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Knowledge Representation and Rule Mining in Entity-Centric Knowledge Bases

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Reasoning Web. Explainable Artificial Intelligence

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11810))

Abstract

Entity-centric knowledge bases are large collections of facts about entities of public interest, such as countries, politicians, or movies. They find applications in search engines, chatbots, and semantic data mining systems. In this paper, we first discuss the knowledge representation that has emerged as a pragmatic consensus in the research community of entity-centric knowledge bases. Then, we describe how these knowledge bases can be mined for logical rules. Finally, we discuss how entities can be represented alternatively as vectors in a vector space, by help of neural networks.

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Notes

  1. 1.

    http://blogs.bing.com/search/2013/03/21/understand-your-world-with-bing/.

  2. 2.

    https://wolframalpha.com.

  3. 3.

    We can even say type(class, class), i.e., class is an instance of class.

  4. 4.

    Let \(|\mathcal {K}|\) be the number of facts and \(|r(\mathcal {K})|\) the number of relations in a KB \(\mathcal {K}\). Let d be the maximal length of a rule. The size of the search space is reduced from \(O(|\mathcal {K}|^d)\) to \(O(|r(\mathcal {K})|^d)\) when we remove the addInstantiatedAtom operator.

  5. 5.

    Instead of: if a rule is not frequent, none of its refinements can be frequent.

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Authors and Affiliations

  1. Telecom Paris, Institut Polytechnique de Paris, Paris, France

    Fabian M. Suchanek, Jonathan Lajus & Armand Boschin

  2. Max Planck Institute for Informatics, Saarbrücken, Germany

    Gerhard Weikum

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Correspondence to Fabian M. Suchanek .

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  1. Technische Universität Dresden, Dresden, