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{{Short description|Learning logic programs from data}}
[[File:ILP family2.png|thumb|A photo of Family sample for Inductive Logic Programming article]]
'''Inductive logic programming''' ('''ILP''') is a subfield of [[symbolic artificial intelligence]] which uses [[logic programming]] as a uniform representation for examples, background knowledge and hypotheses. The term "''inductive''" here refers to [[Inductive reasoning|philosophical]] (i.e. suggesting a theory to explain observed facts) rather than [[mathematical induction|mathematical]] (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical [[database]] of facts, an ILP system will derive a hypothesised logic program which [[Entailment|entails]] all the positive and none of the negative examples.
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== History ==
Building on earlier work on [[Inductive inference]], [[Gordon Plotkin]] was the first to formalise induction in a [[Horn clause|clausal]] setting around 1970, adopting an approach of generalising from examples.<ref name=":02">{{Cite book |last1=Nienhuys-Cheng |first1=Shan-hwei |title=Foundations of inductive logic programming |last2=Wolf |first2=Ronald de |date=1997 |publisher=
Several inductive logic programming systems that proved influential appeared in the early 1990s. [[First-order inductive learner|FOIL]], introduced by [[Ross Quinlan]] in 1990<ref>{{Cite journal |last=Quinlan |first=J. R. |date=August 1990 |title=Learning logical definitions from relations |journal=Machine Learning |volume=5 |issue=3 |pages=239–266 |doi=10.1007/bf00117105 |issn=0885-6125|doi-access=free }}</ref> was based on upgrading [[Propositional calculus|propositional]] learning algorithms [[AQ (machine learning)|AQ]] and [[ID3 algorithm|ID3]].<ref name=":12">{{Cite book |last1=Nienhuys-Cheng |first1=Shan-hwei |title=Foundations of inductive logic programming |last2=Wolf |first2=Ronald de |date=1997 |publisher=
At around the same time, the first practical applications emerged, particularly in [[bioinformatics]], where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins.<ref>{{Citation |last=Džeroski |first=Sašo |title=Relational Data Mining Applications: An Overview |date=2001 |url=http://link.springer.com/10.1007/978-3-662-04599-2_14 |work=Relational Data Mining |pages=339–364 |editor-last=Džeroski |editor-first=Sašo |access-date=2023-11-27 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-662-04599-2_14 |isbn=978-3-642-07604-6 |editor2-last=Lavrač |editor2-first=Nada|editor2-link=Nada Lavrač|url-access=subscription }}</ref> Unlike the focus on [[automatic programming]] inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of [[relational data mining]]. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field.<ref>{{Citation |last=De Raedt |first=Luc |editor-first1=<!-- Deny Citation Bot--> |editor-last1=<!-- Deny Citation Bot--> |url=http://dx.doi.org/10.1007/978-3-540-68856-3 |title=Logical and Relational Learning |series=Cognitive Technologies |page=14|place=Berlin, Heidelberg |publisher=Springer |year=2008 |doi=10.1007/978-3-540-68856-3 |bibcode=2008lrl..book.....D |isbn=978-3-540-20040-6|url-access=subscription }}</ref>
Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the [[Metagol]] system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014.<ref>{{Cite journal |last1=Muggleton |first1=Stephen H. |last2=Lin |first2=Dianhuan |last3=Pahlavi |first3=Niels |last4=Tamaddoni-Nezhad |first4=Alireza |date=2013-05-01 |title=Meta-interpretive learning: application to grammatical inference |url=http://dx.doi.org/10.1007/s10994-013-5358-3 |journal=Machine Learning |volume=94 |issue=1 |pages=25–49 |doi=10.1007/s10994-013-5358-3 |s2cid=254738603 |issn=0885-6125|url-access=subscription }}</ref> This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms.<ref>{{Cite journal |last1=Cropper |first1=Andrew |last2=Dumančić |first2=Sebastijan |last3=Evans |first3=Richard |last4=Muggleton |first4=Stephen |date=2022 |title=Inductive logic programming at 30 |journal=Machine Learning |language=en |volume=111 |issue=1 |pages=147–172 |doi=10.1007/s10994-021-06089-1 |issn=0885-6125|doi-access=free }}</ref>
== Setting ==
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& \textit{false}
\end{array}</math>
Completeness requires any generated hypothesis ''{{mvar|
In Muggleton's setting of concept learning,<ref name="setting2">{{cite journal |last1=Muggleton |first1=Stephen |year=1999 |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 |doi=10.1016/s0004-3702(99)00067-3 |doi-access=}}; here: Sect.2.1</ref> "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "''Necessity''", which postulates that ''{{mvar|B}}'' does not entail <math display="inline">E^+</math>, does not impose a restriction on ''{{mvar|
=== Learning from interpretations ===
In learning from interpretations, the ''positive'' and ''negative'' examples are given as a set of complete or partial [[Herbrand structure]]s, each of which are themselves a finite set of ground literals.
== Approaches to ILP ==
An ''inductive logic programming system'' is a program that takes as an input logic theories <math>B, E^+, E^-</math> and outputs a correct hypothesis {{mvar|H}} with respect to theories <math>B, E^+, E^-</math>. A system is ''complete'' if and only if for any input logic theories <math>B, E^+, E^-</math> any correct hypothesis {{mvar|H}} with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems.
Search-based systems exploit that the space of possible clauses forms a [[complete lattice]] under the [[Theta-subsumption|subsumption]] relation, where one clause <math display="inline">C_1</math> subsumes another clause <math display="inline">C_2</math> if there is a [[Substitution (logic)|substitution]] <math display="inline">\theta</math> such that <math display="inline">C_1\theta</math>, the result of applying <math display="inline">\theta</math> to <math display="inline">C_1</math>, is a subset of <math display="inline">C_2</math>. This lattice can be traversed either bottom-up or top-down.
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==== Least general generalisation ====
A ''least general generalisation algorithm'' takes as input two clauses <math display="inline">C_1</math> and <math display="inline">C_2</math> and outputs the least general generalisation of <math display="inline">C_1</math> and <math display="inline">C_2</math>, that is, a clause <math display="inline">C</math> that subsumes <math display="inline">C_1</math> and <math display="inline">C_2</math>, and that is subsumed by every other clause that subsumes <math display="inline">C_1</math> and <math display="inline">C_2</math>. The least general generalisation can be computed by first computing all ''selections'' from <math display="inline">C</math> and <math display="inline">D</math>, which are pairs of literals <math>(L,M) \in (C_1, C_2)</math> sharing the same predicate symbol and negated/unnegated status. Then, the least general generalisation is obtained as the disjunction of the least general generalisations of the individual selections, which can be obtained by [[first-order syntactical anti-unification]].<ref>{{Cite book |last1=Nienhuys-Cheng |first1=Shan-hwei |title=Foundations of inductive logic programming |last2=Wolf |first2=Ronald de |date=1997 |publisher=
To account for background knowledge, inductive logic programming systems employ ''relative least general generalisations'', which are defined in terms of subsumption relative to a background theory. In general, such relative least general generalisations are not guaranteed to exist; however, if the background theory ''{{mvar|B}}'' is a finite set of [[Ground expression|ground]] [[Literal (mathematical logic)|literals]], then the negation of ''{{mvar|B}}'' is itself a clause. In this case, a relative least general generalisation can be computed by disjoining the negation of ''{{mvar|B}}'' with both <math display="inline">C_1</math> and <math display="inline">C_2</math> and then computing their least general generalisation as before.<ref>{{Cite book |last1=Nienhuys-Cheng |first1=Shan-hwei |title=Foundations of inductive logic programming |last2=Wolf |first2=Ronald de |date=1997 |publisher=
Relative least general generalisations are the foundation of the bottom-up system [[Golem (ILP)|Golem]].<ref name=":12"/><ref name="Springer/Ohmsha"/>
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Inverse resolution is an [[inductive reasoning]] technique that involves [[wiktionary:invert|inverting]] the [[Resolution (logic)|resolution operator]].
Inverse resolution takes information about the [[Resolvent (logic)|resolvent]] of a resolution step to compute possible resolving clauses. Two types of inverse resolution operator are in use in inductive logic programming: V-operators and W-operators. A V-operator takes clauses <math display="inline">R</math> and <math display="inline">C_1</math>as input and returns a clause <math display="inline">C_2</math> such that <math display="inline">R</math> is the resolvent of <math display="inline">C_1</math> and <math display="inline">C_2</math>. A W-operator takes two clauses <math display="inline">R_1</math> and <math display="inline">R_2</math> and returns
Inverse resolution was first introduced by [[Stephen Muggleton]] and Wray Buntine in 1988 for use in the inductive logic programming system Cigol.<ref name="Proceedings of the 5th Internationa"/> By 1993, this spawned a surge of research into inverse resolution operators and their properties.<ref name="invres" />
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And example of a Prolog-based system is [[Metagol]], which is based on a [[Meta-interpreters in Prolog|meta-interpreter in Prolog]], while ASPAL and ILASP are based on an encoding of the inductive logic programming problem in answer set programming.<ref name=":0" />
=== Evolutionary learning ===
[[Evolutionary algorithm]]s in ILP use a population-based approach to evolve hypotheses, refining them through selection, crossover, and mutation. Methods like [[EvoLearner]] have been shown to outperform traditional approaches on structured machine learning benchmarks. <ref>{{cite conference |last1=Heindorf |first1=Stefan |last2=Blübaum |first2=Lukas |last3=Düsterhus |first3= Nick |last4=Werner |first4=Till |last5=Golani |first5=Varun Nandkumar |last6=Demir |first6=Caglar |last7=Ngonga Ngomo |first7=Axel-Cyrille |title=EvoLearner: Learning Description Logics with Evolutionary Algorithms |conference=WWW |date=2022|arxiv=2111.04879 }}</ref>
== List of implementations ==
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* [http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/ Aleph]
* [http://www.ahlgren.info/research/atom/ Atom] {{Webarchive|url=https://web.archive.org/web/20140326152728/http://www.ahlgren.info/research/atom |date=2014-03-26 }}
* [https://archive.today/20121231073113/http://dtai.cs.kuleuven.be/claudien/ Claudien]
* [http://dl-learner.org DL-Learner] {{Webarchive|url=https://web.archive.org/web/20190815184411/http://dl-learner.org/ |date=2019-08-15 }}
* [http://dtai.cs.kuleuven.be/dmax/ DMax]
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* [https://archive.today/20130219215544/http://libra.msra.cn/Publication/3392493/mio-user-s-manual Mio]
* MIS (Model Inference System) by Ehud Shapiro
* [https://github.com/dice-group/Ontolearn Ontolearn]
* [https://github.com/logic-and-learning-lab/Popper Popper]
* [[PROGOL]]
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== Probabilistic inductive logic programming ==
Probabilistic inductive logic programming adapts the setting of inductive logic programming to learning [[Probabilistic logic programming|probabilistic logic programs]]. It can be considered as a form of [[statistical relational learning]] within the formalism of probabilistic logic programming.<ref>{{Citation |last1=De Raedt |first1=Luc |title=Probabilistic Inductive Logic Programming |date=2008 |url=http://dx.doi.org/10.1007/978-3-540-78652-8_1 |pages=1–27 |access-date=2023-12-09 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-540-78651-1 |last2=Kersting |first2=Kristian|doi=10.1007/978-3-540-78652-8_1 |url-access=subscription }}</ref><ref name="pilp">{{Cite journal |last1=Riguzzi |first1=Fabrizio |last2=Bellodi |first2=Elena |last3=Zese |first3=Riccardo |date=2014-09-18 |title=A History of Probabilistic Inductive Logic Programming |journal=Frontiers in Robotics and AI |volume=1 |doi=10.3389/frobt.2014.00006 |issn=2296-9144 |doi-access=free }}</ref>
Given
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Structure learning was pioneered by [[Daphne Koller]] and Avi Pfeffer in 1997,<ref>{{Cite conference |last1=Koller |first1=Daphne |last2=Pfeffer |first2=Avi |date=August 1997 |title=Learning probabilities for noisy first-order rules |url=http://www.robotics.stanford.edu/~koller/Papers/Koller+Pfeffer:IJCAI97.pdf |conference=[[IJCAI]]}}</ref> where the authors learn the structure of [[First-order logic|first-order]] rules with associated probabilistic uncertainty parameters. Their approach involves generating the underlying [[graphical model]] in a preliminary step and then applying expectation-maximisation.<ref name="pilp" />
