Inductive programming: Difference between revisions

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{{Short description|Area of automatic programming}}

{{Programming paradigms}}

'''Inductive programming''' ('''IP''') is a special area of [[automatic programming]], covering research from [[artificial intelligence]] and [[Computer programming|programming]], which addresses [[machine learning|learning]] of typically [[declarative programming|declarative]] ([[logic programming|logic]] or [[functional programming|functional]]) and often [[recursion|recursive]] programs from incomplete specifications, such as input/output examples or constraints.

Depending on the programming language used, there are several kinds of inductive programming. '''Inductive functional programming''', which uses functional programming languages such as [[Lisp (programming language)|Lisp]] or [[Haskell (programming language)|Haskell]], and most especially [[inductive logic programming]], which uses logic programming languages such as [[Prolog]] and other logical representations such as [[description logics]], have been more prominent, but other (programming) language paradigms have also been used, such as [[constraint programming]] or [[probabilistic programming language|probabilistic programming]].

== Definition ==

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Output of an IP system is a program in some arbitrary programming language containing conditionals and loop or recursive control structures, or any other kind of [[Turing completeness|Turing-complete]] [[Knowledge representation and reasoning|representation]] language.

In many applications the output program must be correct with respect to the examples and partial specification, and this leads to the consideration of inductive programming as a special area inside automatic programming or [[program synthesis]],<ref>{{cite journal|first1=A.W.|last1=Biermann|title=Automatic programming|editor1-first=S.C.|editor1-last=Shapiro|publisher=Wiley|journal=Encyclopedia of artificialArtificial intelligenceIntelligence|pages=18–35|year=1992}}</ref><ref>{{cite journalbook|first1=C.|last1=Rich|first2=R.C.|last2=Waters|title=Approaches to automatic programming|editor1-first=M.C.|editor1-last=Yovits|publisher=Academic Press|journalseries=Advances in computersComputers|volume=37|pages=1–57|year=1993|url=http://www.merl.com/publications/docs/TR92-04.pdf|doi=10.1016/S0065-2458(08)60402-7|isbn=9780120121373}}</ref> usually opposed to 'deductive' program synthesis,<ref>{{cite book|editor1-first=M.L.|editor1-last=Lowry|editor2-first=R.D.|editor2-last=McCarthy|title=Automatic software design|year=1991}}</ref><ref>{{cite journal|first1=Z.|last1=Manna|first2=R.|last2=Waldinger|title=Fundamentals of deductive program synthesis|journal=IEEE Trans Softw Eng|volume=18 | issue = 8|pages=674–704|year=1992|doi=10.1109/32.153379|citeseerx=10.1.1.51.817}}</ref><ref>{{citeCite journalbook|first1=P.|last1=Flener|titlechapter=Achievements and prospectsProspects of programProgram synthesisSynthesis |title=Computational Logic: Logic Programming and Beyond; Essays in Honour of Robert A. Kowalski|editor1-first=A.|editor1-last=Kakas|editor2-first=F.|editor2-last=Sadri|publisher=Springer|journal=Computational logic: logic programming and beyond; essays in honour of Robert A. Kowalski|volume=LNAI 2407|pages=310–346|year=2002|doi=10.1007/3-540-45628-7_13|series=Lecture Notes in Computer Science|isbn=978-3-540-43959-2}}</ref> where the specification is usually complete.

In other cases, inductive programming is seen as a more general area where any declarative programming or representation language can be used and we may even have some degree of error in the examples, as in general [[machine learning]], the more specific area of [[structure mining]] or the area of [[symbolic artificial intelligence]]. A distinctive feature is the number of examples or partial specification needed. Typically, inductive programming techniques can learn from just a few examples.

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== History ==

Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers<ref>{{cite journal|first1=P.D.|last1=Summers|title=A methodology for LISP program construction from examples|journal=J ACM|volume=24 | issue = 1|pages=161–175|year=1977|doi=10.1145/321992.322002|s2cid=7474210|doi-access=free}}</ref> and work of Biermann.<ref>{{cite journal|first1=A.W.|last1=Biermann|title=The inference of regular LISP programs from examples|journal=IEEE Trans Syst Man Cybern|volume=8 | issue = 8|pages=585–600|year=1978|doi=10.1109/tsmc.1978.4310035|s2cid=15277948}}</ref>

These approaches were split into two phases: first, input-output examples are transformed into non-recursive programs (traces) using a small set of basic operators; second, regularities in the traces are searched for and used to fold them into a recursive program. The main results until the mid -1980s are surveyed by Smith.<ref>{{cite journal|first1=D.R.|last1=Smith|title=The synthesis of LISP programs from examples: a survey|editor1-first=A.W.|editor1-last=Biermann|editor2-first=G.|editor2-last=Guiho|publisher=Macmillan|journal=Automatic programProgram constructionConstruction techniquesTechniques|pages=307–324|year=1984|url=https://www.researchgate.net/publication/239059541}}</ref> Due to limited progress with respect to the range of programs that could be synthesized, research activities decreased significantly in the next decade.

