In
mathematics and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, an algorithm () is a finite sequence of
rigorous instructions, typically used to solve a class of specific
problem
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
s or to perform a
computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, historically, people) that perform computations are known as '' computers''. An esp ...
. Algorithms are used as specifications for performing
calculation
A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to t ...
s and
data processing
Data processing is the collection and manipulation of digital data to produce meaningful information.
Data processing is a form of '' information processing'', which is the modification (processing) of information in any manner detectable by ...
. More advanced algorithms can perform automated deductions (referred to as
automated reasoning
In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer progr ...
) and use mathematical and logical tests to divert the code execution through various routes (referred to as
automated decision-making
Automated decision-making (ADM) involves the use of data, machines and algorithms to make decisions in a range of contexts, including public administration, business, health, education, law, employment, transport, media and entertainment, with var ...
). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical c ...
with terms such as "memory", "search" and "stimulus".
In contrast, a
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...
is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result.
As an
effective method
In logic, mathematics and computer science, especially metalogic and computability theory, an effective method Hunter, Geoffrey, ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', University of California Press, 1971 ...
, an algorithm can be expressed within a finite amount of space and time, and in a well-defined formal language for calculating a
function. Starting from an initial state and initial input (perhaps
empty), the instructions describe a
computation
Computation is any type of arithmetic or non-arithmetic calculation that follows a well-defined model (e.g., an algorithm).
Mechanical or electronic devices (or, historically, people) that perform computations are known as '' computers''. An esp ...
that, when
executed
Capital punishment, also known as the death penalty, is the State (polity), state-sanctioned practice of deliberately killing a person as a punishment for an actual or supposed crime, usually following an authorized, rule-governed process to ...
, proceeds through a finite number of well-defined successive states, eventually producing "output" and terminating at a final ending state. The transition from one state to the next is not necessarily
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
; some algorithms, known as
randomized algorithms
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
, incorporate random input.
History
The concept of algorithms has existed since antiquity.
Arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
algorithms, such as a
division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Di ...
, were used by ancient
Babylonian mathematicians c. 2500 BC and
Egyptian mathematicians c. 1550 BC.
Greek mathematicians later used algorithms in 240 BC in the
sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime ...
for finding prime numbers, and the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ...
for finding the
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...
of two numbers.
Arabic mathematicians such as
al-Kindi
Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (; ar, أبو يوسف يعقوب بن إسحاق الصبّاح الكندي; la, Alkindus; c. 801–873 AD) was an Arab Muslim philosopher, polymath, mathematician, physician ...
in the 9th century used
cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
algorithms for
code-breaking
Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic se ...
, based on
frequency analysis
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analysis is based on ...
.
The word ''algorithm'' is derived from the name of the 9th-century Persian mathematician
Muhammad ibn Musa al-Khwarizmi.
Al-Khwarizmi was a mathematician,
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either o ...
,
geographer
A geographer is a physical scientist, social scientist or humanist whose area of study is geography, the study of Earth's natural environment and human society, including how society and nature interacts. The Greek prefix "geo" means "earth" a ...
, and scholar in the
House of Wisdom
The House of Wisdom ( ar, بيت الحكمة, Bayt al-Ḥikmah), also known as the Grand Library of Baghdad, refers to either a major Abbasid public academy and intellectual center in Baghdad or to a large private library belonging to the Abba ...
in
Baghdad
Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesipho ...
, whose name means "the native of
Khwarazm
Khwarazm (; Old Persian: ''Hwârazmiya''; fa, خوارزم, ''Xwârazm'' or ''Xârazm'') or Chorasmia () is a large oasis region on the Amu Darya river delta in western Central Asia, bordered on the north by the (former) Aral Sea, on the e ...
", a region that was part of
Greater Iran
Greater Iran ( fa, ایران بزرگ, translit=Irān-e Bozorg) refers to a region covering parts of Western Asia, Central Asia, South Asia, Xinjiang, and the Caucasus, where both Iranian culture and Iranian languages have had ...
and is now in
Uzbekistan
Uzbekistan (, ; uz, Ozbekiston, italic=yes / , ; russian: Узбекистан), officially the Republic of Uzbekistan ( uz, Ozbekiston Respublikasi, italic=yes / ; russian: Республика Узбекистан), is a doubly landlocked co ...
.
In about the year 825, al-Khwarizmi wrote an
Arabic language
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
treatise on the
Hindu–Arabic numeral system
The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...
, which was translated into
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...
during the 12th century. The manuscript starts with the phrase ''Dixit Algorizmi'' ("Thus spake Al-Khwarizmi"), where "Algorizmi" was the translator's
Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through another of his books, the
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
. In late medieval Latin, ''algorismus'', English "
algorism", the corruption of his name, meant the "
decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός (''arithmos''), "number" (''cf.'' "arithmetic"), the Latin word was altered to ''algorithmus'', and the corresponding English term "algorithm" is first attested in the 17th century; the modern sense was introduced in the 19th century.
Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupt ...
was predominantly algorithmic.
Algorithms that are representative of the Indian mathematical tradition range from the ancient ''
Śulbasūtrās'' to the medieval texts of the
Kerala School.
In English, the word ''algorithm'' was first used in about 1230 and then by
Chaucer
Geoffrey Chaucer (; – 25 October 1400) was an English poet, author, and civil servant best known for '' The Canterbury Tales''. He has been called the "father of English literature", or, alternatively, the "father of English poetry". He w ...
in 1391. English adopted the French term, but it was not until the late 19th century that "algorithm" took on the meaning that it has in modern English.
Another early use of the word is from 1240, in a manual titled ''Carmen de Algorismo'' composed by
Alexandre de Villedieu. It begins with:
which translates to:
The poem is a few hundred lines long and summarizes the art of calculating with the new styled Indian dice (''Tali Indorum''), or Hindu numerals.
A partial formalization of the modern concept of algorithm began with attempts to solve the ''
Entscheidungsproblem '' (decision problem) posed by
David Hilbert in 1928. Later formalizations were framed as attempts to define "
effective calculability" or "effective method". Those formalizations included the
Gödel–
Herbrand–
Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch o ...
recursive functions of 1930, 1934 and 1935,
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scie ...
's
lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation t ...
of 1936,
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Gove ...
's
Formulation 1
Formulation is a term used in various senses in various applications, both the material and the Abstract object, abstract or wikt:formal, formal. Its fundamental meaning is the putting together of components in appropriate relationships or struc ...
of 1936, and
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical c ...
's
Turing machines
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
of 1936–37 and 1939.
Informal definition
An informal definition could be "a set of rules that precisely defines a sequence of operations", which would include all
computer programs
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components.
A computer progra ...
(including programs that do not perform numeric calculations), and (for example) any prescribed
bureaucratic
The term bureaucracy () refers to a body of non-elected governing officials as well as to an administrative policy-making group. Historically, a bureaucracy was a government administration managed by departments staffed with non-elected offi ...
procedure
or
cook-book recipe
A recipe is a set of instructions that describes how to prepare or make something, especially a dish of prepared food. A sub-recipe or subrecipe is a recipe for an ingredient that will be called for in the instructions for the main recipe.
