Algorithm: Difference between revisions

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== Legal status ==

By themselves, algorithms 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), so algorithms are not patentable (as in ''[[Gottschalk v. Benson]]''). However practical applications of algorithms are sometimes patentable. For example, in ''[[Diamond v. Diehr]]'', the application of a simple [[feedback]] algorithm to aid in the curing of [[synthetic rubber]] was deemed patentable. The [[Software patent debate|patenting of software]] is controversial,<ref>{{Cite news |date=2013-05-16 |title=The Experts: Does the Patent System Encourage Innovation? |url=https://www.wsj.com/articles/SB10001424127887323582904578487200821421958 |access-date=2017-03-29 |work=[[The Wall Street Journal]] |issn=0099-9660}}</ref> and there are criticized patents involving algorithms, especially [[data compression]] algorithms, such as [[Unisys]]'s [[Graphics Interchange Format#Unisys and LZW patent enforcement|LZW patent]]. Additionally, some cryptographic algorithms have export restrictions (see [[export of cryptography]]).

{{see also|Software patent}}

By themselves, algorithms 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), so algorithms are not patentable (as in ''[[Gottschalk v. Benson]]''). However practical applications of algorithms are sometimes patentable. For example, in ''[[Diamond v. Diehr]]'', the application of a simple [[feedback]] algorithm to aid in the curing of [[synthetic rubber]] was deemed patentable. The [[Software patent debate|patenting of software]] is controversial,<ref>{{Cite news |date=2013-05-16 |title=The Experts: Does the Patent System Encourage Innovation? |url=https://www.wsj.com/articles/SB10001424127887323582904578487200821421958 |access-date=2017-03-29 |work=[[The Wall Street Journal]] |issn=0099-9660}}</ref> and there are criticized patents involving algorithms, especially [[data compression]] algorithms, such as [[Unisys]]'s [[Graphics Interchange Format#Unisys and LZW patent enforcement|LZW patent]]. Additionally, some cryptographic algorithms have export restrictions (see [[export of cryptography]]).

== Classification ==

=== By implementation ===

; Recursion

: A [[recursive algorithm]] invokes itself repeatedly until meeting a termination condition, and is a common [[functional programming]] method. [[Iteration|Iterative]] algorithms use repetitions such as [[Program loops|loop]]s or data structures like [[Stack (data structure)|stack]]sstacks to solve problems. Problems may be suited for one implementation or the other.[[Towers of Hanoi]] is a puzzle commonly solved using recursive implementation. Every recursive version has an equivalent (but possibly more or less complex) iterative version, and vice versa.

; Serial, parallel or distributed

: Algorithms are usually discussed with the assumption that computers execute one instruction of an algorithm at a time on serial computers. Serial algorithms are designed for these environments, unlike [[parallel algorithm|parallel]] or [[distributed algorithm|distributed]] algorithms. Parallel algorithms take advantage of computer architectures where multiple processors can work on a problem at the same time. Distributed algorithms use multiple machines connected via a computer network. Parallel and distributed algorithms divide the problem into subproblems and collect the results back together. Resource consumption in these 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, but some problems have no parallel algorithms and are called inherently serial problems.

; Deterministic or non-deterministic

: [[Deterministic algorithm]]salgorithms solve the problem with exact decision at every step; whereas [[non-deterministic algorithm]]salgorithms solve problems via guessing. Guesses are typically made more accurate through the use of [[heuristics]].

; Exact or approximate

: While many algorithms reach an exact solution, [[approximation algorithm]]salgorithms seek an approximation that is close to the true solution. Such algorithms have practical value for many hard problems. For example, the [[Knapsack problem]], where there is a set of items and the goal is to pack the knapsack to get the maximum total value. Each item has some weight and some value. The 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.<ref>{{Cite book|url=https://www.springer.com/us/book/9783540402862|title=Knapsack Problems {{!}} Hans Kellerer {{!}} Springer|language=en|isbn=978-3-540-40286-2|publisher=Springer|year=2004|doi=10.1007/978-3-540-24777-7|access-date=September 19, 2017|archive-url=https://web.archive.org/web/20171018181055/https://www.springer.com/us/book/9783540402862|archive-date=October 18, 2017|url-status=live|last1=Kellerer|first1=Hans|last2=Pferschy|first2=Ulrich|last3=Pisinger|first3=David|s2cid=28836720 }}</ref>

; Quantum algorithm

: [[Quantum algorithm]]salgorithms run on a realistic model of [[quantum computation]]. The term is usually used for those algorithms which seem inherently quantum or use some essential feature of [[Quantum computing]] such as [[quantum superposition]] or [[quantum entanglement]].

