SchorrKon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p. 28 cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v. 42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS 6051085.]
 Computation: Finite and Infinite Machines Minsky expands his "...idea of an algorithm — an effective procedure..." in chapter 5.1 ''Computability, Effective Procedues and Algorithms. Infinite machines."
 Reprinted in The Undecidable, p. 289ff. Post defines a simple algorithmiclike process of a man writing marks or erasing marks and going from box to box and eventually halting, as he follows a list of simple instructions. This is cited by Kleene as one source of his "Thesis I", the socalled ChurchTuring thesis.
 Reprinted in The Undecidable, p. 223ff. Herein is Rosser's famous definition of "effective method": "...a method each step of which is precisely predetermined and which is certain to produce the answer in a finite number of steps... a machine which will then solve any problem of the set with no human intervention beyond inserting the question and (later) reading the answer" (p. 225–226, The Undecidable)
 Introduction to the Theory of Computation
 Introduction to Computer Organization and Data Structures Cf in particular the first chapter titled: Algorithms, Turing Machines, and Programs. His succinct informal definition: "...any sequence of instructions that can be obeyed by a robot, is called an algorithm" (p. 4).
 . Corrections, ibid, vol. 43(1937) pp.544546. Reprinted in The Undecidable, p. 116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.
 Reprinted in The Undecidable, p. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton USA.
 United States Patent and Trademark Office (2006), 2106.02 **>Mathematical Algorithms<  2100 Patentability, Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
Secondary references
 Turing's Man: Western Culture in the Computer Age, ISBN 0807841080 pbk.
 The Abacus, ISBN 031210409X (pbk.)
 From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, 3rd edition 1976[?], ISBN 0674324498 (pbk.)
 Alan Turing: The Enigma, ISBN 0671492071. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
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Synonyms, Antonyms and Related Words
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