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Inductive logic programming

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Inductive logic programming (ILP) is a subfield of <a href="Symbolic_artificial_intelligence" title="Symbolic artificial intelligence">symbolic artificial intelligence</a> which uses <a href="Logic_programming" title="Logic programming">logic programming</a> as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to <a href="Inductive_reasoning" title="Inductive reasoning">philosophical</a> (i.e. suggesting a theory to explain observed facts) rather than <a href="Mathematical_induction" title="Mathematical induction">mathematical</a> (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 <a href="Database" title="Database">database</a> of facts, an ILP system will derive a hypothesised logic program which <a href="Entailment" class="mw-redirect" title="Entailment">entails</a> all the positive and none of the negative examples.

<ul><li>Schema: positive examples + negative examples + background knowledge ⇒ hypothesis.</li></ul>

Inductive logic programming is particularly useful in <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a> and <a href="Natural_language_processing" title="Natural language processing">natural language processing</a>.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Building on earlier work on <a href="Inductive_inference" class="mw-redirect" title="Inductive inference">Inductive inference</a>, <a href="Gordon_Plotkin" title="Gordon Plotkin">Gordon Plotkin</a> was the first to formalise induction in a <a href="Horn_clause" title="Horn clause">clausal</a> setting around 1970, adopting an approach of generalising from examples.<a href="#cite_note-:02-1">[1]</a><a href="#cite_note-2">[2]</a> In 1981, <a href="Ehud_Shapiro" title="Ehud Shapiro">Ehud Shapiro</a> introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples.<a href="#cite_note-:02-1">[1]</a><a href="#cite_note-3">[3]</a> His first implementation was the Model Inference System in 1981:<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> a <a href="Prolog" title="Prolog">Prolog</a> program that inductively inferred <a href="Horn_clause" title="Horn clause">Horn clause</a> logic programs from positive and negative examples.<a href="#cite_note-:02-1">[1]</a> The term Inductive Logic Programming was first introduced in a paper by <a href="Stephen_Muggleton" title="Stephen Muggleton">Stephen Muggleton</a> in 1990, defined as the intersection of machine learning and logic programming.<a href="#cite_note-:02-1">[1]</a> Muggleton and Wray Buntine introduced predicate invention and <a href="Inverse_resolution" class="mw-redirect" title="Inverse resolution">inverse resolution</a> in 1988.<a href="#cite_note-:02-1">[1]</a><a href="#cite_note-Proceedings_of_the_5th_Internationa-6">[6]</a>

Several inductive logic programming systems that proved influential appeared in the early 1990s. <a href="First-order_inductive_learner" title="First-order inductive learner">FOIL</a>, introduced by <a href="Ross_Quinlan" title="Ross Quinlan">Ross Quinlan</a> in 1990<a href="#cite_note-7">[7]</a> was based on upgrading <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional</a> learning algorithms AQ and <a href="ID3_algorithm" title="ID3 algorithm">ID3</a>.<a href="#cite_note-:12-8">[8]</a> <a href="Golem_(ILP)" title="Golem (ILP)">Golem</a>, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm.<a href="#cite_note-:12-8">[8]</a><a href="#cite_note-Springer/Ohmsha-9">[9]</a> The <a href="Progol" title="Progol">Progol</a> system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems.<a href="#cite_note-:12-8">[8]</a><a href="#cite_note-:3-10">[10]</a><a href="#cite_note-:2-11">[11]</a> <a href="Aleph_(ILP)" title="Aleph (ILP)">Aleph</a>, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems as of 2022.<a href="#cite_note-:3-10">[10]</a>

At around the same time, the first practical applications emerged, particularly in <a href="Bioinformatics" title="Bioinformatics">bioinformatics</a>, 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.<a href="#cite_note-12">[12]</a> Unlike the focus on <a href="Automatic_programming" title="Automatic programming">automatic programming</a> inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of <a href="Relational_data_mining" title="Relational data mining">relational data mining</a>. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field.<a href="#cite_note-13">[13]</a>

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.<a href="#cite_note-14">[14]</a> This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms.<a href="#cite_note-15">[15]</a>

<div class="mw-heading mw-heading2"><h2 id="Setting">Setting</h2></div>

Inductive logic programming has adopted several different learning settings, the most common of which are learning from <a href="Entailment" class="mw-redirect" title="Entailment">entailment</a> and learning from interpretations.<a href="#cite_note-setting-16">[16]</a> In both cases, the input is provided in the form of background knowledge B, a logical theory (commonly in the form of <a href="Clause_(logic)" title="Clause (logic)">clauses</a> used in <a href="Logic_programming" title="Logic programming">logic programming</a>), as well as positive and negative examples, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{+}}">

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The two settings differ in the format of examples presented.