In 2008, [[Luc De Raedt|De Raedt]] et al. presented an algorithm for performing [[theory compression]] on [[ProbLog]] programs, where theory compression refers to a process of removing as many clauses as possible from the theory in order to maximize the probability of a given set of positive and negative examples. No new clause can be added to the theory.<ref name="pilp" /><ref>{{Cite journal |last1=De Raedt |first1=L. |last2=Kersting |first2=K. |last3=Kimmig |first3=A. |last4=Revoredo |first4=K. |last5=Toivonen |first5=H. |date=March 2008 |title=Compressing probabilistic Prolog programs |url=http://link.springer.com/10.1007/s10994-007-5030-x |journal=Machine Learning |language=en |volume=70 |issue=2–3 |pages=151–168 |doi=10.1007/s10994-007-5030-x |issn=0885-6125|hdl=10138/143971 |hdl-access=free }}</ref>
In the same year, Meert, W. et al. introduced a method for learning parameters and structure of [[Ground term|ground]] probabilistic logic programs by considering the [[Bayesian network]]s equivalent to them and applying techniques for learning Bayesian networks.<ref>{{Citation |last1=Blockeel |first1=Hendrik |title=Towards Learning Non-recursive LPADs by Transforming Them into Bayesian Networks |url=http://dx.doi.org/10.1007/978-3-540-73847-3_16 |work=Inductive Logic Programming |pages=94–108 |access-date=2023-12-09 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-540-73846-6 |last2=Meert |first2=Wannes|series=Lecture Notes in Computer Science |date=2007 |volume=4455 |doi=10.1007/978-3-540-73847-3_16 |url-access=subscription }}</ref><ref name="pilp" />
ProbFOIL, introduced by De Raedt and Ingo Thon in 2010, combined the inductive logic programming system [[First-order inductive learner|FOIL]] with [[ProbLog]]. Logical rules are learned from probabilistic data in the sense that both the examples themselves and their classifications can be probabilistic. The set of rules has to allow one to predict the probability of the examples from their description. In this setting, the parameters (the probability values) are fixed and the structure has to be learned.<ref>{{Citation |last1=De Raedt |first1=Luc |title=Probabilistic Rule Learning |date=2011 |url=http://link.springer.com/10.1007/978-3-642-21295-6_9 |work=Inductive Logic Programming |volume=6489 |pages=47–58 |editor-last=Frasconi |editor-first=Paolo |access-date=2023-12-09 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-21295-6_9 |isbn=978-3-642-21294-9 |last2=Thon |first2=Ingo |s2cid=11727522 |editor2-last=Lisi |editor2-first=Francesca A.|url-access=subscription }}</ref><ref name="pilp" />
In 2011, Elena Bellodi and Fabrizio Riguzzi introduced SLIPCASE, which performs a beam search among probabilistic logic programs by iteratively refining probabilistic theories and optimizing the parameters of each theory using expectation-maximisation.<ref>{{Citation |last1=Bellodi |first1=Elena |title=Learning the Structure of Probabilistic Logic Programs |date=2012 |url=http://dx.doi.org/10.1007/978-3-642-31951-8_10 |work=Inductive Logic Programming |pages=61–75 |access-date=2023-12-09 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-642-31950-1 |last2=Riguzzi |first2=Fabrizio|doi=10.1007/978-3-642-31951-8_10 |url-access=subscription }}</ref>
Its extension SLIPCOVER, proposed in 2014, uses bottom clauses generated as in [[Progol]] to guide the refinement process, thus reducing the number of revisions and exploring the search space more effectively. Moreover, SLIPCOVER separates the search for promising clauses from that of the theory: the space of clauses is explored with a [[beam search]], while the space of theories is searched [[Greedy search|greedily]].<ref>{{Cite journal |last1=Bellodi |first1=Elena |last2=Riguzzi |first2=Fabrizio |date=2014-01-15 |title=Structure learning of probabilistic logic programs by searching the clause space |url=http://dx.doi.org/10.1017/s1471068413000689 |journal=Theory and Practice of Logic Programming |volume=15 |issue=2 |pages=169–212 |doi=10.1017/s1471068413000689 |arxiv=1309.2080 |s2cid=17669522 |issn=1471-0684}}</ref><ref name="pilp" />
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