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ProbLog

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</style><table class="infobox vevent"><tbody><tr><th colspan="2" class="infobox-above summary">ProbLog</th></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;"><a href="Programmer" title="Programmer">Original author(s)</a></th><td class="infobox-data">DTAI research lab (KU Leuven)</td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Initial release</th><td class="infobox-data">November 11, 2007 (2007-11-11)</td></tr><tr style="display: none;"><td colspan="2" class="infobox-full-data"></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;"><a href="Software_release_life_cycle" title="Software release life cycle">Stable release</a></th><td class="infobox-data"><div style="margin:0px;">2.1

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</td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Written in</th><td class="infobox-data"><a href="Python_(programming_language)" title="Python (programming language)">Python</a></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;"><a href="Operating_system" title="Operating system">Operating system</a></th><td class="infobox-data"><a href="Linux" title="Linux">Linux</a>, <a href="Mac_OS_X" class="mw-redirect" title="Mac OS X">Mac OS X</a>, <a href="Microsoft_Windows" title="Microsoft Windows">Microsoft Windows</a></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;"><a href="Software_categories#Categorization_approaches" title="Software categories">Type</a></th><td class="infobox-data"><a href="Probabilistic_logic" title="Probabilistic logic">Probabilistic logic</a></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;"><a href="Software_license" title="Software license">License</a></th><td class="infobox-data"><a href="Apache_License" title="Apache License"> Apache License, Version 2.0</a></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Website</th><td class="infobox-data"><a rel="nofollow" class="external text" href="https://dtai.cs.kuleuven.be/problog/">dtai.cs.kuleuven.be/problog/</a></td></tr></tbody></table>

ProbLog is a <a href="Probabilistic_logic_programming" title="Probabilistic logic programming">probabilistic logic programming</a> language that extends <a href="Prolog" title="Prolog">Prolog</a> with probabilities.<a href="#cite_note-ProbLogLinkDiscovery-1">[1]</a><a href="#cite_note-PLP-2">[2]</a><a href="#cite_note-ProbLogConcepts-3">[3]</a> It minimally extends <a href="Prolog" title="Prolog">Prolog</a> by adding the notion of a probabilistic fact, which combines the idea of <a href="Atomic_formula" title="Atomic formula">logical atoms</a> and <a href="Random_variable" title="Random variable">random variables</a>. Similarly to Prolog, ProbLog can query an <a href="Atomic_formula" title="Atomic formula">atom</a>. While Prolog returns the <a href="Truth_value" title="Truth value">truth value</a> of the queried <a href="Atomic_formula" title="Atomic formula">atom</a>, ProbLog returns the <a href="Probability" title="Probability">probability</a> of it being true.

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

A probabilistic fact is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,a)}">

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<annotation encoding="application/x-tex">{\displaystyle (p,a)}</annotation>

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</math><img src="./5db544419bdd0558faa9ee8fedf1a3bc33fc0de7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.242ex; height:2.843ex;" alt="{\displaystyle (p,a)}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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</mrow>

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> a <a href="Ground_expression#Ground_atom" title="Ground expression">ground atom</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle p\in [0,1]}</annotation>

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</math><img src="./33c3a52aa7b2d00227e85c641cca67e85583c43c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.752ex; height:2.843ex;" alt="{\displaystyle p\in [0,1]}" loading="lazy"> the probability of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> being true. A rule is defined by an atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, called the head, and a finite set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> <a href="Literal_(mathematical_logic)" title="Literal (mathematical logic)">literals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{1},b_{2},...,b_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{b_{1},b_{2},...,b_{n}\}}</annotation>

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</math><img src="./818c2b6b79c8f588a31b98d8dab8f1b1d0049fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.848ex; height:2.843ex;" alt="{\displaystyle \{b_{1},b_{2},...,b_{n}\}}" loading="lazy">, called the body.