<div class="mw-heading mw-heading3"><h3 id="Learning_from_entailment">Learning from entailment</h3></div>

As of 2022, learning from entailment is by far the most popular setting for inductive logic programming.<a href="#cite_note-setting-16">[16]</a> In this setting, the positive and negative examples are given as finite sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{+}}">

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<annotation encoding="application/x-tex">{\textstyle E^{-}}</annotation>

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</math><img src="./5f2de04a8716d1450603ddb2760521155c1f2c6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.305ex; height:2.343ex;" alt="{\textstyle E^{-}}" loading="lazy"> of positive and negated <a href="Ground_expression" title="Ground expression">ground</a> <a href="Literal_(mathematical_logic)" title="Literal (mathematical logic)">literals</a>, respectively. A correct hypothesis H is a set of clauses satisfying the following requirements, where the turnstile symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \models }">

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<annotation encoding="application/x-tex">{\displaystyle \models }</annotation>

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</math><img src="./89dbad9a523091069a540122aeb15e41d1fe18b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.015ex; height:2.843ex;" alt="{\displaystyle \models }" loading="lazy"> stands for <a href="Logical_entailment" class="mw-redirect" title="Logical entailment">logical entailment</a>:<a href="#cite_note-setting-16">[16]</a><a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a>

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<mtext>Completeness:</mtext>

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<mtext>Consistency: </mtext>

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<mtext class="MJX-tex-mathit" mathvariant="italic">false</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{llll}{\text{Completeness:}}&B\cup H&\models &E^{+}\\{\text{Consistency: }}&B\cup H\cup E^{-}&\not \models &{\textit {false}}\end{array}}}</annotation>

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Completeness requires any generated hypothesis H to explain all positive examples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{+}}">

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</math><img src="./80124bbf80b76c335176811694ce9871869fbf4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.305ex; height:2.343ex;" alt="{\textstyle E^{+}}" loading="lazy">, and consistency forbids generation of any hypothesis H that is inconsistent with the negative examples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{-}}">

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In Muggleton's setting of concept learning,<a href="#cite_note-setting2-19">[19]</a> "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "Necessity", which postulates that B does not entail <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{+}}">

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</math><img src="./80124bbf80b76c335176811694ce9871869fbf4c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.305ex; height:2.343ex;" alt="{\textstyle E^{+}}" loading="lazy">, does not impose a restriction on H, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Weak consistency", which states that no contradiction can be derived from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\land H}">

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<annotation encoding="application/x-tex">{\textstyle B\land H}</annotation>

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</math><img src="./be716b070931a6f86692723ecfe4d9c69d702e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle B\land H}" loading="lazy">, forbids generation of any hypothesis H that contradicts the background knowledge B. Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed.<a href="#cite_note-setting2-19">[19]</a>

<div class="mw-heading mw-heading3"><h3 id="Learning_from_interpretations">Learning from interpretations</h3></div>

In learning from interpretations, the positive and negative examples are given as a set of complete or partial <a href="Herbrand_structure" title="Herbrand structure">Herbrand structures</a>, each of which are themselves a finite set of ground literals. Such a structure e is said to be a model of the set of clauses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\cup H}">

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<annotation encoding="application/x-tex">{\textstyle B\cup H}</annotation>

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</math><img src="./10e9d40eb468da8360644a107701572fda8bb182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle B\cup H}" loading="lazy"> if for any <a href="Substitution_(logic)" title="Substitution (logic)">substitution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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<annotation encoding="application/x-tex">{\textstyle \theta }</annotation>

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\textstyle \theta }" loading="lazy"> and any clause <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {head} \leftarrow \mathrm {body} }">

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<annotation encoding="application/x-tex">{\textstyle \mathrm {head} \leftarrow \mathrm {body} }</annotation>

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</math><img src="./959886e24f1a89c16469089509e0b3d145186faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.369ex; height:2.509ex;" alt="{\textstyle \mathrm {head} \leftarrow \mathrm {body} }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\cup H}">

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<annotation encoding="application/x-tex">{\textstyle B\cup H}</annotation>

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</math><img src="./10e9d40eb468da8360644a107701572fda8bb182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle B\cup H}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathrm {body} \theta \subseteq e}">

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<annotation encoding="application/x-tex">{\textstyle \mathrm {body} \theta \subseteq e}</annotation>

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</math><img src="./d82b7e033d9f9c62cf7d479459c910c42737444f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.247ex; height:2.509ex;" alt="{\textstyle \mathrm {body} \theta \subseteq e}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {head} \theta \subseteq e}">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {head} \theta \subseteq e}</annotation>

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</math><img src="./6d44c74712d222d1221f05b5314d15d53ae74fd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.052ex; height:2.343ex;" alt="{\displaystyle \mathrm {head} \theta \subseteq e}" loading="lazy"> also holds. The goal is then to output a hypothesis that is complete, meaning every positive example is a model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\cup H}">

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<annotation encoding="application/x-tex">{\textstyle B\cup H}</annotation>

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</math><img src="./10e9d40eb468da8360644a107701572fda8bb182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle B\cup H}" loading="lazy">, and consistent, meaning that no negative example is a model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\cup H}">

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<annotation encoding="application/x-tex">{\textstyle B\cup H}</annotation>

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</math><img src="./10e9d40eb468da8360644a107701572fda8bb182.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle B\cup H}" loading="lazy">.<a href="#cite_note-setting-16">[16]</a>

<div class="mw-heading mw-heading2"><h2 id="Approaches_to_ILP">Approaches to ILP</h2></div>

An inductive logic programming system is a program that takes as an input logic theories <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,E^{+},E^{-}}">

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</math><img src="./1df5a807aae71d080dd1a00418823bc1858a3b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle B,E^{+},E^{-}}" loading="lazy"> and outputs a correct hypothesis H with respect to theories <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,E^{+},E^{-}}">

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<annotation encoding="application/x-tex">{\displaystyle B,E^{+},E^{-}}</annotation>

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</math><img src="./1df5a807aae71d080dd1a00418823bc1858a3b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle B,E^{+},E^{-}}" loading="lazy">. A system is complete if and only if for any input logic theories <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B,E^{+},E^{-}}">

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</math><img src="./1df5a807aae71d080dd1a00418823bc1858a3b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.441ex; height:2.843ex;" alt="{\displaystyle B,E^{+},E^{-}}" loading="lazy"> any correct hypothesis 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 <a href="Complete_lattice" title="Complete lattice">complete lattice</a> under the <a href="Theta-subsumption" title="Theta-subsumption">subsumption</a> relation, where one clause <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

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</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> subsumes another clause <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

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</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> if there is a <a href="Substitution_(logic)" title="Substitution (logic)">substitution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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<annotation encoding="application/x-tex">{\textstyle \theta }</annotation>

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\textstyle \theta }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}\theta }">

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<annotation encoding="application/x-tex">{\textstyle C_{1}\theta }</annotation>

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</math><img src="./07ecd729cf7ef736a83e90fce791389a004f8acb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.807ex; height:2.509ex;" alt="{\textstyle C_{1}\theta }" loading="lazy">, the result of applying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \theta }">

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</math><img src="./a11744bd71a5eb6efe4f28e12ca57f874d82658c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\textstyle \theta }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

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<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

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</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy">, is a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

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<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Bottom-up_search">Bottom-up search</h3></div>

Bottom-up methods to search the subsumption lattice have been investigated since Plotkin's first work on formalising induction in clausal logic in 1970.<a href="#cite_note-:02-1">[1]</a><a href="#cite_note-20">[20]</a> Techniques used include least general generalisation, based on <a href="Anti-unification_(computer_science)" class="mw-redirect" title="Anti-unification (computer science)">anti-unification</a>, and inverse resolution, based on inverting the <a href="Resolution_(logic)" title="Resolution (logic)">resolution</a> inference rule.

<div class="mw-heading mw-heading4"><h4 id="Least_general_generalisation">Least general generalisation</h4></div>

A least general generalisation algorithm takes as input two clauses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

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<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

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</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> and outputs the least general generalisation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy">, that is, a clause <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C}</annotation>

</semantics>

</math><img src="./6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\textstyle C}" loading="lazy"> that subsumes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy">, and that is subsumed by every other clause that subsumes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy">. The least general generalisation can be computed by first computing all selections from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C}</annotation>

</semantics>

</math><img src="./6dca76d9ff4b48256b6a4a99bcb234b64b2fa72b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\textstyle C}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle D}</annotation>

</semantics>

</math><img src="./4e5200f518cb5afe304ec42ffdd4f6c63c702f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\textstyle D}" loading="lazy">, which are pairs of literals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (L,M)\in (C_{1},C_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (L,M)\in (C_{1},C_{2})}</annotation>

</semantics>

</math><img src="./64dda10f71a5dd2273d8342ec3abae301c33f018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.984ex; height:2.843ex;" alt="{\displaystyle (L,M)\in (C_{1},C_{2})}" loading="lazy"> 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 <a href="First-order_syntactical_anti-unification" class="mw-redirect" title="First-order syntactical anti-unification">first-order syntactical anti-unification</a>.<a href="#cite_note-21">[21]</a>

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 B is a finite set of <a href="Ground_expression" title="Ground expression">ground</a> <a href="Literal_(mathematical_logic)" title="Literal (mathematical logic)">literals</a>, then the negation of B is itself a clause. In this case, a relative least general generalisation can be computed by disjoining the negation of B with both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> and then computing their least general generalisation as before.<a href="#cite_note-22">[22]</a>

Relative least general generalisations are the foundation of the bottom-up system <a href="Golem_(ILP)" title="Golem (ILP)">Golem</a>.<a href="#cite_note-:12-8">[8]</a><a href="#cite_note-Springer/Ohmsha-9">[9]</a>

<div class="mw-heading mw-heading4"><h4 id="Inverse_resolution">Inverse resolution</h4></div>

Inverse resolution is an <a href="Inductive_reasoning" title="Inductive reasoning">inductive reasoning</a> technique that involves <a href="https://en.wiktionary.org/wiki/invert" class="extiw external" title="wiktionary:invert">inverting</a> the <a href="Resolution_(logic)" title="Resolution (logic)">resolution operator</a>.

Inverse resolution takes information about the <a href="Resolvent_(logic)" class="mw-redirect" title="Resolvent (logic)">resolvent</a> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R}</annotation>

</semantics>

</math><img src="./197e66194eb64577670e2a100026bff6fb15d236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle R}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy">as input and returns a clause <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R}</annotation>

</semantics>

</math><img src="./197e66194eb64577670e2a100026bff6fb15d236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle R}" loading="lazy"> is the resolvent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy">. A W-operator takes two clauses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R_{1}}</annotation>

</semantics>

</math><img src="./c115730d178ae9c5ffe2747f97a29c73698f9d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\textstyle R_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R_{2}}</annotation>

</semantics>

</math><img src="./eeb9f9b1a756c21ec0dd01be6df73c2fe1776753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\textstyle R_{2}}" loading="lazy"> and returns three clauses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{3}}</annotation>

</semantics>

</math><img src="./c569c5cbf0f21d2eb5d5bede70f56db75f4949bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{3}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R_{1}}</annotation>

</semantics>

</math><img src="./c115730d178ae9c5ffe2747f97a29c73698f9d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\textstyle R_{1}}" loading="lazy"> is the resolvent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{1}}</annotation>

</semantics>

</math><img src="./677a99cc6a1ac23ff88fc13fa384646a09857d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle R_{2}}</annotation>

</semantics>

</math><img src="./eeb9f9b1a756c21ec0dd01be6df73c2fe1776753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\textstyle R_{2}}" loading="lazy"> is the resolvent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{2}}</annotation>

</semantics>

</math><img src="./c2dc7c3155640d6e3d86593fc6b62048a4b61a45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle C_{3}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle C_{3}}</annotation>

</semantics>

</math><img src="./c569c5cbf0f21d2eb5d5bede70f56db75f4949bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\textstyle C_{3}}" loading="lazy">.<a href="#cite_note-invres-23">[23]</a>

Inverse resolution was first introduced by <a href="Stephen_Muggleton" title="Stephen Muggleton">Stephen Muggleton</a> and Wray Buntine in 1988 for use in the inductive logic programming system Cigol.<a href="#cite_note-Proceedings_of_the_5th_Internationa-6">[6]</a> By 1993, this spawned a surge of research into inverse resolution operators and their properties.<a href="#cite_note-invres-23">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Top-down_search">Top-down search</h3></div>

The ILP systems Progol,<a href="#cite_note-:2-11">[11]</a> Hail <a href="#cite_note-24">[24]</a> and Imparo <a href="#cite_note-25">[25]</a> find a hypothesis H using the principle of the inverse entailment<a href="#cite_note-:2-11">[11]</a> for theories B, E, H: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\land H\models E\iff B\land \neg E\models \neg H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\land H\models E\iff B\land \neg E\models \neg H}</annotation>

</semantics>

</math><img src="./e2d8fc90c4a014390b9a10c71890eb00b38e00fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.98ex; height:2.843ex;" alt="{\displaystyle B\land H\models E\iff B\land \neg E\models \neg H}" loading="lazy">. First they construct an intermediate theory F called a bridge theory satisfying the conditions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\land \neg E\models F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\land \neg E\models F}</annotation>

</semantics>

</math><img src="./d46ce07c3f1d4f2b3e2c086f6b94a1fc2692c9dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.718ex; height:2.843ex;" alt="{\displaystyle B\land \neg E\models F}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\models \neg H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\models \neg H}</annotation>

</semantics>

</math><img src="./f9e70df65de3d1a89a688393d452aaf3e130762d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle F\models \neg H}" loading="lazy">. Then as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\models \neg F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\models \neg F}</annotation>

</semantics>

</math><img src="./6e51b63437d525fc3b6caf34693fc180b0803ab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle H\models \neg F}" loading="lazy">, they generalize the negation of the bridge theory F with anti-entailment.<a href="#cite_note-26">[26]</a> However, the operation of anti-entailment is computationally more expensive since it is highly nondeterministic. Therefore, an alternative hypothesis search can be conducted using the inverse subsumption (anti-subsumption) operation instead, which is less non-deterministic than anti-entailment.

Questions of completeness of a hypothesis search procedure of specific inductive logic programming system arise. For example, the Progol hypothesis search procedure based on the inverse entailment inference rule is not complete by Yamamoto's example.<a href="#cite_note-27">[27]</a> On the other hand, Imparo is complete by both anti-entailment procedure <a href="#cite_note-kimber2009induction-28">[28]</a> and its extended inverse subsumption <a href="#cite_note-29">[29]</a> procedure.

<div class="mw-heading mw-heading3"><h3 id="Metainterpretive_learning">Metainterpretive learning</h3></div>

Rather than explicitly searching the hypothesis graph, metainterpretive or meta-level systems encode the inductive logic programming program as a meta-level logic program which is then solved to obtain an optimal hypothesis. Formalisms used to express the problem specification include <a href="Prolog" title="Prolog">Prolog</a> and <a href="Answer_set_programming" title="Answer set programming">answer set programming</a>, with existing Prolog systems and answer set solvers used for solving the constraints.<a href="#cite_note-:0-30">[30]</a>

And example of a Prolog-based system is Metagol, which is based on a <a href="Meta-interpreters_in_Prolog" class="mw-redirect" title="Meta-interpreters in Prolog">meta-interpreter in Prolog</a>, while ASPAL and ILASP are based on an encoding of the inductive logic programming problem in answer set programming.<a href="#cite_note-:0-30">[30]</a>

<div class="mw-heading mw-heading3"><h3 id="Evolutionary_learning">Evolutionary learning</h3></div>

<a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithms</a> 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. <a href="#cite_note-31">[31]</a>

<div class="mw-heading mw-heading2"><h2 id="List_of_implementations">List of implementations</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.cs.bris.ac.uk/Research/MachineLearning/1BC/">1BC and 1BC2: first-order naive Bayesian classifiers:</a></li>

<li><a rel="nofollow" class="external text" href="http://dtai.cs.kuleuven.be/ACE/">ACE (A Combined Engine)</a></li>

<li><a rel="nofollow" class="external text" href="http://web.comlab.ox.ac.uk/oucl/research/areas/machlearn/Aleph/">Aleph</a></li>

<li><a rel="nofollow" class="external text" href="http://www.ahlgren.info/research/atom/">Atom</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140326152728/http://www.ahlgren.info/research/atom">Archived</a> 2014-03-26 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="https://archive.today/20121231073113/http://dtai.cs.kuleuven.be/claudien/">Claudien</a></li>

