English
Noun
algorithms
 Plural of algorithm
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"

 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.
 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.
"It may be that some of these change necessarily
invoke a change of state of mind. The most general single operation
must therefore be taken to be one of the following:

 "(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
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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