Hist ...
.
In general, a program is only an algorithm if it stops eventually—even though
infinite loop
In computer programming, an infinite loop (or endless loop) is a sequence of instructions that, as written, will continue endlessly, unless an external intervention occurs ("pull the plug"). It may be intentional.
Overview
This differs from:
* ...
s may sometimes prove desirable.
A prototypical example of an algorithm is the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ...
, which is used to determine the maximum common divisor of two integers; an example (there are others) is described by the
flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of va ...
above and as an example in a later section.
offer an informal meaning of the word "algorithm" in the following quotation:
An
"enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. 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 arbitrarily large. For example, an algorithm can be an algebraic equation such as ''y = m + n'' (i.e., two arbitrary "input variables" ''m'' and ''n'' that produce an output ''y''), but various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
:Precise instructions (in a 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 internally contained information and capabilities) to find, decode, and then process arbitrary input integers/symbols ''m'' and ''n'', symbols ''+'' and ''='' ... and "effectively" produce, in a "reasonable" time, output-integer ''y'' at a specified place and in a specified format.
The concept of ''algorithm'' is also used to define the notion of
decidability—a notion that is central for explaining how
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A for ...
s come into being starting from a small set of
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
s and rules. In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
, the time that an algorithm requires to complete cannot be measured, as it is not apparently related to the 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.
Most algorithms are intended to be
implemented as
computer programs
A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components.
A computer progra ...
. However, algorithms are also implemented by other means, such as in a
biological neural network
A neural circuit is a population of neurons interconnected by synapses to carry out a specific function when activated. Neural circuits interconnect to one another to form large scale brain networks.
Biological neural networks have inspired t ...
(for example, the
human brain
The human brain is the central organ (anatomy), organ of the human nervous system, and with the spinal cord makes up the central nervous system. The brain consists of the cerebrum, the brainstem and the cerebellum. It controls most of the act ...
implementing
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
or an insect looking for food), in an
electrical circuit
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sour ...
, or in a mechanical device.
Formalization
Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform—in a specific 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 simulated by a
Turing-complete
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Tu ...
system. Authors who assert this thesis include Minsky (1967), Savage (1987), and Gurevich (2000):
Turing machines can define computational processes that do not terminate. The informal definitions of algorithms generally require that the algorithm always terminates. This requirement renders the task of deciding whether a formal procedure is an algorithm impossible in the general case—due to a major theorem of
computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ...
known as the
halting problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
.
Typically, when an algorithm is associated with processing information, data can be read from an input source, written to an output device and 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 one or more
data structure
In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...
s.
For some of these computational processes, the algorithm must be rigorously defined: and specified in the way it applies in all possible circumstances that could arise. This means that any conditional steps must be systematically dealt with, case by case; the criteria for each case must be clear (and computable).
Because an algorithm is a precise list of precise steps, the order of computation is always crucial 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, the discussion on the formalization of an algorithm has assumed the premises of
imperative programming
In computer science, imperative programming is a programming paradigm of software that uses statements that change a program's state. In much the same way that the imperative mood in natural languages expresses commands, an imperative program co ...
. This is the most common conception—one which attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the
assignment operation
In computer programming, an assignment statement sets and/or re-sets the value (computer science), value stored in the storage location(s) denoted by a variable (programming), variable name; in other words, it copies a value into the variable. In ...
, which sets the value of a variable. It derives from the intuition of "
memory
Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...
" as a scratchpad. An example of such an assignment can be found below.
For some alternate conceptions of what constitutes an algorithm, see
functional programming
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...
and
logic programming
Logic programming is a programming paradigm which is largely based on formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of log ...
.
Expressing algorithms
Algorithms can be expressed in many kinds of notation, including
natural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languag ...
s,
pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
,
flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of va ...
s,
drakon-charts,
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming l ...
s or
control tables (processed by
interpreters
Interpreting is a translational activity in which one produces a first and final target-language output on the basis of a one-time exposure to an expression in a source language.
The most common two modes of interpreting are simultaneous interp ...
). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts,
drakon-charts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in the statements based on natural language. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are also 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
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...
program as a sequence of machine tables (see
finite-state machine
A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number ...
,
state transition table and
control table for more), as flowcharts and
drakon-charts (see
state diagram for more), or as a form of rudimentary
machine code
In computer programming, machine code is any low-level programming language, consisting of machine language instructions, which are used to control a computer's central processing unit (CPU). Each instruction causes the CPU to perform a ver ...
or
assembly code called "sets of quadruples" (see
Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...
for more).
Representations of algorithms can be classed into three accepted levels of Turing machine description, as follows:
; 1 High-level 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
Examples
Example may refer to:
* '' exempli gratia'' (e.g.), usually read out in English as "for example"
* .example, reserved as a domain name that may not be installed as a top-level domain of the Internet
** example.com, example.net, example.org, e ...
.
Design
Algorithm design refers to a method or a mathematical process for problem-solving and engineering algorithms. The design of algorithms is part of many solution theories, such as
divide-and-conquer or
dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...
within
operation research. Techniques for designing and implementing algorithm designs are also called algorithm design patterns, with examples including the template method pattern and the decorator pattern.
One of the most important aspects of algorithm design is resource (run-time, memory usage) efficiency; the
big O notation is used to describe e.g. an algorithm's run-time growth as the size of its input increases.
Typical steps in the development of algorithms:
# Problem definition
# Development of a model
# Specification of the algorithm
# Designing an algorithm
# Checking the
correctness of the algorithm
# Analysis of algorithm
# Implementation of algorithm
# Program testing
# Documentation preparation
Computer algorithms
''"Elegant" (compact) programs, "good" (fast) programs '': The notion of "simplicity and elegance" appears informally in
Knuth and precisely in
Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine- American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a compute ...
:
:Knuth: " ... we want ''good'' algorithms in some loosely defined aesthetic sense. One criterion ... is the length of time taken to perform the algorithm .... Other criteria are adaptability of the algorithm to computers, its simplicity, and elegance, etc."
:Chaitin: " ... a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant—such a proof would solve the
Halting problem
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a ...
(ibid).
''Algorithm versus function computable by an algorithm'': For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of ''algorithm'', i.e. procedure and the notion of ''function computable by algorithm'', i.e. mapping yielded by procedure. The same function may have several different algorithms".
Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below.
''Computers (and computors), models of computation'': A computer (or human "computer") is a restricted type of machine, a "discrete deterministic mechanical device" that blindly follows its instructions. Melzak's and Lambek's primitive models reduced this notion to four elements: (i) discrete, distinguishable ''locations'', (ii) discrete, indistinguishable ''counters'' (iii) an agent, and (iv) a list of instructions that are ''effective'' relative to the capability of the agent.
Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for
Computability
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is clo ...
".
Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions unless either a conditional IF-THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three ''assignment'' (replacement, substitution) operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1). Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is
Turing complete
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
with only four general ''types'' of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is convenient; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ← 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional.