<li><a rel="nofollow" class="external text" href="http://dl-learner.org">DL-Learner</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190815184411/http://dl-learner.org/">Archived</a> 2019-08-15 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="https://spike-imperial.github.io/FastLAS/">FastLAS (Fast Learning from Answer Sets)</a></li>

<li><a href="First_Order_Inductive_Learner" class="mw-redirect" title="First Order Inductive Learner">FOIL (First Order Inductive Learner)</a></li>

<li><a href="Golem_(ILP)" title="Golem (ILP)">Golem</a></li>

<li><a rel="nofollow" class="external text" href="http://www.ilasp.com/">ILASP (Inductive Learning of Answer Set Programs)</a></li>

<li>Imparo<a href="#cite_note-kimber2009induction-28">[28]</a></li>

<li><a rel="nofollow" class="external text" href="http://lacam.di.uniba.it:8000/systems/inthelex/">Inthelex (INcremental THEory Learner from EXamples)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111128110512/http://lacam.di.uniba.it:8000/systems/inthelex/">Archived</a> 2011-11-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/metagol/metagol">Metagol</a></li>

<li>MIS (Model Inference System) by Ehud Shapiro</li>

<li><a rel="nofollow" class="external text" href="https://github.com/dice-group/Ontolearn">Ontolearn</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/logic-and-learning-lab/Popper">Popper</a></li>

<li><a href="PROGOL" class="mw-redirect" title="PROGOL">PROGOL</a></li>

<li>Warmr (now included in ACE)</li>

<li><a rel="nofollow" class="external text" href="http://ilp.doc.ic.ac.uk/ProGolem/">ProGolem</a> <a href="#cite_note-32">[32]</a><a href="#cite_note-33">[33]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Probabilistic_inductive_logic_programming">Probabilistic inductive logic programming</h2></div>

Probabilistic inductive logic programming adapts the setting of inductive logic programming to learning <a href="Probabilistic_logic_programming" title="Probabilistic logic programming">probabilistic logic programs</a>. It can be considered as a form of <a href="Statistical_relational_learning" title="Statistical relational learning">statistical relational learning</a> within the formalism of probabilistic logic programming.<a href="#cite_note-34">[34]</a><a href="#cite_note-pilp-35">[35]</a>

Given

<ol><li>background knowledge as a probabilistic logic program B, and</li>

<li>a set of positive and negative examples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle E^{+}}</annotation>

</semantics>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle E^{-}}</annotation>

</semantics>

the goal of probabilistic inductive logic programming is to find a probabilistic logic program <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle H}</annotation>

</semantics>

</math><img src="./2435ce1b360a5f6f5a0ab5acbb3c672bf229dac7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\textstyle H}" loading="lazy"> such that the probability of positive examples according to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {H\cup B}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {H\cup B}}</annotation>

</semantics>

</math><img src="./d714a9dbb34847cf0e1fecf816b651a96c899938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.41ex; height:2.176ex;" alt="{\textstyle {H\cup B}}" loading="lazy"> is maximized and the probability of negative examples is minimized.<a href="#cite_note-pilp-35">[35]</a>

This problem has two variants: parameter learning and structure learning. In the former, one is given the structure (the clauses) of H and the goal is to infer the probabilities annotations of the given clauses, while in the latter the goal is to infer both the structure and the probability parameters of H. Just as in classical inductive logic programming, the examples can be given as examples or as (partial) interpretations.<a href="#cite_note-pilp-35">[35]</a>

<div class="mw-heading mw-heading3"><h3 id="Parameter_Learning">Parameter Learning</h3></div>

Parameter learning for languages following the distribution semantics has been performed by using an <a href="Expectation-maximisation_algorithm" class="mw-redirect" title="Expectation-maximisation algorithm">expectation-maximisation algorithm</a> or by <a href="Gradient_descent" title="Gradient descent">gradient descent</a>.

An expectation-maximisation algorithm consists of a cycle in which the steps of expectation and maximization are repeatedly performed. In the expectation step, the distribution of the hidden variables is computed according to the current values of the probability parameters, while in the maximisation step, the new values of the parameters are computed.

Gradient descent methods compute the gradient of the target function and iteratively modify the parameters moving in the direction of the gradient.<a href="#cite_note-pilp-35">[35]</a>

<div class="mw-heading mw-heading3"><h3 id="Structure_Learning">Structure Learning</h3></div>

Structure learning was pioneered by <a href="Daphne_Koller" title="Daphne Koller">Daphne Koller</a> and Avi Pfeffer in 1997,<a href="#cite_note-36">[36]</a> where the authors learn the structure of <a href="First-order_logic" title="First-order logic">first-order</a> rules with associated probabilistic uncertainty parameters. Their approach involves generating the underlying <a href="Graphical_model" title="Graphical model">graphical model</a> in a preliminary step and then applying expectation-maximisation.<a href="#cite_note-pilp-35">[35]</a>

In 2008, De Raedt et al. presented an algorithm for performing theory compression on <a href="ProbLog" title="ProbLog">ProbLog</a> 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.<a href="#cite_note-pilp-35">[35]</a><a href="#cite_note-37">[37]</a>