The advent of logic programming brought a new elan but also a new direction in the early 1980s, especially due to the MIS system of Shapiro<ref>{{cite book|first1=E.Y.|last1=Shapiro|title=Algorithmic program debugging|publisher=The MIT Press|year=1983}}</ref> eventually spawning the new field of inductive logic programming (ILP).<ref>{{Cite journal | last1 = Muggleton | first1 = S. | title = Inductive logic programming | doi = 10.1007/BF03037089 | journal = New Generation Computing | volume = 8 | issue = 4 | pages = 295–318 | year = 1991 | pmidciteseerx = 10.1.1.329.5312 | pmcs2cid = 5462416 }}</ref> The early works of Plotkin,<ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|publisher=Edinburgh University Press|journal=Machine Intelligence|volume=5|pages=153–163|year=1970|url=http://homepages.inf.ed.ac.uk/gdp/publications/MI5_note_ind_gen.pdf}}</ref><ref>{{cite journal|first1=Gordon D.|last1=Plotkin|title=A Further Note on Inductive Generalization|editor1-first=B.|editor1-last=Meltzer|editor2-first=D.|editor2-last=Michie|publisher=Edinburgh University Press|journal=Machine Intelligence|volume=6|pages=101–124|year=1971}}</ref> and his "''relative least general generalization (rlgg)''", had an enormous impact in inductive logic programming. Most of ILP work addresses a wider class of problems, as the focus is not only on recursive logic programs but on machine learning of symbolic hypotheses from logical representations. However, there were some encouraging results on learning recursive Prolog programs such as quicksort from examples together with suitable background knowledge, for example with GOLEM.<ref>{{cite journal|first1=S.H.|last1=Muggleton|first2=C.|last2=Feng|s2cid=14992676|title=Efficient induction of logic programs|publisher=the Japanese Society for AI|journal=Proceedings of the Workshop on Algorithmic Learning Theory|volume=6|pages=368–381|year=1990|url=https://pdfs.semanticscholar.org/6a66/9636e0ada62a0fb444e95435e24fdbdf4dbd.pdf}}</ref> But again, after initial success, the community got disappointed by limited progress about the induction of recursive programs<ref>{{cite journal

|first1=J.R.|last1=Quinlan|first2=R.M.|last2=Cameron-Jones

|s2cid=11138624|title=Avoiding Pitfalls When Learning Recursive Theories

|publisher=

|journal=IJCAI

|pages=1050–1057

|year=1993

}}

|url=https://pdfs.semanticscholar.org/4af3/18f1d267889faec6ecf1be6bc5fe570838dd.pdf}}

</ref><ref>{{cite journal|first1=J.R.|last1=Quinlan|first2=R.M.|last2=Cameron-Jones|title=Induction of logic programs: FOIL and related systems|publisher=Springer|volume=13|issue=3–4|pages=287–312|year=1995|url=http://dottorato.di.uniba.it/dottoratoXXVI/dm/FOILvsRelatedSystems.pdf|access-date=2017-09-07|archive-date=2017-09-07|archive-url=https://web.archive.org/web/20170907080358/http://dottorato.di.uniba.it/dottoratoXXVI/dm/FOILvsRelatedSystems.pdf|url-status=dead}}</ref><ref>