=== By design paradigm ===

Another way of classifying algorithms is by their design methodology or [[algorithmic paradigm|paradigm]]. Some common paradigms are:

; [[Brute-force search|Brute-force]] or exhaustive search

: Brute force is a problem-solving method of systematically trying every possible option until the optimal solution is found. This approach can be very time-consuming, testing every possible combination of variables. It is often used when other methods are unavailable or too complex. Brute force can solve a variety of problems, including finding the shortest path between two points and cracking passwords.

; Divide and conquer

: A [[divide-and-conquer algorithm]] repeatedly reduces a problem to one or more smaller instances of itself (usually [[recursion|recursively]]) until the instances are small enough to solve easily. [[mergesort|Merge sorting]] is an example of divide and conquer, where an unordered list can be divided into segments containing one item and sorting of entire list can be obtained by merging the segments. A simpler variant of divide and conquer is called a ''decrease-and-conquer algorithm'', which solves one smaller instance of itself, and uses the solution 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.{{Citation needed|date=October 2024}} An example of a decrease and conquer algorithm is the [[binary search algorithm]].

; Search and enumeration

: Many problems (such as playing [[Chess|ches]]s) can be modelled as problems on [[graph theory|graph]]sgraphs. A [[graph exploration algorithm]] specifies rules for moving around a graph and is useful for such problems. This category also includes [[search algorithm]]salgorithms, [[branch and bound]] enumeration, and [[backtracking]].

;[[Randomized algorithm]]

: Such algorithms make some choices randomly (or pseudo-randomly). They find approximate solutions when finding exact solutions may be impractical (see heuristic method below). For some problems the fastest approximations must involve some [[randomness]].<ref>For instance, the [[volume]] of a [[convex polytope]] (described using a membership oracle) can be approximated to high accuracy by a randomized polynomial time algorithm, but not by a deterministic one: see {{cite journal

| last1 = Dyer | first1 = Martin

| last2 = Frieze | first2 = Alan

Line 128 ⟶ 126:

| title = A Random Polynomial-time Algorithm for Approximating the Volume of Convex Bodies

| volume = 38| citeseerx = 10.1.1.145.4600| s2cid = 13268711

}}</ref> Whether randomized algorithms with [[P (complexity)|polynomial time complexity]] can be the fastest algorithm for some problems is an open question known as the [[P versus NP problem]]. There are two large classes of such algorithms:

# [[Monte Carlo algorithm]]salgorithms return a correct answer with high probability. E.g. [[RP (complexity)|RP]] is the subclass of these that run in [[polynomial time]].

# [[Las Vegas algorithm]]salgorithms always return the correct answer, but their running time is only probabilistically bound, e.g. [[Zero-error Probabilistic Polynomial time|ZPP]].

; [[Reduction (complexity)|Reduction of complexity]]

: This technique transforms difficult problems into better-known problems solvable with (hopefully) [[asymptotically optimal]] algorithms. The goal is to find a reducing algorithm whose [[Computational complexity theory|complexity]] is not dominated by the resulting reduced algorithms. For example, one [[selection algorithm]] finds the median of an unsorted list by first sorting the list (the expensive portion), then pulling out the middle element in the sorted list (the cheap portion). This technique is also known as ''[[Transform and conquer algorithm|transform and conquer]]''.