In the same year, Meert, W. et al. introduced a method for learning parameters and structure of <a href="Ground_term" class="mw-redirect" title="Ground term">ground</a> probabilistic logic programs by considering the <a href="Bayesian_network" title="Bayesian network">Bayesian networks</a> equivalent to them and applying techniques for learning Bayesian networks.<a href="#cite_note-38">[38]</a><a href="#cite_note-pilp-35">[35]</a>

ProbFOIL, introduced by De Raedt and Ingo Thon in 2010, combined the inductive logic programming system <a href="First-order_inductive_learner" title="First-order inductive learner">FOIL</a> with <a href="ProbLog" title="ProbLog">ProbLog</a>. 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.<a href="#cite_note-39">[39]</a><a href="#cite_note-pilp-35">[35]</a>

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.<a href="#cite_note-40">[40]</a>

Its extension SLIPCOVER, proposed in 2014, uses bottom clauses generated as in <a href="Progol" title="Progol">Progol</a> 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 <a href="Beam_search" title="Beam search">beam search</a>, while the space of theories is searched <a href="Greedy_search" class="mw-redirect" title="Greedy search">greedily</a>.<a href="#cite_note-41">[41]</a><a href="#cite_note-pilp-35">[35]</a>

<ul><li><a href="Commonsense_reasoning" title="Commonsense reasoning">Commonsense reasoning</a></li>

<li><a href="Formal_concept_analysis" title="Formal concept analysis">Formal concept analysis</a></li>

<li><a href="Inductive_reasoning" title="Inductive reasoning">Inductive reasoning</a></li>

<li><a href="Inductive_programming" title="Inductive programming">Inductive programming</a></li>

<li><a href="Inductive_probability" title="Inductive probability">Inductive probability</a></li>

<li><a href="Statistical_relational_learning" title="Statistical relational learning">Statistical relational learning</a></li>

<li><a href="Version_space_learning" title="Version space learning">Version space learning</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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This article incorporates text from a <a href="Free_content" title="Free content">free content</a> work. Licensed under CC-BY 4.0 (<a rel="nofollow" class="external text" href="http://creativecommons.org/licenses/by/4.0/">license statement/permission</a>). Text taken from <a rel="nofollow" class="external text" href="http://journal.frontiersin.org/article/10.3389/frobt.2014.00006/">A History of Probabilistic Inductive Logic Programming</a>, Fabrizio Riguzzi, Elena Bellodi and Riccardo Zese, <a href="Frontiers_Media" title="Frontiers Media">Frontiers Media</a>.

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li><cite id="CITEREFMuggletonDe_Raedt1994" class="citation journal cs1 cs1-prop-long-vol">Muggleton, S.; De Raedt, L. (1994). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0743-1066%2894%2990035-3">"Inductive Logic Programming: Theory and methods"</a>. The Journal of Logic Programming. 19–20: 629–679. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0743-1066%2894%2990035-3">10.1016/0743-1066(94)90035-3</a>.</cite></li>

<li><cite id="CITEREFLavracDzeroski1994" class="citation book cs1">Lavrac, N.; Dzeroski, S. (1994). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040906084947/http://www-ai.ijs.si/SasoDzeroski/ILPBook/">Inductive Logic Programming: Techniques and Applications</a>. New York: Ellis Horwood. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-457870-5</bdi>. Archived from <a rel="nofollow" class="external text" href="http://www-ai.ijs.si/SasoDzeroski/ILPBook/">the original</a> on 2004-09-06. Retrieved 2004-09-22.</cite></li>

<li>Visual example of inducing the grandparenthood relation by the <a href="#Implementations">Atom system</a>. <a rel="nofollow" class="external free" href="http://john-ahlgren.blogspot.com/2014/03/inductive-reasoning-visualized.html">http://john-ahlgren.blogspot.com/2014/03/inductive-reasoning-visualized.html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140326101408/http://john-ahlgren.blogspot.com/2014/03/inductive-reasoning-visualized.html">Archived</a> 2014-03-26 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>

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</style><div id="Programming_paradigms_(Comparison_by_language)368" style="font-size:114%;margin:0 4em"><a href="Programming_paradigm" title="Programming paradigm">Programming paradigms</a> (<a href="Comparison_of_multi-paradigm_programming_languages" title="Comparison of multi-paradigm programming languages">Comparison by language</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Imperative_programming" title="Imperative programming">Imperative</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Structured_programming" title="Structured programming">Structured</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Jackson_structured_programming" title="Jackson structured programming">Jackson structures</a></li>

<li><a href="Block_(programming)" title="Block (programming)">Block-structured</a></li>

<li><a href="Modular_programming" title="Modular programming">Modular</a></li>

<li><a href="Non-structured_programming" title="Non-structured programming">Non-structured</a></li>

<li><a href="Procedural_programming" title="Procedural programming">Procedural</a></li>

<li><a href="Programming_in_the_large_and_programming_in_the_small" title="Programming in the large and programming in the small">Programming in the large and in the small</a></li>

<li><a href="Design_by_contract" title="Design by contract">Design by contract</a></li>

<li><a href="Invariant-based_programming" title="Invariant-based programming">Invariant-based</a></li>