''Simulation of an algorithm: computer (computor) language'': Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example". But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can ''effectively'' execute. Stone gives an example of this: when computing the roots of a quadratic equation the computer must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.
This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).
But what model should be used for the simulation? Van Emde Boas observes "even if we base
complexity theory on abstract instead of concrete machines, the arbitrariness of the choice of a model remains. It is at this point that the notion of ''simulation'' enters". When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "
modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement").
''Structured programming, canonical structures'': Per the
Church–Turing thesis
In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of co ...
, any algorithm can be computed by a model known to be
Turing complete
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical co ...
, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "
spaghetti code
Spaghetti code is a pejorative phrase for unstructured and difficult-to-maintain source code. Spaghetti code can be caused by several factors, such as volatile project requirements, lack of programming style rules, and software engineers with insuf ...
", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language". Tausworthe augments the three
Böhm-Jacopini canonical structures: SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE. An additional benefit of a structured program is that it lends itself to
proofs of correctness using
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
.
''Canonical flowchart symbols'': The graphical aide called a
flowchart
A flowchart is a type of diagram that represents a workflow or process. A flowchart can also be defined as a diagrammatic representation of an algorithm, a step-by-step approach to solving a task.
The flowchart shows the steps as boxes of va ...
offers a way to describe and document an algorithm (and a computer program corresponding to it). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.
Examples
Algorithm example
One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as:
''High-level description:''
# If there are no numbers in the set, then there is no highest number.
# Assume the first number in the set is the largest number in the set.
# For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set.
# When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set.
''(Quasi-)formal description:''
Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in
pseudocode
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
or
pidgin code:
Input: A list of numbers ''L''.
Output: The largest number in the list ''L''.
if ''L.size'' = 0 return null
''largest'' ← ''L''
for each ''item'' in ''L'', do
if ''item'' > ''largest'', then
''largest'' ← ''item''
return ''largest''
Euclid's algorithm
In
mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...
(GCD) of two integers (numbers), the largest number that divides them both without a
remainder
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algeb ...
. It is named after the ancient Greek
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ...
, who first described it in
his ''Elements'' (). It is one of the oldest algorithms in common use. It can be used to reduce
fractions
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
to their
simplest form, and is a part of many other number-theoretic and cryptographic calculations.
Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number
o bea multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length ''s'' successively (''q'' times) along longer length ''l'' until the remaining portion ''r'' is less than the shorter length ''s''. In modern words, remainder ''r'' = ''l'' − ''q''×''s'', ''q'' being the quotient, or remainder ''r'' is the "modulus", the integer-fractional part left over after the division.
For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be "proper"; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero).
Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a
reductio ad absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
proof that the two numbers' common measure is in fact the ''greatest''. While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm.
Computer language for Euclid's algorithm
Only a few instruction ''types'' are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction.
* A ''location'' is symbolized by upper case letter(s), e.g. S, A, etc.
* The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number ''l'' = 3009.
An inelegant program for Euclid's algorithm
The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length ''s'' from the remaining length ''r'' until ''r'' is less than ''s''. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4:
INPUT:
nto two locations L and S put the numbers ''l'' and ''s'' that represent the two lengths
INPUT L, S
nitialize R: make the remaining length ''r'' equal to the starting/initial/input length ''l''
R ← L
E0:
nsure ''r'' ≥ ''s''.''
nsure the smaller of the two numbers is in S and the larger in R
IF R > S THEN
the contents of L is the larger number so skip over the exchange-steps
4,
5 and
6:
GOTO step
7
ELSE
swap the contents of R and S.
L ← R (this first step is redundant, but is useful for later discussion).
R ← S
S ← L
E1:
ind remainder
Ind or IND may refer to:
General
* Independent (politician), a politician not affiliated to any political party
* Independent station, used within television program listings and the television industry for a station that is not affiliated with ...
'': Until the remaining length ''r'' in R is less than the shorter length ''s'' in S, repeatedly subtract the measuring number ''s'' in S from the remaining length ''r'' in R.
IF S > R THEN
done measuring so
GOTO
10
ELSE
measure again,
R ← R − S
emainder-loop
GOTO
7.
E2:
s the remainder zero?'': EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S.
IF R = 0 THEN
done so
GOTO
step 15
ELSE
CONTINUE TO
step 11,
E3:
nterchange ''s'' and ''r'''': The nut of Euclid's algorithm. Use remainder ''r'' to measure what was previously smaller number ''s''; L serves as a temporary location.
L ← R
R ← S
S ← L
epeat the measuring process The Electronic Product Environmental Assessment Tool (EPEAT) is a method for purchasers (governments, institutions, consumers, etc.) to evaluate the effect of a product on the environment. It assesses various lifecycle environmental aspects of a dev ...
GOTO
7
OUTPUT:
[Done. S contains the
greatest common divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...
]:
PRINT S
DONE:
HALT, END, STOP.
An elegant program for Euclid's algorithm
The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction
LET [] = [] is the assignment instruction symbolized by ←.
5 REM Euclid's algorithm for greatest common divisor
6 PRINT "Type two integers greater than 0"
10 INPUT A,B
20 IF B=0 THEN GOTO 80
30 IF A > B THEN GOTO 60
40 LET B=B-A
50 GOTO 20
60 LET A=A-B
70 GOTO 20
80 PRINT A
90 END
''How "Elegant" works'': In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ← A − B, and a B ≤ A loop that computes B ← B − A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend − Subtrahend), the minuend can become ''s'' (the new measuring length) and the subtrahend can become the new ''r'' (the length to be measured); in other words the "sense" of the subtraction reverses.
The following version can be used with
programming languages from the C-family:
// Euclid's algorithm for greatest common divisor
int euclidAlgorithm (int A, int B)
Testing the Euclid algorithms
Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two
relatively prime numbers 14157 and 5950.
But "exceptional cases" must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if ''negative'' numbers are entered? Fractional numbers? If the input numbers, i.e. the
domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that
instantiates it) is a
partial function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
rather than a
total function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
. A notable failure due to exceptions is the
Ariane 5 Flight 501 rocket failure (June 4, 1996).
''Proof of program correctness by use of mathematical induction'': Knuth demonstrates the application of
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
to an "extended" version of Euclid's algorithm, and he proposes "a general method applicable to proving the validity of any algorithm". Tausworthe proposes that a measure of the complexity of a program be the length of its correctness proof.
Measuring and improving the Euclid algorithms
''Elegance (compactness) versus goodness (speed)'': With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is ''faster'' (it arrives at HALT in fewer steps).
Algorithm analysis indicates why this is the case: "Elegant" does ''two'' conditional tests in every subtraction loop, whereas "Inelegant" only does one. As the algorithm (usually) requires many loop-throughs, ''on average'' much time is wasted doing a "B = 0?" test that is needed only after the remainder is computed.
''Can the algorithms be improved?'': Once the programmer judges a program "fit" and "effective"—that is, it computes the function intended by its author—then the question becomes, can it be improved?
The compactness of "Inelegant" can be improved by the elimination of five steps. But Chaitin proved that compacting an algorithm cannot be automated by a generalized algorithm; rather, it can only be done
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediat ...
ally; i.e., by exhaustive search (examples to be found at
Busy beaver), trial and error, cleverness, insight, application of
inductive reasoning, etc. Observe that steps 4, 5 and 6 are repeated in steps 11, 12 and 13. Comparison with "Elegant" provides a hint that these steps, together with steps 2 and 3, can be eliminated. This reduces the number of core instructions from thirteen to eight, which makes it "more elegant" than "Elegant", at nine steps.
The speed of "Elegant" can be improved by moving the "B=0?" test outside of the two subtraction loops. This change calls for the addition of three instructions (B = 0?, A = 0?, GOTO). Now "Elegant" computes the example-numbers faster; whether this is always the case for any given A, B, and R, S would require a detailed analysis.
Algorithmic analysis
It is frequently important to know how much of a particular resource (such as time or storage) is theoretically required for a given algorithm. Methods have been developed for the
analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that r ...
to obtain such quantitative answers (estimates); for example, an algorithm which adds up the elements of a list of ''n'' numbers would have a time requirement of ''O(n)'', using
big O notation. At all times the algorithm only needs to remember two values: the sum of all the elements 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(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, a
binary search
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the m ...
algorithm (with cost ''O(log n)'') outperforms a sequential search (cost ''O(n)'' ) when used for
table lookups on sorted lists or arrays.
Formal versus empirical
The
analysis, and study of algorithms is a discipline of
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, and is often practiced abstractly without the use of a specific
programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming l ...
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
In computer science, pseudocode is a plain language description of the steps in an algorithm or another system. Pseudocode often uses structural conventions of a normal programming language, but is intended for human reading rather than machine re ...
is used for analysis as it is the simplest and most general representation. However, ultimately, most algorithms are usually implemented on particular hardware/software platforms and their
algorithmic efficiency
In computer science, algorithmic efficiency is a property of an algorithm which relates to the amount of computational resources used by the algorithm. An algorithm must be analyzed to determine its resource usage, and the efficiency of an al ...
is eventually put to the test using real code. For the solution of a "one off" problem, the efficiency of a particular algorithm may not have significant consequences (unless n is extremely large) but for algorithms designed for fast interactive, commercial or long life scientific usage it may be critical. Scaling from small n to large n frequently exposes inefficient algorithms that are otherwise benign.
Empirical testing is useful because it may uncover unexpected interactions that affect performance.
Benchmarks may be used to compare before/after potential improvements to an algorithm after program optimization.
Empirical tests cannot replace formal analysis, though, and are not trivial to perform in a fair manner.
Execution efficiency
To illustrate the potential improvements possible even in well-established algorithms, a recent significant innovation, relating to
FFT algorithms (used heavily in the field of image processing), can decrease processing time up to 1,000 times for applications like medical imaging. In general, speed improvements depend on special properties of the problem, which are very common in practical applications.
[Haitham Hassanieh, Piotr Indyk, Dina Katabi, and Eric Price,]
ACM-SIAM Symposium On Discrete Algorithms (SODA)
, Kyoto, January 2012. See also th
sFFT Web Page
. Speedups of this magnitude enable computing devices that make extensive use of image processing (like digital cameras and medical equipment) to consume less power.
Classification
There are various ways to classify algorithms, each with its own merits.
By implementation
One way to classify algorithms is by implementation means.
; Recursion
: A
recursive algorithm
In computer science, recursion is a method of solving a computational problem where the solution depends on solutions to smaller instances of the same problem. Recursion solves such recursive problems by using functions that call themselves ...
is one that invokes (makes reference to) itself repeatedly until a certain condition (also known as termination condition) matches, which is a method common to
functional programming
In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...
.
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 using 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''. 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
Logic programming is a programming paradigm which is largely based on formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of log ...
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
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and compu ...
: a change in the axioms produces a well-defined change in the algorithm.
; Serial, 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 algorithm
In computer science, a parallel algorithm, as opposed to a traditional serial algorithm, is an algorithm which can do multiple operations in a given time. It has been a tradition of computer science to describe serial algorithms in abstract machine ...
s 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 use multiple machines connected with a
computer network
A computer network is a set of computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections ar ...
. 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. Some 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 non-deterministic
:
Deterministic algorithm
In computer science, a deterministic algorithm is an algorithm that, given a particular input, will always produce the same output, with the underlying machine always passing through the same sequence of states. Deterministic algorithms are by far ...
s solve the problem with exact decision at every step of the algorithm whereas
non-deterministic algorithms solve problems via guessing although typical guesses are made more accurate through the use of
heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
.
; Exact or approximate
: While many algorithms reach an exact solution,
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
s seek an approximation that is closer to the true solution. The approximation can be reached by either using a deterministic or a random strategy. Such algorithms have practical value for many hard problems. One of the examples of an approximate algorithm is the
Knapsack problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit a ...
, where there is a set of given items. Its goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. Total weight that can be carried is no more than some fixed number X. So, the solution must consider weights of items as well as their value.
;
Quantum algorithm
In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical (or non-quantum) algorithm is a finite seq ...
: They run on a realistic model of
quantum computation
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
. The term is usually used for those algorithms which seem inherently quantum, or use some essential feature of
Quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thou ...
such as
quantum superposition
Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum ...
or
quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state o ...
.
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 includes many different types of algorithms. Some common paradigms are:
;
Brute-force or exhaustive search
: This is the
naive method of trying every possible solution to see which is best.
; Divide and conquer
: A
divide-and-conquer algorithm
In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved dire ...
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 the conquer phase by merging the segments. A simpler variant of divide and conquer is called a ''decrease-and-conquer algorithm'', which 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 the conquer stage is more complex than decrease and conquer algorithms. An example of a decrease and conquer algorithm is the
binary search algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the ...
.
; Search and enumeration
: Many problems (such as playing
chess
Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to dist ...
) 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 algorithm
In computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the Feasible region, search space of a problem do ...
s,
branch and bound
Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutio ...
enumeration and
backtracking
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it de ...
.
;
Randomized algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performa ...
: Such algorithms make some choices randomly (or pseudo-randomly). They can be very useful in finding approximate solutions for problems where finding exact solutions can be impractical (see heuristic method below). For some of these problems, it is known that the fastest approximations must involve some
randomness
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
. Whether randomized algorithms with
polynomial time complexity can be the fastest algorithms for some problems is an open question known as the
P versus NP problem. There are two large classes of such algorithms:
#
Monte Carlo algorithms return a correct answer with high-probability. E.g.
RP is the subclass of these that run in
polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
.
#
Las Vegas algorithm In computing, a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the ...
s always return the correct answer, but their running time is only probabilistically bound, e.g.
ZPP.