; [[Back tracking]]

: 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]]sproblems 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]]

: When searching for optimal solutions to a linear function bound by linear equality and inequality constraints, the constraints can be used directly to produce optimal solutions. There are algorithms that can solve any problem in this category, such as the popular [[simplex algorithm]].<ref>

[[George B. Dantzig]] and Mukund N. Thapa. 2003. ''Linear Programming 2: Theory and Extensions''. Springer-Verlag.</ref> Problems that can be solved with linear programming include the [[maximum flow problem]] for directed graphs. If a problem also requires that any of the unknowns be [[integer|integers]], then it is classified in [[integer programming]]. 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]]

: When a problem shows optimal substructures—meaning the optimal solution can be constructed from optimal solutions to subproblems—and [[overlapping subproblem]]ssubproblems, meaning the same subproblems are used to solve many different problem instances, a quicker approach called ''dynamic programming'' avoids recomputing solutions. For example, [[Floyd–Warshall algorithm]], the shortest path between a start and goal vertex in a weighted [[graph (discrete mathematics)|graph]] can be found using the shortest path to the goal from all adjacent vertices. Dynamic programming and [[memoization]] go together. Unlike divide and conquer, dynamic programming subproblems often overlap. The difference between dynamic programming and simple recursion is the caching or memoization of recursive calls. When subproblems are independent and do not repeat, memoization does not help; hence dynamic programming is not applicable to all complex problems. Using memoization dynamic programming reduces the complexity of many problems from exponential to polynomial.

; The greedy method

: [[greedy algorithm|Greedy algorithms]], similarly to a dynamic programming, work by examining substructures, in this case not of the problem but of a given solution. Such algorithms start with some solution and improve it by making small modifications. For some problems they always find the optimal solution but for others they may stop at [[local optimum|local optima]]. The most popular use of greedy algorithms is finding minimal spanning trees of graphs without negative cycles. [[Huffman coding|Huffman Tree]], [[kruskal's algorithm|Kruskal]], [[Prim's algorithm|Prim]], [[Sollin's algorithm|Sollin]] are greedy algorithms that can solve this optimization problem.

;The heuristic method

:In [[optimization problem]]sproblems, [[heuristic algorithm]]s find solutions close to the optimal solution when finding the optimal solution is impractical. These algorithms get 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. They can ideally find a solution very close to the optimal solution in a relatively short time. These algorithms include [[local search (optimization)|local search]], [[tabu search]], [[simulated annealing]], and [[genetic algorithm|genetic algoAArithm]]s. Some, 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]].

== Examples ==

{{Further|List of algorithms}}

One of the simplest algorithms finds 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 described in plain English as:

Line 161 ⟶ 157:

''(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]] or [[pidgin code]]:111111

{{algorithm-begin|name=LargestNumber}}

Input: A list of numbers ''L''.

Output: The largest number in the list ''L''.

'''if''' ''L.size'' = 0 '''return''' null

''largest'' ← ''L''[0]

'''for each''' ''item'' '''in''' ''L'', '''do'''

'''if''' ''item'' > ''largest'', '''then'''

''largest'' ← ''item''

'''return''' ''largest''

{{algorithm-end}}

== See also ==

Line 203 ⟶ 187:

== Notes ==

{{Reflist}}

== Bibliography ==

{{refbegin|30em}}

* {{cite journal | last1 = Axt | first1 = P | year = 1959 | title = On a Subrecursive Hierarchy and Primitive Recursive Degrees | journal = Transactions of the American Mathematical Society | volume = 92 | issue = 1| pages = 85–105 | doi=10.2307/1993169| jstor = 1993169 | doi-access = free}}

* Bell, C. Gordon and Newell, Allen (1971), ''Computer Structures: Readings and Examples'', McGraw–Hill Book Company, New York. {{ISBN|0-07-004357-4}}.

* {{Cite journal|author1-link=Andreas Blass|first1=Andreas|last1=Blass|author2-link=Yuri Gurevich|first2=Yuri|last2=Gurevich|year=2003|url=http://research.microsoft.com/~gurevich/Opera/164.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://research.microsoft.com/~gurevich/Opera/164.pdf |archive-date=2022-10-09 |url-status=live|title=Algorithms: A Quest for Absolute Definitions|journal= Bulletin of European Association for Theoretical Computer Science|volume= 81}} Includes a bibliography of 56 references.