<li><a href="Nested_function" title="Nested function">Nested function</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Object-oriented_programming" title="Object-oriented programming">Object-oriented</a> (<a href="Comparison_of_programming_languages_(object-oriented_programming)" title="Comparison of programming languages (object-oriented programming)">comparison</a>, <a href="List_of_object-oriented_programming_languages" title="List of object-oriented programming languages">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Class-based_programming" title="Class-based programming">Class-based</a>, <a href="Prototype-based_programming" title="Prototype-based programming">Prototype-based</a>, <a href="Object-based_language" title="Object-based language">Object-based</a></li>

<li><a href="Agent-oriented_programming" title="Agent-oriented programming">Agent</a></li>

<li><a href="Immutable_object" title="Immutable object">Immutable object</a></li>

<li><a href="Persistent_programming_language" title="Persistent programming language">Persistent</a></li>

<li><a href="Uniform_function_call_syntax" title="Uniform function call syntax">Uniform function call syntax</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Declarative_programming" title="Declarative programming">Declarative</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Functional_programming" title="Functional programming">Functional</a> (<a href="Comparison_of_functional_programming_languages" title="Comparison of functional programming languages">comparison</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Recursion_(computer_science)" title="Recursion (computer science)">Recursive</a></li>

<li><a href="Anonymous_function" title="Anonymous function">Anonymous function</a> (<a href="Partial_application" title="Partial application">Partial application</a>)</li>

<li><a href="Higher-order_programming" title="Higher-order programming">Higher-order</a></li>

<li><a href="Purely_functional_programming" title="Purely functional programming">Purely functional</a></li>

<li><a href="Total_functional_programming" title="Total functional programming">Total</a></li>

<li><a href="Strict_programming_language" title="Strict programming language">Strict</a></li>

<li><a href="Generalized_algebraic_data_type" title="Generalized algebraic data type">GADTs</a></li>

<li><a href="Dependent_type" title="Dependent type">Dependent types</a></li>

<li><a href="Functional_logic_programming" title="Functional logic programming">Functional logic</a></li>

<li><a href="Tacit_programming" title="Tacit programming">Point-free style</a></li>

<li><a href="Expression-oriented_programming_language" title="Expression-oriented programming language">Expression-oriented</a></li>

<li><a href="Applicative_programming_language" title="Applicative programming language">Applicative</a>, <a href="Concatenative_programming_language" title="Concatenative programming language">Concatenative</a></li>

<li><a href="Function-level_programming" title="Function-level programming">Function-level</a>, <a href="Value-level_programming" title="Value-level programming">Value-level</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dataflow_programming" title="Dataflow programming">Dataflow</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Flow-based_programming" title="Flow-based programming">Flow-based</a></li>

<li><a href="Reactive_programming" title="Reactive programming">Reactive</a> (<a href="Functional_reactive_programming" title="Functional reactive programming">Functional reactive</a>)</li>

<li><a href="Signal_programming" class="mw-redirect" title="Signal programming">Signals</a></li>

<li><a href="Stream_processing" title="Stream processing">Streams</a></li>

<li><a href="Synchronous_programming_language" title="Synchronous programming language">Synchronous</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic_programming" title="Logic programming">Logic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abductive_logic_programming" title="Abductive logic programming">Abductive logic</a></li>

<li><a href="Answer_set_programming" title="Answer set programming">Answer set</a></li>

<li><a href="Constraint_programming" title="Constraint programming">Constraint</a> (<a href="Constraint_logic_programming" title="Constraint logic programming">Constraint logic</a>)</li>

<li><a href="Nondeterministic_programming" title="Nondeterministic programming">Nondeterministic</a></li>

<li><a href="Ontology_language" title="Ontology language">Ontology</a></li>

<li><a href="Probabilistic_logic_programming" title="Probabilistic logic programming">Probabilistic logic</a></li>

<li><a href="Query_language" title="Query language">Query</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Domain-specific_language" title="Domain-specific language">DSL</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algebraic_modeling_language" title="Algebraic modeling language">Algebraic modeling</a></li>

<li><a href="Array_programming" title="Array programming">Array</a></li>

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<li><a href="Command_language" title="Command language">Command</a> (<a href="Spacecraft_command_language" title="Spacecraft command language">Spacecraft</a>)</li>

<li><a href="Differentiable_programming" title="Differentiable programming">Differentiable</a></li>

<li><a href="Grammar-oriented_programming" title="Grammar-oriented programming">Grammar-oriented</a></li>

<li><a href="Interface_description_language" title="Interface description language">Interface description</a></li>

<li><a href="Language-oriented_programming" title="Language-oriented programming">Language-oriented</a></li>

<li><a href="List_comprehension" title="List comprehension">List comprehension</a></li>

<li><a href="Modeling_language" title="Modeling language">Modeling</a></li>

<li><a href="Natural-language_programming" class="mw-redirect" title="Natural-language programming">Natural language</a></li>

<li><a href="Non-English-based_programming_languages" title="Non-English-based programming languages">Non-English-based</a></li>

<li><a href="Page_description_language" title="Page description language">Page description</a></li>

<li><a href="Pipeline_(software)" title="Pipeline (software)">Pipes</a> and <a href="Filter_(software)" title="Filter (software)">filters</a></li>

<li><a href="Probabilistic_programming" title="Probabilistic programming">Probabilistic</a></li>

<li><a href="Quantum_programming" title="Quantum programming">Quantum</a></li>

<li><a href="Scientific_programming_language" title="Scientific programming language">Scientific</a></li>

<li><a href="Scripting_language" title="Scripting language">Scripting</a></li>

<li><a href="Set_theoretic_programming" title="Set theoretic programming">Set-theoretic</a></li>