;
Reduction of complexity
: This technique involves solving a difficult problem by transforming it into a better-known problem for which we have (hopefully)
asymptotically optimal
In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor (independent of the input size) worse than the best possible algorithm. It is a term commonly en ...
algorithms. The goal is to find a reducing algorithm whose
complexity
Complexity characterises the behaviour of a system or model whose components interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generally used to c ...
is not dominated by the resulting reduced algorithm's. For example, one
selection algorithm
In computer science, a selection algorithm is an algorithm for finding the ''k''th smallest number in a list or array; such a number is called the ''k''th ''order statistic''. This includes the cases of finding the minimum, maximum, and median e ...
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''.
;
Back tracking
Backtracking is a class of algorithms for finding solutions to some computational problems, notably constraint satisfaction problems, that incrementally builds candidates to the solutions, and abandons a candidate ("backtracks") as soon as it de ...
: In this approach, multiple solutions are built incrementally and abandoned when it is determined that they cannot lead to a valid full solution.
Optimization problems
For
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
s there is a more specific classification of algorithms; an algorithm for such problems may fall into one or more of the general categories described above as well as into one of the following:
;
Linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...
: When searching for optimal solutions to a linear function bound to linear equality and inequality constraints, the constraints of the problem can be used directly in producing the optimal solutions. There are algorithms that can solve any problem in this category, such as the popular
simplex algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are no ...
. Problems that can be solved with linear programming include the
maximum flow problem
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
The maximum flow problem can be seen as a special case of more complex network flow problems, such ...
for directed graphs. If a problem additionally requires that one or more of the unknowns must be an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
then it is classified in
integer programming
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective ...
. A linear programming algorithm can solve such a problem if it can be proved that all restrictions for integer values are superficial, i.e., the solutions satisfy these restrictions anyway. In the general case, a specialized algorithm or an algorithm that finds approximate solutions is used, depending on the difficulty of the problem.
;
Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...
: When a problem shows
optimal substructures—meaning the optimal solution to a problem can be constructed from optimal solutions to subproblems—and
overlapping subproblems In computer science, a problem is said to have overlapping subproblems if the problem can be broken down into subproblems which are reused several times or a recursive algorithm for the problem solves the same subproblem over and over rather than al ...
, 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,
Floyd–Warshall algorithm
In computer science, the Floyd–Warshall algorithm (also known as Floyd's algorithm, the Roy–Warshall algorithm, the Roy–Floyd algorithm, or the WFI algorithm) is an algorithm for finding shortest paths in a directed weighted graph with p ...
, the shortest path to a goal from a vertex in a weighted
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
can be found by using the shortest path to the goal from all adjacent vertices. Dynamic programming and
memoization
In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. 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
Table may refer to:
* Table (furniture), a piece of furniture with a flat surface and one or more legs
* Table (landform), a flat area of land
* Table (information), a data arrangement with rows and columns
* Table (database), how the table data ...
of subproblems already solved, dynamic programming reduces the exponential nature of many problems to polynomial complexity.
; The greedy method
: A
greedy algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locall ...
is similar to a dynamic programming algorithm in that it works by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution, which may be given or have been constructed in some way, and improve it by making small modifications. For some problems they can find the optimal solution while for others they stop at
local optima, that is, at solutions that cannot be improved by the algorithm but are not optimum. The most popular use of greedy algorithms is for finding the minimal spanning tree where finding the optimal solution is possible with this method.
Huffman Tree,
Kruskal,
Prim,
Sollin are greedy algorithms that can solve this optimization problem.
;The heuristic method
:In
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
s,
heuristic algorithm
In mathematical optimization and computer science, heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow for finding an approximate solution, or whe ...
s can be used to find a solution close to the optimal solution in cases where finding the optimal solution is impractical. These algorithms work by getting closer and closer to the optimal solution as they progress. In principle, if run for an infinite amount of time, they will find the optimal solution. Their merit is that they can find a solution very close to the optimal solution in a relatively short time. Such algorithms include
local search,
tabu search Tabu search is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986
and formalized in 1989.
Local (neighborhood) searches take a potential solution to a prob ...
,
simulated annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. ...
, and
genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gen ...
s. Some of them, like simulated annealing, are non-deterministic algorithms while others, like tabu search, are deterministic. When a bound on the error of the non-optimal solution is known, the algorithm is further categorized as an
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
.
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 algorithm
In computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the Feasible region, search space of a problem do ...
s,
sorting algorithm
In computer science, a sorting algorithm is an algorithm that puts elements of a list into an order. The most frequently used orders are numerical order and lexicographical order, and either ascending or descending. Efficient sorting is importan ...
s,
merge algorithm
Merge algorithms are a family of algorithms that take multiple sorted lists as input and produce a single list as output, containing all the elements of the inputs lists in sorted order. These algorithms are used as subroutines in various sorting ...
s,
numerical algorithms,
graph algorithms
The following is a list of well-known algorithms along with one-line descriptions for each.
Automated planning
Combinatorial algorithms
General combinatorial algorithms
* Brent's algorithm: finds a cycle in function value iterations using on ...
,
string algorithms
In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation ...
,
computational geometric algorithms,
combinatorial algorithms
The following is a list of well-known algorithms along with one-line descriptions for each.
Automated planning
Combinatorial algorithms
General combinatorial algorithms
* Brent's algorithm: finds a cycle in function value iterations using on ...
,
medical algorithms,
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
,
data compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressi ...
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 invented for optimization of resource consumption in industry but is now used in solving a broad range of problems in many fields.
By complexity
Algorithms can be classified by the amount of time they need to complete compared to their input size:
* Constant time: if the time needed by the algorithm is the same, regardless of the input size. E.g. an access to an
array
An array is a systematic arrangement of similar objects, usually in rows and columns.
Things called an array include:
{{TOC right
Music
* In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...
element.
* Logarithmic time: if the time is a logarithmic function of the input size. E.g.
binary search algorithm
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array. Binary search compares the target value to the ...
.
* Linear time: if the time is proportional to the input size. E.g. the traverse of a list.
* Polynomial time: if the time is a power of the input size. E.g. the
bubble sort
Bubble sort, sometimes referred to as sinking sort, is a simple sorting algorithm that repeatedly steps through the input list element by element, comparing the current element with the one after it, swapping their values if needed. These passes ...
algorithm has quadratic time complexity.
* Exponential time: if the time is an exponential function of the input size. E.g.
Brute-force search
In computer science, brute-force search or exhaustive search, also known as generate and test, is a very general problem-solving technique and algorithmic paradigm that consists of systematically enumerating all possible candidates for the soluti ...
.
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.
Continuous algorithms
The adjective "continuous" when applied to the word "algorithm" can mean:
* An algorithm operating on data that represents continuous quantities, even though this data is represented by discrete approximations—such algorithms are studied in
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
; or
* An algorithm in the form of a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
that operates continuously on the data, running on an
analog computer
An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. ...
.