* {{cite book| last = Bolter| first = David J.| title = Turing's Man: Western Culture in the Computer Age| edition = 1984| year = 1984| publisher = The University of North Carolina Press|___location= Chapel Hill, NC| isbn = 978-0-8078-1564-9 }}, {{ISBN|0-8078-4108-0}}

* {{cite book| last1 = Boolos| first1 = George| last2 = Jeffrey| first2 = Richard| title = Computability and Logic| url = https://archive.org/details/computabilitylog0000bool_r8y9| url-access = registration| edition = 4th| orig-year = 1974| year = 1999| publisher = Cambridge University Press, London| isbn = 978-0-521-20402-6| author1-link = George Boolos| author2-link = Richard Jeffrey }}: cf. Chapter 3 ''Turing machines'' where they discuss "certain enumerable sets not effectively (mechanically) enumerable".

* {{cite book| last = Burgin| first = Mark| title = Super-Recursive Algorithms| year = 2004| publisher = Springer| isbn = 978-0-387-95569-8 }}

* Campagnolo, M.L., [[Cris Moore|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.&nbsp;91–109

* {{Cite journal|last=Church|first=Alonzo|author-link=Alonzo Church|title=An Unsolvable Problem of Elementary Number Theory|journal=American Journal of Mathematics|volume=58|pages= 345–363|year=1936|doi=10.2307/2371045|issue=2|jstor=2371045}} Reprinted in ''The Undecidable'', p.&nbsp;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.

* {{Cite journal|last=Church|first=Alonzo|author-link=Alonzo Church|title=A Note on the Entscheidungsproblem|journal=The Journal of Symbolic Logic|volume=1|year=1936|pages=40–41|doi=10.2307/2269326|issue=1|jstor=2269326|s2cid=42323521 }} {{cite journal|last=Church|first=Alonzo|title=Correction to a Note on the Entscheidungsproblem|journal=The Journal of Symbolic Logic|volume=1|year=1936|pages=101–102|doi=10.2307/2269030|issue=3|jstor=2269030|s2cid=5557237 }} Reprinted in ''The Undecidable'', p.&nbsp;110ff. Church shows that the Entscheidungsproblem is unsolvable in about 3 pages of text and 3 pages of footnotes.

* {{cite book| last = Daffa'| first = Ali Abdullah al-| title = The Muslim contribution to mathematics| year = 1977| publisher = Croom Helm| ___location = London| isbn = 978-0-85664-464-1 }}

* {{cite book| last = Davis| first = Martin| author-link = Martin Davis (mathematician)| title = The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions| url = https://archive.org/details/undecidablebasic0000davi| url-access = registration| year = 1965| publisher = Raven Press| ___location = New York| isbn = 978-0-486-43228-1 }} Davis gives commentary before each article. Papers of [[Gödel]], [[Alonzo Church]], [[Alan Turing|Turing]], [[J. Barkley Rosser|Rosser]], [[Kleene]], and [[Emil Post]] are included; those cited in the article are listed here by author's name.

* {{cite book| last = Davis| first = Martin| author-link = Martin Davis (mathematician)| title = Engines of Logic: Mathematicians and the Origin of the Computer| year = 2000| publisher = W.W. Nortion| ___location = New York| isbn = 978-0-393-32229-3 }} Davis offers concise biographies of [[Gottfried Leibniz|Leibniz]], [[George Boole|Boole]], [[Gottlob Frege|Frege]], [[Georg Cantor|Cantor]], [[David Hilbert|Hilbert]], Gödel and Turing with [[John von Neumann|von Neumann]] as the show-stealing villain. Very brief bios of [[Joseph-Marie Jacquard]], [[Babbage]], [[Ada Lovelace]], [[Claude Shannon]], [[Howard Aiken]], etc.

* {{DADS|algorithm|algorithm}}

* {{cite journal|title= Evolution and moral diversity |author=Dean, Tim |journal=Baltic International Yearbook of Cognition, Logic and Communication|year=2012|volume=7|doi=10.4148/biyclc.v7i0.1775 |doi-access=free}}

* {{cite book| last = Dennett| first = Daniel| author-link = Daniel Dennett| title = Darwin's Dangerous Idea| pages = [https://archive.org/details/darwinsdangerous0000denn/page/32 32]–36| year = 1995| publisher = Touchstone/Simon & Schuster| ___location = New York| isbn = 978-0-684-80290-9| url = https://archive.org/details/darwinsdangerous0000denn| url-access = registration}}

