Editing Algorithm (section)

== 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|publisher=McGraw-Hill, New York|isbn=9780070617261|year=1971}} 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