Legal issues
Algorithms, by themselves, are not usually patentable. In the United States, a claim consisting solely of simple manipulations of abstract concepts, numbers, or signals does not constitute "processes" (USPTO 2006), and hence algorithms are not patentable (as in
Gottschalk v. Benson
''Gottschalk v. Benson'', 409 U.S. 63 (1972), was a United States Supreme Court case in which the Court ruled that a process claim directed to a numerical algorithm, as such, was not patentable because "the patent would ''wholly pre-empt'' the ...
). However practical applications of algorithms are sometimes patentable. For example, in
Diamond v. Diehr, the application of a simple
feedback
Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
algorithm to aid in the curing of
synthetic rubber
A synthetic rubber is an artificial elastomer. They are polymers synthesized from petroleum byproducts. About 32-million metric tons of rubbers are produced annually in the United States, and of that amount two thirds are synthetic. Synthetic rubbe ...
was deemed patentable. The
patenting of software is highly controversial, and there are highly criticized patents involving algorithms, especially
data compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compressi ...
algorithms, such as
Unisys
Unisys Corporation is an American multinational information technology (IT) services and consulting company headquartered in Blue Bell, Pennsylvania. It provides digital workplace solutions, cloud, applications, and infrastructure solutions ...
'
LZW patent.
Additionally, some cryptographic algorithms have export restrictions (see
export of cryptography
The export of cryptography is the transfer from one country to another of devices and technology related to cryptography.
In the early days of the Cold War, the United States and its allies developed an elaborate series of export control regulat ...
).
History: Development of the notion of "algorithm"
Ancient Near East
The earliest evidence of algorithms is found in the
Babylonian mathematics
Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Bab ...
of ancient
Mesopotamia
Mesopotamia ''Mesopotamíā''; ar, بِلَاد ٱلرَّافِدَيْن or ; syc, ܐܪܡ ܢܗܪ̈ܝܢ, or , ) is a historical region of Western Asia situated within the Tigris–Euphrates river system, in the northern part of the F ...
(modern Iraq). A
Sumerian clay tablet found in
Shuruppak
Shuruppak ( sux, , "the healing place"), modern Tell Fara, was an ancient Sumerian city situated about 55 kilometres (35 mi) south of Nippur on the banks of the Euphrates in Iraq's Al-Qādisiyyah Governorate. Shuruppak was dedicated t ...
near
Baghdad
Baghdad (; ar, بَغْدَاد , ) is the capital of Iraq and the second-largest city in the Arab world after Cairo. It is located on the Tigris near the ruins of the ancient city of Babylon and the Sassanid Persian capital of Ctesipho ...
and dated to circa 2500 BC described the earliest
division algorithm
A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software.
Di ...
.
During the
Hammurabi dynasty circa 1800-1600 BC,
Babylonia
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state ...
n clay tablets described algorithms for computing
formulas
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
. Algorithms were also used in
Babylonian astronomy
Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia.
Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations ...
. Babylonian clay tablets describe and employ algorithmic procedures to compute the time and place of significant astronomical events.
Algorithms for arithmetic are also found in ancient
Egyptian mathematics
Ancient Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt 3000 to c. , from the Old Kingdom of Egypt until roughly the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for coun ...
, dating back to the
Rhind Mathematical Papyrus
The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...
circa 1550 BC.
Algorithms were later used in ancient
Hellenistic mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek math ...
. Two examples are the
Sieve of Eratosthenes
In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit.
It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime ...
, which was described in the ''
Introduction to Arithmetic'' by
Nicomachus
Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician and music theorist, best known for his works '' Introduction to Arithmetic'' and '' Manual of Harmonics'' in Greek. He was born ...
,
[ and the ]Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ...
, which was first described in ''Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
'' ().[
]
Discrete and distinguishable symbols
Tally-marks: 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
The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the H ...
evolved (Dilson, p. 16–41). Tally marks appear prominently in unary numeral system
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times.
In the unary system, the number 0 (zero) is represented by the empty string, th ...
arithmetic used in Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...
and Post–Turing machine computations.
Manipulation of symbols as "place holders" for numbers: algebra
Muhammad ibn Mūsā al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astrono ...
, a Persian mathematician, wrote the ''Al-jabr
''The Compendious Book on Calculation by Completion and Balancing'' ( ar, كتاب المختصر في حساب الجبر والمقابلة, ; la, Liber Algebræ et Almucabola), also known as ''Al-Jabr'' (), is an Arabic mathematics, Arabic ...
'' in the 9th century. The terms " algorism" and "algorithm" are derived from the name al-Khwārizmī, while the term "algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
" is derived from the book ''Al-jabr''. In Europe, the word "algorithm" was originally used to refer to the sets of rules and techniques used by Al-Khwarizmi to solve algebraic equations, before later being generalized to refer to any set of rules or techniques. This eventually culminated in Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
's notion of the calculus ratiocinator ():
Cryptographic algorithms
The first cryptographic
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
algorithm for deciphering encrypted code was developed by Al-Kindi
Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (; ar, أبو يوسف يعقوب بن إسحاق الصبّاح الكندي; la, Alkindus; c. 801–873 AD) was an Arab Muslim philosopher, polymath, mathematician, physician ...
, a 9th-century Arab mathematician, in ''A Manuscript On Deciphering Cryptographic Messages''. He gave the first description of cryptanalysis by frequency analysis
In cryptanalysis, frequency analysis (also known as counting letters) is the study of the frequency of letters or groups of letters in a ciphertext. The method is used as an aid to breaking classical ciphers.
Frequency analysis is based on ...
, the earliest codebreaking
Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic se ...
algorithm.
Mechanical contrivances with discrete states
''The clock'': Bolter credits the invention of the weight-driven clock
A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and t ...
as "The key invention f Europe in the Middle Ages
F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''.
Hi ...
, in particular, the verge escapement
The verge (or crown wheel) escapement is the earliest known type of mechanical escapement, the mechanism in a mechanical clock that controls its rate by allowing the gear train to advance at regular intervals or 'ticks'. Its origin is unknown. V ...
that provides us with the tick and tock of a mechanical clock. "The accurate automatic machine" led immediately to "mechanical automata
An automaton (; plural: automata or automatons) is a relatively self-operating machine, or control mechanism designed to automatically follow a sequence of operations, or respond to predetermined instructions.Automaton – Definition and Mor ...
" beginning in the 13th century and finally to "computational machines"—the difference engine
A difference engine is an automatic mechanical calculator designed to tabulate polynomial functions. It was designed in the 1820s, and was first created by Charles Babbage. The name, the difference engine, is derived from the method of divided d ...
and analytical engine
The Analytical Engine was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage. It was first described in 1837 as the successor to Babbage's difference engine, which was a design ...
s of Charles Babbage
Charles Babbage (; 26 December 1791 – 18 October 1871) was an English polymath. A mathematician, philosopher, inventor and mechanical engineer, Babbage originated the concept of a digital programmable computer.
Babbage is considered ...
and Countess Ada Lovelace
Augusta Ada King, Countess of Lovelace ('' née'' Byron; 10 December 1815 – 27 November 1852) was an English mathematician and writer, chiefly known for her work on Charles Babbage's proposed mechanical general-purpose computer, the ...