* {{cite book| last = Dilson| first = Jesse| title = The Abacus| edition = (1968, 1994)| year = 2007| publisher = St. Martin's Press, NY| isbn = 978-0-312-10409-2| url = https://archive.org/details/abacusworldsfirs0000dils}}, {{ISBN|0-312-10409-X}}

* [[Yuri Gurevich]], [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.3017&rep=rep1&type=pdf ''Sequential Abstract State Machines Capture Sequential Algorithms''], ACM Transactions on Computational Logic, Vol 1, no 1 (July 2000), pp.&nbsp;77–111. Includes bibliography of 33 sources.

* {{cite book| last = van Heijenoort| first = Jean| author-link = Jean van Heijenoort| title = From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931| edition = (1967)| year = 2001| publisher = Harvard University Press, Cambridge| isbn = 978-0-674-32449-7 }}, 3rd edition 1976[?], {{ISBN|0-674-32449-8}} (pbk.)

* {{cite book| last = Hodges| first = Andrew| author-link = Andrew Hodges| title = Alan Turing: The Enigma| year = 1983| publisher = [[Simon and Schuster]]| ___location = New York| isbn = 978-0-671-49207-6| title-link = Alan Turing: The Enigma}}, {{ISBN|0-671-49207-1}}. Cf. Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.

* {{Cite journal|last=Kleene|first=Stephen C.|author-link=Stephen Kleene|title=General Recursive Functions of Natural Numbers|journal=Mathematische Annalen|volume=112|pages=727–742|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1|year=1936|doi=10.1007/BF01565439|issue=5|s2cid=120517999|access-date=September 30, 2013|archive-url=https://web.archive.org/web/20140903092121/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002278499&L=1|archive-date=September 3, 2014|url-status=dead}} Presented to the American Mathematical Society, September 1935. Reprinted in ''The Undecidable'', p.&nbsp;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).

* {{Cite journal|last=Kleene|first=Stephen C.|author-link=Stephen Kleene |title= Recursive Predicates and Quantifiers|journal= Transactions of the American Mathematical Society|volume=53|pages=41–73|year=1943 |doi= 10.2307/1990131|issue=1|jstor=1990131|doi-access=free}} Reprinted in ''The Undecidable'', p.&nbsp;255ff. Kleene refined his definition of "general recursion" and proceeded in his chapter "12. Algorithmic theories" to posit "Thesis I" (p.&nbsp;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]]).

* {{cite book| last = Kleene| first = Stephen C.| author-link = Kleene| title = Introduction to Metamathematics| edition = Tenth|year= 1991| orig-year = 1952| publisher = North-Holland Publishing Company| isbn = 978-0-7204-2103-3 }}

* {{cite book| last = Knuth| first = Donald| author-link = Donald Knuth| title = Fundamental Algorithms, Third Edition| year = 1997| publisher = Addison–Wesley| ___location = Reading, Massachusetts| isbn = 978-0-201-89683-1 }}

* {{Cite book|last=Knuth|first=Donald|author-link=Donald Knuth|title=Volume 2/Seminumerical Algorithms, The Art of Computer Programming First Edition|publisher=Addison–Wesley|___location=Reading, Massachusetts|year=1969}}

* Kosovsky, N.K. ''Elements of Mathematical Logic and its Application to the theory of Subrecursive Algorithms'', LSU Publ., Leningrad, 1981

* {{Cite journal|last=Kowalski|first=Robert|author-link=Robert Kowalski|title=Algorithm=Logic+Control|journal=[[Communications of the ACM]]|volume=22|issue=7|pages=424–436|year=1979|doi=10.1145/359131.359136|s2cid=2509896|doi-access=free}}

* [[A.A. Markov]] (1954) ''Theory of algorithms''. [Translated by Jacques J. Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of the USSR, 1954 [i.e., Jerusalem, Israel Program for Scientific Translations, 1961; available from the Office of Technical Services, U.S. Dept. of Commerce, Washington] Description 444 p.&nbsp;28&nbsp;cm. Added t.p. in Russian Translation of Works of the Mathematical Institute, Academy of Sciences of the USSR, v.&nbsp;42. Original title: Teoriya algerifmov. [QA248.M2943 Dartmouth College library. U.S. Dept. of Commerce, Office of Technical Services, number OTS {{not a typo|60-51085}}.]