ProbLog programs consist of a set of probabilistic facts <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

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

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</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> and a set of rules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}">

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

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</math><img src="./74532dc308c806964b832df0d0d73352195c2f2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}" loading="lazy">. Using the distribution semantics, a probability distribution is defined over the two-valued <a href="Well-founded_semantics" title="Well-founded semantics">well-founded models</a> of the atoms in the program. The probability of a model is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(M)=\prod _{l\in M}P(l)}">

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle P(M)=\prod _{l\in M}P(l)}</annotation>

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</math> where the product runs over all the literals in the model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. For a query atom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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</mrow>

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

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> the distribution semantics defines a probability for the query

</mrow>

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle P(q)=\sum _{M\models q}P(M)=\sum _{M\models q}\prod _{l\in M}P(l)}</annotation>

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</math>

in which the sum runs over all the models where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

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

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ProbLog supports multiple tasks:

<ul><li>Probabilistic inference: calculate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(q)}">

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<annotation encoding="application/x-tex">{\displaystyle P(q)}</annotation>

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</math><img src="./7076229c299e38dcffbc3bca2144e5f64fac9537.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.624ex; height:2.843ex;" alt="{\displaystyle P(q)}" loading="lazy"></li>

<li>Most probable explanation: calculate the most probable model probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max _{M\models q}P(M)}">

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</munder>

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<annotation encoding="application/x-tex">{\displaystyle \max _{M\models q}P(M)}</annotation>

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</math><img src="./045cceb7cf6801d633618d351a0759b87f8e328c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.71ex; height:4.509ex;" alt="{\displaystyle \max _{M\models q}P(M)}" loading="lazy"></li>

<li>Sampling: generate samples of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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

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<li>Learning from interpretations: learn the probabilities of ProbLog programs from data</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

ProbLog can for example be used to calculate the probability of getting wet given the probabilities for rain and the probabilities that someone brings an umbrella as follows:

<div class="mw-highlight mw-highlight-lang-prolog mw-content-ltr" dir="ltr"><pre>0.4 :: rain(weekday).

0.9 :: rain(weekend).

0.8 :: umbrella_if_rainy(Day).

0.2 :: umbrella_if_dry(Day).

umbrella(Day) :- rain(Day), umbrella_if_rainy(Day).

umbrella(Day) :- \+rain(Day), umbrella_if_dry(Day).

wet(Day) :- rain(Day), \+umbrella(Day).

query(\+wet(weekend)).

</pre></div>

The last rule before the query states that someone gets wet if it rains and no umbrella was brought. When ProbLog is asked to solve the "probabilistic inference" task, the query asks for the probability to stay dry on a weekend day. When solving the "most probable explanation" task, ProbLog will return the most likely reason for staying dry, i.e. because it is not raining or because the person has an umbrella.

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

The ProbLog language has been implemented as a YAP Prolog library (ProbLog 1).<a href="#cite_note-ProbLog1-4">[4]</a> and as a stand-alone Python framework (ProbLog 2)<a href="#cite_note-ProbLogHomepage-5">[5]</a>

The source code of ProbLog 2 is licensed under <a href="Apache_License" title="Apache License"> Apache License, Version 2.0</a> and available on <a href="GitHub" title="GitHub">GitHub</a>.<a href="#cite_note-ProbLogRepository-6">[6]</a> The ProbLog language has also been implemented as part of the cplint probabilistic logic programming package for SWI-Prolog, YAP and XSB.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="ProbLog_variants">ProbLog variants</h2></div>

ProbLog has been extended or used as inspiration for several different variants, including:

<ul><li>DeepProbLog extends ProbLog by allowing the probability to be parametrized by a neural network.<a href="#cite_note-DeepProbLog-8">[8]</a></li>

<li>DTProblog extends ProbLog with <a href="Decision_theory" title="Decision theory">decision theory</a>. The utility of a strategy is defined as the expected reward for its execution in the presence of probabilistic effects.<a href="#cite_note-DTProbLog-9">[9]</a></li>

<li>DC-ProbLog extends ProbLog with distributional facts, meaning that instead of probabilities, a logic atom has a corresponding continuous probability distribution instead.</li>

<li>aProbLog generalizes ProbLog by allowing any <a href="Commutative_semiring" class="mw-redirect" title="Commutative semiring">commutative semiring</a> instead of just probabilities.<a href="#cite_note-aProbLog-10">[10]</a></li>

<li>ProbFOIL: given a set of ProbLog facts as a probabilistic relational database, ProbFOIL finds a set of probabilistic rules to predict the facts of one relation based on all other relations.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Related_languages">Related languages</h2></div>