, mid-19th century. Lovelace is credited with the first creation of an algorithm intended for processing on a computer—Babbage's analytical engine, the first device considered a real Turing-complete
In computability theory, a system of data-manipulation rules (such as a computer's instruction set, a programming language, or a cellular automaton) is said to be Turing-complete or computationally universal if it can be used to simulate any Tu ...
computer instead of just a calculator
An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics.
The first solid-state electronic calculator was created in the early 1960s. Pocket-size ...
—and is sometimes called "history's first programmer" as a result, though a full implementation of Babbage's second device would not be realized until decades after her lifetime.
''Logical machines 1870 – Stanley Jevons' "logical abacus" and "logical machine"'': The technical problem was to reduce Boolean equation
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
s when presented in a form similar to what is now known as Karnaugh map
The Karnaugh map (KM or K-map) is a method of simplifying Boolean algebra expressions. Maurice Karnaugh introduced it in 1953 as a refinement of Edward W. Veitch's 1952 Veitch chart, which was a rediscovery of Allan Marquand's 1881 ''logic ...
s. Jevons (1880) describes first a simple "abacus" of "slips of wood furnished with pins, contrived so that any part or class of the ogicalcombinations can be picked out mechanically ... More recently, however, I have reduced the system to a completely mechanical form, and have thus embodied the whole of the indirect process of inference in what may be called a ''Logical Machine''" His machine came equipped with "certain moveable wooden rods" and "at the foot are 21 keys like those of a piano tc....". With this machine he could analyze a "syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...
or any other simple logical argument".
This machine he displayed in 1870 before the Fellows of the Royal Society. Another logician John Venn
John Venn, FRS, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in logic, set theory, probability, statistics, and computer science. In 1866, V ...
, however, in his 1881 ''Symbolic Logic'', turned a jaundiced eye to this effort: "I have no high estimate myself of the interest or importance of what are sometimes called logical machines ... it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines"; see more at Algorithm characterizations. But not to be outdone he too presented "a plan somewhat analogous, I apprehend, to Prof. Jevon's ''abacus'' ... nd ain, corresponding to Prof. Jevons's logical machine, the following contrivance may be described. I prefer to call it merely a logical-diagram machine ... but I suppose that it could do very completely all that can be rationally expected of any logical machine".
''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
A punched card (also punch card or punched-card) is a piece of stiff paper that holds digital data represented by the presence or absence of holes in predefined positions. Punched cards were once common in data processing applications or to di ...
(punch cards, 1887), and "telephone switching technologies" were the roots of a tree leading to the development of the first computers. By the mid-19th century the telegraph
Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas ...
, 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 19th century the ticker tape
Ticker tape was the earliest electrical dedicated financial communications medium, transmitting stock price information over telegraph lines, in use from around 1870 through 1970. It consisted of a paper strip that ran through a machine called ...
() was in use, as was the use of Hollerith cards in the 1890 U.S. census. Then came the teleprinter
A teleprinter (teletypewriter, teletype or TTY) is an electromechanical device that can be used to send and receive typed messages through various communications channels, in both point-to-point (telecommunications), point-to-point and point- ...
() with its punched-paper use of Baudot code
The Baudot code is an early character encoding for telegraphy invented by Émile Baudot in the 1870s. It was the predecessor to the International Telegraph Alphabet No. 2 (ITA2), the most common teleprinter code in use until the advent of ASCII. ...
on tape.
''Telephone-switching networks'' of electromechanical relay
A relay
Electromechanical relay schematic showing a control coil, four pairs of normally open and one pair of normally closed contacts
An automotive-style miniature relay with the dust cover taken off
A relay is an electrically operated swit ...
s (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".
The mathematician Martin Davis Martin Davis may refer to:
* Martin Davis (Australian footballer) (born 1936), Australian rules footballer
* Martin Davis (Jamaican footballer) (born 1996), Jamaican footballer
* Martin Davis (mathematician)
Martin David Davis (March 8, 1928 ...
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."
Mathematics during the 19th century up to the mid-20th century
''Symbols and rules'': In rapid succession, the mathematics of George Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
(1847, 1854), Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic ph ...
(1879), and Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
(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".
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". The work of Frege was further simplified and amplified by Alfred North Whitehead
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applic ...
and Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ar ...
in their Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
(1910–1913).
''The paradoxes'': At the same time a number of disturbing paradoxes appeared in the literature, in particular, the Burali-Forti paradox (1897), the Russell paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
(1902–03), and the Richard Paradox
In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully betwee ...
. The resultant considerations led to Kurt Gödel
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imm ...
's paper (1931)—he specifically cites the paradox of the liar—that completely reduces rules of recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematic ...
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
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scie ...
, Stephen Kleene and J.B. Rosser's λ-calculus a finely honed definition of "general recursion" from the work of Gödel acting on suggestions of Jacques Herbrand
Jacques Herbrand (12 February 1908 – 27 July 1931) was a French mathematician. Although he died at age 23, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse and Richard Cour ...
(cf. Gödel's Princeton lectures of 1934) and subsequent simplifications by Kleene. Church's proof that the Entscheidungsproblem was unsolvable, Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Gove ...
'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 yes-no decision about the next instruction. Alan Turing's proof of that the Entscheidungsproblem was unsolvable by use of his "a- utomatic-machine"—in effect almost identical to Post's "formulation", J. Barkley Rosser's definition of "effective method" in terms of "a machine". Kleene's proposal of a precursor to " Church thesis" that he called "Thesis I", and a few years later Kleene's renaming his Thesis "Church's Thesis" and proposing "Turing's Thesis".
Emil Post (1936) and Alan Turing (1936–37, 1939)
Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Gove ...
(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 .e., INPUTbeing marked with a stroke. Likewise, the answer .e., OUTPUTis 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 terminates only when it comes to the direction of type (C ) .e., STOP. See more at Post–Turing machine
Alan Turing
Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical c ...
's work 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 typewriter-like 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. Given the prevalence at the time of Morse code, telegraphy, ticker tape machines, and teletypewriters, it is quite possible that all were influences on Turing during his youth.
Turing—his model of computation is now called a Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...
—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.
:"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 one-dimensional 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 that 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 1936–37 in Davis 1965:136]
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."
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. Though it is fairly easy to get an intuitive grasp of this idea, it is nevertheless desirable to have some more definite, mathematical expressible definition ... e 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 calculable" refer to the intuitive idea without particular identification with any one of these definitions".[Turing 1939 in Davis 1965:160]
J. B. Rosser (1939) and S. C. Kleene (1943)
J. Barkley Rosser defined an "effective athematicalmethod" in the following manner (italicization 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. is 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 1939:225–226)
Rosser's footnote No. 5 references the work of (1) Church and Kleene and their definition of λ-definability, 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 (1936–37) in their mechanism-models of computation.
Stephen C. Kleene defined as his now-famous "Thesis I" known as the Church–Turing thesis
In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of co ...
. 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 on-going because of issues surrounding, in particular, foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
(especially the Church–Turing thesis
In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of co ...