* {{cite book| last = Minsky| first = Marvin| author-link = Marvin Minsky| title = Computation: Finite and Infinite Machines| url = https://archive.org/details/computationfinit0000mins| url-access = registration| edition = First| year = 1967| publisher = Prentice-Hall, Englewood Cliffs, NJ| isbn = 978-0-13-165449-5 }} Minsky expands his "...idea of an algorithm – an effective procedure..." in chapter 5.1 ''Computability, Effective Procedures and Algorithms. Infinite machines.''

* {{Cite journal|last=Post|first=Emil|author-link=Emil Post|title=Finite Combinatory Processes, Formulation I |journal=The Journal of Symbolic Logic |volume=1 |year=1936 |pages=103–105 |doi=10.2307/2269031 |issue=3 |jstor=2269031|s2cid=40284503 }} Reprinted in ''The Undecidable'', pp.&nbsp;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]].

* {{Cite book|last=Rogers|first=Hartley Jr.|title=Theory of Recursive Functions and Effective Computability|publisher=The MIT Press|year=1987|isbn=978-0-262-68052-3}}

* {{Cite journal|last=Rosser|first=J.B.|author-link=J. B. Rosser|title=An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem|journal=Journal of Symbolic Logic|volume= 4 |issue=2|year=1939|doi=10.2307/2269059|pages=53–60|jstor=2269059|s2cid=39499392 }} Reprinted in ''The Undecidable'', p.&nbsp;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.&nbsp;225–226, ''The Undecidable'')

* {{cite book |last=Santos-Lang |first=Christopher |editor1-first=Simon |editor1-last=van Rysewyk |editor2-first=Matthijs |editor2-last=Pontier |title=Machine Medical Ethics |volume=74 |publisher=Springer | ___location=Switzerland | pages=111–127 | chapter=Moral Ecology Approaches to Machine Ethics| chapter-url=http://grinfree.com/MoralEcology.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://grinfree.com/MoralEcology.pdf |archive-date=2022-10-09 |url-status=live | doi=10.1007/978-3-319-08108-3_8|series=Intelligent Systems, Control and Automation: Science and Engineering |date=2015 |isbn=978-3-319-08107-6 }}

* {{Cite book|last=Scott|first=Michael L.|title=Programming Language Pragmatics |edition=3rd |publisher=Morgan Kaufmann Publishers/Elsevier|year=2009|isbn=978-0-12-374514-9}}

* {{cite book| last = Sipser| first = Michael| title = Introduction to the Theory of Computation| year = 2006| publisher = PWS Publishing Company| isbn = 978-0-534-94728-6| url = https://archive.org/details/introductiontoth00sips}}

* {{cite book |last1=Sober |first1=Elliott |last2=Wilson |first2=David Sloan |year=1998 |title=Unto Others: The Evolution and Psychology of Unselfish Behavior |url=https://archive.org/details/untoothersevolut00sobe |url-access=registration |___location=Cambridge |publisher=Harvard University Press|isbn=9780674930469 }}

* {{Cite book|last=Stone|first=Harold S.|title=Introduction to Computer Organization and Data Structures|edition=1972|publisher=McGraw-Hill, New York|isbn=978-0-07-061726-1|year=1972}} 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.&nbsp;4).

* {{cite book| last = Tausworthe| first = Robert C| title = Standardized Development of Computer Software Part 1 Methods| year = 1977| publisher = Prentice–Hall, Inc.| ___location = Englewood Cliffs NJ| isbn = 978-0-13-842195-3 }}

* {{Cite journal|last=Turing|first=Alan M.|author-link=A. M. Turing|title=On Computable Numbers, With An Application to the Entscheidungsproblem|journal=[[Proceedings of the London Mathematical Society]]|series=Series 2|volume=42|pages= 230–265 |year=1936–37|doi=10.1112/plms/s2-42.1.230 |s2cid=73712 }}. Corrections, ibid, vol. 43(1937) pp.&nbsp;544–546. Reprinted in ''The Undecidable'', p.&nbsp;116ff. Turing's famous paper completed as a Master's dissertation while at King's College Cambridge UK.