<ul><li>PRISM: Programming in statistical modeling<a href="#cite_note-PRISM-11">[11]</a></li>

<li>ICL: Independent Choice Logic<a href="#cite_note-ICL-12">[12]</a></li>

<li>CP-Logic: Language of causal probabilistic events<a href="#cite_note-CP-Logic-13">[13]</a></li>

<li>LPAD: Logic programs with annotated disjunctions<a href="#cite_note-PITA-14">[14]</a></li>

<li>Distributional clauses: A probabilistic logic language for hybrid relational domains<a href="#cite_note-DC-15">[15]</a></li></ul>

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

<ul><li>ProbLog homepage<a href="#cite_note-ProbLogHomepage-5">[5]</a></li>

<li>ProbLog docs<a href="#cite_note-ProbLogDocs-16">[16]</a></li>

<li>ProbLog repository<a href="#cite_note-ProbLogRepository-6">[6]</a></li></ul>

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

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</style><cite id="CITEREFDe_RaedtKimmigToivonen2007" class="citation conference cs1">De Raedt, Luc; Kimmig, Angelika; Toivonen, Hannu (November 2007). ProbLog: A Probabilistic Prolog and Its Application in Link Discovery. IJCAI. Vol. 7.</cite>

</li>

<li id="cite_note-PLP-2"><a href="#cite_ref-PLP_2-0">^</a> <cite id="CITEREFFierensVan_den_BroeckBruynoogheDe_Raedt2012" class="citation conference cs1">Fierens, D; Van den Broeck, G.; Bruynooghe, M.; De Raedt, L. (2012). Constraints for probabilistic logic programming. Proceedings of the NIPS Probabilistic Programming Workshop. pp. 1–4.</cite>

</li>

<li id="cite_note-ProbLogConcepts-3"><a href="#cite_ref-ProbLogConcepts_3-0">^</a> <cite id="CITEREFDe_RaedtKimmig2015" class="citation journal cs1">De Raedt, Luc; Kimmig, Angelika (2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10994-015-5494-z">"Probabilistic (logic) programming concepts"</a>. Machine Learning. 100 (1): 5–47. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10994-015-5494-z">10.1007/s10994-015-5494-z</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3166992">3166992</a>.</cite>

</li>

<li id="cite_note-ProbLog1-4"><a href="#cite_ref-ProbLog1_4-0">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dtai.cs.kuleuven.be/problog/problog1/installation.html">"ProbLog1"</a>. dtai.cs.kuleuven.be.</cite>

</li>

<li id="cite_note-ProbLogHomepage-5">^ <a href="#cite_ref-ProbLogHomepage_5-0">a</a> <a href="#cite_ref-ProbLogHomepage_5-1">b</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://dtai.cs.kuleuven.be/problog">"ProbLog: Probabilistic Programming"</a>. dtai.cs.kuleuven.be.</cite>

</li>

<li id="cite_note-ProbLogRepository-6">^ <a href="#cite_ref-ProbLogRepository_6-0">a</a> <a href="#cite_ref-ProbLogRepository_6-1">b</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/ML-KULeuven/problog">"ProbLog GitHub repository"</a>. github.com. 12 October 2022.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ai.unife.it/cplint/">"cplint – AI@UNIFE"</a>. Retrieved 2023-11-13.</cite>

</li>

<li id="cite_note-DeepProbLog-8"><a href="#cite_ref-DeepProbLog_8-0">^</a> <cite id="CITEREFManhaeveDumancicKimmigDemeester2018" class="citation conference cs1">Manhaeve, Robin; Dumancic, Sebastijan; Kimmig, Angelika; Demeester, Thomas; De Raedt, Luc (2018). DeepProbLog: Neural Probabilistic Logic Programming. NeurIPS 2018, Thirty-second Conference on Neural Information Processing Systems. pp. 3753–3760.</cite>

</li>

<li id="cite_note-DTProbLog-9"><a href="#cite_ref-DTProbLog_9-0">^</a> <cite id="CITEREFVan_den_BroeckThonVan_OtterloDe_Raedt2010" class="citation conference cs1">Van den Broeck, Guy; Thon, Ingo; Van Otterlo, Martijn; De Raedt, Luc (2010). "DTProbLog: A decision-theoretic probabilistic Prolog". Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 24.</cite>

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