) and philosophy of mind
Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the body. The mind–body problem is a paradigmatic issue in philosophy of mind, although a number of other issues are ad ...
(especially arguments about artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machine
A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, moveme ...
). For more, see Algorithm characterizations.
See also
* Abstract machine
An abstract machine is a computer science theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is analogous to a mathematical function in that it receives inputs and produces outputs based on p ...
* ALGOL
ALGOL (; short for "Algorithmic Language") is a family of imperative computer programming languages originally developed in 1958. ALGOL heavily influenced many other languages and was the standard method for algorithm description used by th ...
* Algorithm engineering Algorithm engineering focuses on the design, analysis, implementation, optimization, profiling and experimental evaluation of computer algorithms, bridging the gap between algorithm theory and practical applications of algorithms in software enginee ...
* Algorithm characterizations
* Algorithmic bias
* Algorithmic composition
Algorithmic composition is the technique of using algorithms to create music.
Algorithms (or, at the very least, formal sets of rules) have been used to compose music for centuries; the procedures used to plot voice-leading in Western counterp ...
* Algorithmic entities Algorithmic entities refer to autonomous algorithms that operate without human control or interference. Recently, attention is being given to the idea of algorithmic entities being granted (partial or full) legal personhood. Professor Shawn Bayern ...
* Algorithmic synthesis
High-level synthesis (HLS), sometimes referred to as C synthesis, electronic system-level (ESL) synthesis, algorithmic synthesis, or behavioral synthesis, is an automated design process that takes an abstract behavioral specification of a digital ...
* Algorithmic technique In mathematics and computer science, an algorithmic technique is a general approach for implementing a process or computation.
General techniques
There are several broadly recognized algorithmic techniques that offer a proven method or process for ...
* Algorithmic topology
* Garbage in, garbage out
In computer science, garbage in, garbage out (GIGO) is the concept that flawed, or nonsense (garbage) input data produces nonsense output. Rubbish in, rubbish out (RIRO) is an alternate wording.
The principle applies to all logical argumentatio ...
* ''Introduction to Algorithms
''Introduction to Algorithms'' is a book on computer programming by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The book has been widely used as the textbook for algorithms courses at many universities and is ...
'' (textbook)
* Government by algorithm
Government by algorithm (also known as algorithmic regulation, regulation by algorithms, algorithmic governance, algocratic governance, algorithmic legal order or algocracy) is an alternative form of government or social ordering, where the usa ...
* List of algorithms
* List of algorithm general topics
* List of important publications in theoretical computer science – Algorithms
* Regulation of algorithms
Regulation of algorithms, or algorithmic regulation, is the creation of laws, rules and public sector policies for promotion and regulation of algorithms, particularly in artificial intelligence and machine learning. For the subset of AI algorith ...
* Theory of computation
In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., ...
** Computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ...
** Computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
* Computational mathematics
Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006 ...
Notes
Bibliography
*
* Bell, C. Gordon and Newell, Allen (1971), ''Computer Structures: Readings and Examples'', McGraw–Hill Book Company, New York. .
* Includes an excellent bibliography of 56 references.
* ,
* : cf. Chapter 3 ''Turing machines'' where they discuss "certain enumerable sets not effectively (mechanically) enumerable".
*
* 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. 91–109
* 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.
*
* Davis gives commentary before each article. Papers of Gödel, Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scie ...
, Turing, Rosser, Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch o ...
, and Emil Post
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Gove ...
are included; those cited in the article are listed here by author's name.
* Davis offers concise biographies of Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...
, Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Irel ...
, Frege, Cantor
A cantor or chanter is a person who leads people in singing or sometimes in prayer. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds.
In Judaism, a cantor sings and lead ...
, Hilbert, Gödel and Turing with von Neumann as the show-stealing villain. Very brief bios of Joseph-Marie Jacquard, Babbage, Ada Lovelace
Augusta Ada King, Countess of Lovelace ('' née'' Byron; 10 December 1815 – 27 November 1852) was an English mathematician and writer, chiefly known for her work on Charles Babbage's proposed mechanical general-purpose computer, the ...
, Claude Shannon
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as a "father of information theory".
As a 21-year-old master's degree student at the Massachusetts In ...
, Howard Aiken, etc.
*
*
*
* ,
* Yuri Gurevich
''Sequential Abstract State Machines Capture Sequential Algorithms''
ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp. 77–111. Includes bibliography of 33 sources.
* , 3rd edition 1976 (pbk.)
* , . Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
* Presented to the American Mathematical Society, September 1935. Reprinted in ''The Undecidable'', p. 237ff. Kleene's definition of "general recursion" (known now as mu-recursion) 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).
*
*
*
* Kosovsky, N.K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981
*
* A.A. Markov
AA, Aa, Double A, or Double-A may refer to:
Arts, entertainment and media
* ''America's Army'', a 2002 computer game published by the U.S. Army
* '' Ancient Anguish'', a computer game in existence since 1992
* Aa!, a J-Pop musical group
* Dou ...
(1954) ''Theory of algorithms''. ranslated by Jacques J. Schorr-Kon and PST staffImprint Moscow, Academy of Sciences of the USSR, 1954 .e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, WashingtonDescription 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. A248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS .* Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 ''Computability, Effective Procedures and Algorithms. Infinite machines.''
* Reprinted in ''The Undecidable'', pp. 289ff. Post defines a simple algorithmic-like 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 so-called Church–Turing thesis
In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of co ...
.
*
* 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'')
*
*
*
*
* 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. 544–546. 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'', pp. 155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.
* United States Patent and Trademark Office
The United States Patent and Trademark Office (USPTO) is an agency in the U.S. Department of Commerce that serves as the national patent office and trademark registration authority for the United States. The USPTO's headquarters are in Alex ...
(2006)
''2106.02 **>Mathematical Algorithms: 2100 Patentability''
Manual of Patent Examining Procedure (MPEP). Latest revision August 2006
Further reading
*
*
*
*
*
*
* Knuth, Donald E. (2000).
Selected Papers on Analysis of Algorithms
'. Stanford, California: Center for the Study of Language and Information.
* Knuth, Donald E. (2010).
'. Stanford, California: Center for the Study of Language and Information.
*
*
External links
*
*
*
Dictionary of Algorithms and Data Structures
– National Institute of Standards and Technology
The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into Outline of p ...
; Algorithm repositories
The Stony Brook Algorithm Repository
– State University of New York at Stony Brook
Stony Brook University (SBU), officially the State University of New York at Stony Brook, is a public research university in Stony Brook, New York. Along with the University at Buffalo, it is one of the State University of New York system's ...
Collected Algorithms of the ACM
– Association for Computing Machinery
The Association for Computing Machinery (ACM) is a US-based international learned society for computing. It was founded in 1947 and is the world's largest scientific and educational computing society. The ACM is a non-profit professional member ...
The Stanford GraphBase
– Stanford University
{{Authority control
Articles with example pseudocode
Mathematical logic
Theoretical computer science