<li><a href="Simulation_language" title="Simulation language">Simulation</a></li>

<li><a href="Stack-oriented_programming" title="Stack-oriented programming">Stack-based</a></li>

<li><a href="System_programming_language" title="System programming language">System</a></li>

<li><a href="Tactile_programming_language" title="Tactile programming language">Tactile</a></li>

<li><a href="Template_processor" title="Template processor">Templating</a></li>

<li><a href="Transformation_language" title="Transformation language">Transformation</a> (<a href="Graph_rewriting" title="Graph rewriting">Graph rewriting</a>, <a href="Production_system_(computer_science)" title="Production system (computer science)">Production</a>, <a href="Pattern_matching" title="Pattern matching">Pattern</a>)</li>

<li><a href="Visual_programming_language" title="Visual programming language">Visual</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Concurrent_computing" title="Concurrent computing">Concurrent</a>, <a href="Distributed_computing" title="Distributed computing">distributed</a>, <a href="Parallel_computing" title="Parallel computing">parallel</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Actor_model" title="Actor model">Actor-based</a></li>

<li><a href="Automatic_mutual_exclusion" title="Automatic mutual exclusion">Automatic mutual exclusion</a></li>

<li><a href="Choreographic_programming" title="Choreographic programming">Choreographic programming</a></li>

<li><a href="Concurrent_logic_programming" title="Concurrent logic programming">Concurrent logic</a> (<a href="Concurrent_constraint_logic_programming" title="Concurrent constraint logic programming">Concurrent constraint logic</a>)</li>

<li><a href="Concurrent_object-oriented_programming" title="Concurrent object-oriented programming">Concurrent OO</a></li>

<li><a href="Macroprogramming" title="Macroprogramming">Macroprogramming</a></li>

<li><a href="Multitier_programming" title="Multitier programming">Multitier programming</a></li>

<li><a href="Organic_computing" title="Organic computing">Organic computing</a></li>

<li><a href="Parallel_programming_model" title="Parallel programming model">Parallel programming models</a></li>

<li><a href="Partitioned_global_address_space" title="Partitioned global address space">Partitioned global address space</a></li>

<li><a href="Process-oriented_programming" title="Process-oriented programming">Process-oriented</a></li>

<li><a href="Relativistic_programming" title="Relativistic programming">Relativistic programming</a></li>

<li><a href="Service-oriented_programming" title="Service-oriented programming">Service-oriented</a></li>

<li><a href="Structured_concurrency" title="Structured concurrency">Structured concurrency</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Metaprogramming" title="Metaprogramming">Metaprogramming</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Attribute-oriented_programming" title="Attribute-oriented programming">Attribute-oriented</a></li>

<li><a href="Automatic_programming" title="Automatic programming">Automatic</a> (<a href="Inductive_programming" title="Inductive programming">Inductive</a>)</li>

<li><a href="Dynamic_programming_language" title="Dynamic programming language">Dynamic</a></li>

<li><a href="Extensible_programming" title="Extensible programming">Extensible</a></li>

<li><a href="Generic_programming" title="Generic programming">Generic</a></li>

<li><a href="Homoiconicity" title="Homoiconicity">Homoiconicity</a></li>

<li><a href="Interactive_programming" title="Interactive programming">Interactive</a></li>

<li><a href="Macro_(computer_science)" title="Macro (computer science)">Macro</a> (<a href="Hygienic_macro" title="Hygienic macro">Hygienic</a>)</li>

<li><a href="Metalinguistic_abstraction" title="Metalinguistic abstraction">Metalinguistic abstraction</a></li>

<li><a href="Multi-stage_programming" title="Multi-stage programming">Multi-stage</a></li>

<li><a href="Program_synthesis" title="Program synthesis">Program synthesis</a> (<a href="Bayesian_program_synthesis" title="Bayesian program synthesis">Bayesian</a>, <a href="Inferential_programming" title="Inferential programming">Inferential</a>, <a href="Programming_by_demonstration" title="Programming by demonstration">by demonstration</a>, <a href="Programming_by_example" title="Programming by example">by example</a>)</li>

<li><a href="Reflective_programming" title="Reflective programming">Reflective</a></li>

<li><a href="Self-modifying_code" title="Self-modifying code">Self-modifying code</a></li>

<li><a href="Symbolic_programming" title="Symbolic programming">Symbolic</a></li>

<li><a href="Template_metaprogramming" title="Template metaprogramming">Template</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Separation_of_concerns" title="Separation of concerns">Separation of concerns</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aspect-oriented_programming" title="Aspect-oriented programming">Aspects</a></li>

<li><a href="Component-based_software_engineering" title="Component-based software engineering">Components</a></li>

<li><a href="Data-driven_programming" title="Data-driven programming">Data-driven</a></li>

<li><a href="Data-oriented_design" title="Data-oriented design">Data-oriented</a></li>

<li><a href="Event-driven_programming" title="Event-driven programming">Event-driven</a></li>

<li><a href="Feature-oriented_programming" title="Feature-oriented programming">Features</a></li>

<li><a href="Literate_programming" title="Literate programming">Literate</a></li>

<li><a href="Role-oriented_programming" title="Role-oriented programming">Roles</a></li>

<li><a href="Subject-oriented_programming" title="Subject-oriented programming">Subjects</a></li></ul>

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