</ref><ref>

{{cite journal

|first1=J.R.|last1=Quinlan|first2=R.M.|last2=Cameron-Jones

|title=Induction of logic programs: FOIL and related systems

|publisher=Springer

|volume=13 |issue=3-4|pages=287–312

|year=1995

|url=http://dottorato.di.uniba.it/dottoratoXXVI/dm/FOILvsRelatedSystems.pdf}}

</ref><ref>

{{cite journal

|first1=P.|last1=Flener|first2=S.|last2=Yilmaz

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|volume=41 | issue = 2

|pages=141–195

|year=1999|doi=10.1016/s0743-1066(99)00028-x|doi-access=free}}

</ref> with ILP less and less focusing on recursive programs and leaning more and more towards a machine learning setting with applications in [[relational data mining]] and knowledge discovery.<ref>{{citation|first1=Sašo|last1=Džeroski|contribution=Inductive Logic Programming and Knowledge Discovery in Databases|pages=117–152|editor1-first=U.M.|editor1-last=Fayyad|editor2-first=G.|editor2-last=Piatetsky-Shapiro|editor3-first=P.|editor3-last=Smith|editor4-first=R.|editor4-last=Uthurusamy|title=Advances in Knowledge Discovery and Data Mining|publisher=MIT Press|year=1996}}</ref>

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|publisher=MIT Press

|year=1992

|url=https://books.google.com/books?id=Bhtxo60BV0EC&printsec|isbn=frontcover#v=onepage&q&f=false}}9780262111706

}}

</ref> proposed genetic programming in the early 1990s as a generate-and-test based approach to learning programs. The idea of genetic programming was further developed into the inductive programming system ADATE<ref>

</ref> proposed [[genetic programming]] in the early 1990s as a generate-and-test based approach to learning programs. The idea of genetic programming was further developed into the inductive programming system ADATE<ref>

{{cite journal

|first1=J.R.|last1=Olsson

|title=Inductive functional programming using incremental program transformation

|publisher=Elsevier

|journal=Artificial Intelligence

|volume=74 | issue = 1|pages=55–83

|year=1995|doi=10.1016/0004-3702(94)00042-y|doi-access=free}}

</ref> and the systematic-search-based system MagicHaskeller.<ref>

{{cite journalbook

|first1=Susumu|last1=Katayama

|title=PRICAI 2008: Trends in Artificial Intelligence

|title=Efficient exhaustive generation of functional programs using Monte-Carlo search with iterative deepening

|chapter=Efficient Exhaustive Generation of Functional Programs Using Monte-Carlo Search with Iterative Deepening

|journal=PRICAI 2008: Trends in Artificial Intelligence

|volume=5351

|pages=199–210

|year=2008

|chapter-url=http://nautilus.cs.miyazaki-u.ac.jp/~skata/skatayama_pricai2008.pdf}}|doi=10.1007/978-3-540-89197-0_21

|citeseerx=10.1.1.606.1447

|series=Lecture Notes in Computer Science

|isbn=978-3-540-89196-3

}}

</ref> Here again, functional programs are learned from sets of positive examples together with an output evaluation (fitness) function which specifies the desired input/output behavior of the program to be learned.

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|first1=D.|last1=Angluin|first2=Smith|last2=C.H.

|title=Inductive inference: Theory and methods

|publisher=ACM

|journal=ACM Computing Surveys

|volume=15|issue=3|pages=237–269

|year=1983|doi=10.1145/356914.356918|s2cid=3209224}}

</ref> The results in terms of learnability were related to classical concepts, such as identification-in-the-limit, as introduced in the seminal work of Gold.<ref>

{{cite journal

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|last1=Gold

|title=Language identification in the limit

|publisher=Elsevier

|journal=Information and Control

|volume=10

|issue=5

|pages=447–474

|url=http://www.isrl.uiuc.edu/~amag/langev/paper/gold67limit.html

|year=1967

|doi=10.1016/s0019-9958(67)91165-5

|deadurldoi-access=yesfree

}}

|archiveurl=https://web.archive.org/web/20090125120159/http://www.isrl.uiuc.edu/~amag/langev/paper/gold67limit.html

</ref> More recently, the language learning problem was addressed by the inductive programming community.<ref>{{cite journal|first1=Stephen|last1=Muggleton|title=Inductive Logic Programming: Issues, Results and the Challenge of Learning Language in Logic|journal=Artificial Intelligence|volume=114|issue=1–2|pages=283–296|year=1999|doi=10.1016/s0004-3702(99)00067-3|doi-access=free}}; here: Sect.2.1</ref><ref>

|archivedate=2009-01-25

|df=

}}

</ref> More recently, the language learning problem was addressed by the inductive programming community.<ref>{{cite journal|first1=Stephen|last1=Muggleton|title=Inductive Logic Programming: Issues, Results and the Challenge of Learning Language in Logic|journal=Artificial Intelligence|volume=114|pages=283–296|year=1999|doi=10.1016/s0004-3702(99)00067-3}}; here: Sect.2.1</ref><ref>