* {{Cite journal|last=Turing|first=Alan M.|author-link=A. M. Turing|title=Systems of Logic Based on Ordinals|journal=Proceedings of the London Mathematical Society|volume=45|pages=161–228|year=1939|doi=10.1112/plms/s2-45.1.161|hdl=21.11116/0000-0001-91CE-3|hdl-access=free}} Reprinted in ''The Undecidable'', pp.&nbsp;155ff. Turing's paper that defined "the oracle" was his PhD thesis while at Princeton.

* [[United States Patent and Trademark Office]] (2006), [http://www.uspto.gov/web/offices/pac/mpep/documents/2100_2106_02.htm ''2106.02 **>Mathematical Algorithms: 2100 Patentability''], Manual of Patent Examining Procedure (MPEP). Latest revision August 2006

{{refend|30em}}

* Zaslavsky, C. (1970). Mathematics of the Yoruba People and of Their Neighbors in Southern Nigeria. The Two-Year College Mathematics Journal, 1(2), 76–99. https://doi.org/10.2307/3027363

==Further reading==

{{refbegin}}

* {{cite book |last=Bellah |first=Robert Neelly |year=1985 |author-link=Robert N. Bellah |title=Habits of the Heart: Individualism and Commitment in American Life |___location=Berkeley |isbn=978-0-520-25419-0 |publisher=University of California Press |url=https://books.google.com/books?id=XsUojihVZQcC }}

* {{cite book |last=Berlinski |first=David |title=The Advent of the Algorithm: The 300-Year Journey from an Idea to the Computer |year=2001 |publisher=Harvest Books |isbn=978-0-15-601391-8 |url=https://archive.org/details/adventofalgorith0000berl }}

* {{cite book |last=Chabert |first=Jean-Luc |title=A History of Algorithms: From the Pebble to the Microchip |year=1999 |publisher=Springer Verlag |isbn=978-3-540-63369-3}}

* {{cite book |author1=Thomas H. Cormen |author2=Charles E. Leiserson |author3=Ronald L. Rivest |author4=Clifford Stein |title=Introduction To Algorithms |edition=3rd |year=2009 |publisher=MIT Press |isbn=978-0-262-03384-8}}

* {{cite book |author=Harel, David |author2=Feldman, Yishai |title=Algorithmics: The Spirit of Computing |year=2004 |publisher=Addison-Wesley |isbn=978-0-321-11784-7}}

* {{cite book |last1=Hertzke |first1=Allen D. |last2=McRorie |first2=Chris |year=1998 |editor1-last=Lawler |editor1-first=Peter Augustine |editor2-last=McConkey |editor2-first=Dale |chapter=The Concept of Moral Ecology |title=Community and Political Thought Today |___location=Westport, CT |publisher=[[Praeger Publishers|Praeger]] }}

* [[Donald Knuth|Knuth, Donald E.]] (2000). ''[http://www-cs-faculty.stanford.edu/~uno/aa.html Selected Papers on Analysis of Algorithms] {{Webarchive|url=https://web.archive.org/web/20170701190647/http://www-cs-faculty.stanford.edu/~uno/aa.html |date=July 1, 2017 }}''. Stanford, California: Center for the Study of Language and Information.

* Knuth, Donald E. (2010). ''[http://www-cs-faculty.stanford.edu/~uno/da.html Selected Papers on Design of Algorithms] {{Webarchive|url=https://web.archive.org/web/20170716225848/http://www-cs-faculty.stanford.edu/~uno/da.html |date=July 16, 2017 }}''. Stanford, California: Center for the Study of Language and Information.

* {{Cite book |first1=Wendell |last1=Wallach |first2=Colin |last2=Allen |date=November 2008 |title=Moral Machines: Teaching Robots Right from Wrong |isbn=978-0-19-537404-9 |publisher=Oxford University Press |___location=US }}

* {{cite book |author=Bleakley, Chris |title=Poems that Solve Puzzles: The History and Science of Algorithms |year=2020 |publisher=Oxford University Press |isbn=978-0-19-885373-2 |url=https://books.google.com/books?id=3pr5DwAAQBAJ }}

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