{{cite journal

|first1=J.R.|last1=Olsson|first2=D.M.W.|last2=Powers

|title=Machine learning of human language through automatic programming

|publisher=University of New South Wales

|journal=Proceedings of the International Conference on Cognitive Science

|pages=507–512

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In the recent years, the classical approaches have been resumed and advanced with great success. Therefore, the synthesis problem has been reformulated on the background of constructor-based term rewriting systems taking into account modern techniques of functional programming, as well as moderate use of search-based strategies and usage of background knowledge as well as automatic invention of subprograms. Many new and successful applications have recently appeared beyond program synthesis, most especially in the area of data manipulation, programming by example and cognitive modelling (see below).

Other ideas have also been explored with the common characteristic of using declarative languages for the representation of hypotheses. For instance, the use of higher-order features, schemes or structured distances have been advocated for a better handling of [[recursive data typestype]]s and structures;<ref>

{{cite journal

|first1=J.W.|last1=Lloyd

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|title=Logic for learning: learning comprehensible theories from structured data

|publisher=Springer

|year=2003}}

|url=https://books.google.com/books?id=8dioCAAAQBAJ|isbn=9783662084069

}}

</ref><ref>

{{cite journal

|first1=V.|last1=Estruch|first2=C.|last2=Ferri|first3=J.|last3=Hernandez-Orallo|first4=M.J.|last4=Ramirez-Quintana

|s2cid=7255690|title=Bridging the gap between distance and generalization

|publisher=Wiley

|journal=Computational Intelligence

|volume=30|issue=3|pages=473–513|year=2014

|doi=10.1111/coin.12004|doi-access=free|hdl=10251/34946|hdl-access=free}}

|url=https://pdfs.semanticscholar.org/48f2/2821220555f8e327c2aa9614fb28c98f9542.pdf}}

</ref> abstraction has also been explored as a more powerful approach to [[cumulative learning]] and function invention.<ref>

{{cite journal

|first1=R.J.|last1=Henderson|first2=S.H.|last2=Muggleton

|title=Automatic invention of functional abstractions

|publisher=Imperial College Press

|journal=Advances in Inductive Logic Programming

|year=2012

|url=http://ilp11.doc.ic.ac.uk/short_papers/ilp2011_submission_62.pdf}}

</ref><ref name="Inducing probabilistic programs by">{{cite arXiv

</ref><ref>

{{cite arXiv

|first1=H.|last1=Irvin|first2=A.|last2=Stuhlmuller|first3=N.D.|last3=Goodman

|title=Inducing probabilistic programs by Bayesian program merging|arxiveprint=1110.5667

|year=2011|class=cs.AI}}</ref>

</ref>

One powerful paradigm that has been recently used for the representation of hypotheses in inductive programming (generally in the form of [[generative model]]s) is [[probabilistic programming language|probabilistic programming]] (and related paradigms, such as stochastic logic programs and Bayesian logic programming).<ref>{{cite journal

|first1=S.

{{cite journal

|first1=S.|last1=Muggleton

|title=Learning stochastic logic programs

|journal=Electron. Trans. Artif. Intell.

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|pages=141–153

|year=2000

|url=https://ocs.aaai.org/Papers/Workshops/2000/WS-00-06/WS00-06-006.pdf}}

|access-date=2017-09-07

</ref><ref>

|archive-date=2017-09-07

|archive-url=https://web.archive.org/web/20170907080041/https://ocs.aaai.org/Papers/Workshops/2000/WS-00-06/WS00-06-006.pdf

|url-status=dead

}}</ref><ref>

{{cite book

|first1=L.|last1=De Raedt|first2=K.|last2=Kersting

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|publisher=Springer

|year=2008}}

</ref><ref name="Inducing probabilistic programs by"/><ref name="Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs">

</ref><ref>

{{cite arXiv

|first1=H.|last1=Irvin|first2=A.|last2=Stuhlmuller|first3=N.D.|last3=Goodman

|title=Inducing probabilistic programs by Bayesian program merging|arxiv=1110.5667

|year=2011}}

</ref><ref>

{{cite journal

|first1=A.|last1=Stuhlmuller|first2=N.D.|last2=Goodman

|s2cid=7602205|title=Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs

|publisher=Elsevier

|journal=Cognitive Systems Research

|year=2012}}

|volume=28|pages=80–99|doi=10.1016/j.cogsys.2013.07.003}}

</ref>

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|publisher=Morgan Kaufmann

|year=2001

|url=https://books.google.com/books?id=wM2JYafw11gC&printsec|isbn=frontcover#v=onepage&q&f=false}}9781558606883

}}

</ref> and [[programming by demonstration]],<ref>

{{citeCite journalbook

|first1=E.|last1=Cypher|first2=D.C.|last2=Halbert

|title=Watch what I do: programming by demonstration |publisher=

|year=1993

|journal=

|publisher=MIT Press |url=https://books.google.com/books?id=Ggzjo0-W1y0C|isbn=9780262032131}}

|volume=|pages=

|year=

|url=https://books.google.com/books?id=Ggzjo0-W1y0C&printsec=frontcover#v=onepage&q&f=false}}

</ref> and [[intelligent tutoring system]]s.

Other areas where inductive inference has been recently applied are [[knowledge acquisition]],<ref>

{{cite journal

|first1=U.|last1=Schmid|author1-link= Ute Schmid |first2=M.|last2=Hofmann|first3=E.|last3=Kitzelmann

|title=Analytical inductive programming as a cognitive rule acquisition devise

|journal=Proceedings of the Second Conference on Artificial General Intelligence

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</ref> [[artificial general intelligence]],<ref>

{{cite journal

|first1=N.|last1=Crossley|first2=E.|last2=Kitzelmann|first3=M.|last3=Hofmann|first4=U.|last4=Schmid|author4-link= Ute Schmid

|title=Combining analytical and evolutionary inductive programming

|journal=Proceedings of the Second Conference on Artificial General Intelligence

|pages=19–24

|year=2009}}

|url=https://download.atlantis-press.com/article/1824.pdf}}

</ref> [[reinforcement learning]] and theory evaluation,<ref>

{{cite journal

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|volume=15 | issue = 3|pages=241–264

|year=2000

|doi=10.1002/(sici)1098-111x(200003)15:3<241::aid-int6>3.0.co;2-z}}|citeseerx=10.1.1.34.8877

|s2cid=123390956

}}

</ref><ref>

{{cite journal

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|journal=Advances in Neural Information Processing Systems

|pages=753–760

|year=2007}}

|url=http://papers.nips.cc/paper/3332-learning-and-using-relational-theories.pdf}}

</ref> and [[cognitive science]] in general.<ref>

{{cite journal

|first1=U.|last1=Schmid|author1-link= Ute Schmid |first2=E.|last2=Kitzelmann

|title=Inductive rule learning on the knowledge level

|journal=Cognitive Systems Research

|volume=12 | issue = 3

|pages=237–248

|year=2011|doi=10.1016/j.cogsys.2010.12.002|s2cid=18613664}}

</ref><ref name="Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs"/> There may also be prospective applications in intelligent agents, games, robotics, personalisation, ambient intelligence and human interfaces.

</ref><ref>

{{cite journal

|first1=A.|last1=Stuhlmuller|first2=N.D.|last2=Goodman

|title=Reasoning about reasoning by nested conditioning: Modeling theory of mind with probabilistic programs

|publisher=Elsevier

|journal=Cognitive Systems Research

|year=2012}}

</ref> There may also be prospective applications in intelligent agents, games, robotics, personalisation, ambient intelligence and human interfaces.

== See also ==

* [[Automatic programming]]

* [[Declarative programming]]

* [[Evolutionary programming]]

* [[Functional programming]]

* [[Genetic programming]]

* [[Grammar induction]]

* [[Inductive reasoning]]

* [[Inductive logic programming]]

* [[Inductive functional programming]]

* [[Logic programming]]

* [[Machine learning]]

* [[Probabilistic programming language]]

* [[Program synthesis]]

* [[Programming by example]]

* [[Programming by demonstration]]

* [[Structure mining]]

* [[Test-driven development]]<!-- starting with input/output examples and manually producing a program that satisfies them -->

== External linksReferences ==

{{reflist}}

* [http://www.inductive-programming.org/ Inductive Programming community page], hosted by the University of Bamberg.

== Further reading ==

{{Refbegin}}

* {{cite journal|first1=P.|last1=Flener|first2=U.|last2=Schmid|author2-link= Ute Schmid |title=An introduction to inductive programming|journal=Artificial Intelligence Review|volume=29 | issue = 1|pages=45–62|year=2008|doi=10.1007/s10462-009-9108-7|s2cid=26314997}}

* {{cite book|first1=E.|last1=Kitzelmann|title=Approaches and Applications of Inductive Programming |chapter=Inductive Programming: A Survey of Program Synthesis Techniques |volume=5812|pages=50–73|year=2010|chapter-url=http://emanuel.kitzelmann.org/documents/publications/Kitzelmann2010.pdf|doi=10.1007/978-3-642-11931-6_3|citeseerx=10.1.1.180.1237|series=Lecture Notes in Computer Science|isbn=978-3-642-11930-9}}

* {{cite journal|first1=P.|last1=Flener|first2=U.|last2=Schmid|title=An introduction to inductive programming|publisher=Springer|journal=Artificial Intelligence Review|volume=29 | issue = 1|pages=45–62|year=2008|doi=10.1007/s10462-009-9108-7}}

* {{cite journal|first1=ED.|last1=KitzelmannPartridge|title=InductiveThe programming:case Afor surveyinductive of program synthesis techniquesprogramming|publisherjournal=SpringerComputer|journalvolume=Approaches30 and| Applicationsissue of Inductive= Programming1|pages=50–7336–41|year=20101997|doi=10.1109/2.562924|s2cid=206403583 }}

* {{cite journal|first1=DP.|last1=Flener|first2=D.|last2=Partridge|title=TheInductive case for inductive programming|publisher=IEEEProgramming|journal=ComputerAutomated Software Engineering|volume=308 | issue = 12|pages=36–41131–137|year=19972001|doi=10.11091023/2.562924a:1008797606116|s2cid=6675212}}

* {{cite journal|first1=P.|last1=Flener|first2=D.|last2=Partridge|title=Inductive Programming|publisher=Springer|journal=Automated Software Engineering|volume=8 | issue = 2|pages=131–137|year=2001|doi=10.1023/a:1008797606116}}

* {{cite journal|first1=M.|last1=Hofmann|first2=E.|last2=Kitzelmann|title=A unifying framework for analysis and evaluation of inductive programming systems|journal=Proceedings of the Second Conference on Artificial General Intelligence|pages=55–60|year=2009|url=http://www.atlantis-press.com/php/download_paper.php?id=1839}}

* {{Cite journal | last1 = Muggleton | first1 = S. | last2 = De Raedt | doi = 10.1016/0743-1066(94)90035-3 | first2 = L. | title = Inductive Logic Programming: Theory and methods | journal = The Journal of Logic Programming | volume = 19-20 | pages = 629–679 | year = 1994 | pmidurl = https://lirias.kuleuven.be/handle/123456789/125406 | pmcdoi-access = free }}

* {{cite book | first1 = N. | last1 = Lavrac |author1-link=Nada Lavrač| first2 = S. | last2 = Dzeroski | title = Inductive Logic Programming: Techniques and Applications | publisher = Ellis Horwood | ___location = New York | year = 1994 | isbn = 978-0-13-457870-85 }} https://web.archive.org/web/20040906084947/http://www-ai.ijs.si/SasoDzeroski/ILPBook/{{deadlink}}

* {{cite journal|first1=S.|last1=Muggleton|first2=Luc.|last2=De Raedt|first3=D.|last3=Poole|first4=I.|last4=Bratko|first5=P.|last5=Flach|first6=K.|last6=Inoue|first7=A.|last7=Srinivasan|title=ILP turns 20|publisher=Springer|journal=Machine Learning|volume=86 | issue = 1|pages=3–23|year=2012|doi=10.1007/s10994-011-5259-2|doi-access=free}}

* {{cite journal|first1=S.|last1=GalwaniGulwani|first2=J.|last2=Hernandez-Orallo|first3=E.|last3=Kitzelmann|first4=S.H.|last4=Muggleton|first5=U.|last5=Schmid|author5-link= Ute Schmid |first6=B.|last6=Zorn|title=Inductive Programming Meets the Real World|publisher=ACM|journal=Communications of the ACM|volume=58 | issue = 11|pages=90–99|year=2015|doi=10.1145/2736282|url=http://cacm.acm.org/magazines/2015/11/193326-inductive-programming-meets-the-real-world/abstract|citeseerx=10.1.1.696.3800|hdl=10251/64984|s2cid=425881}}

{{refend}}

== ReferencesExternal links ==

* [http://www.inductive-programming.org/ Inductive Programming community page], hosted by the University of Bamberg.

{{reflist}}

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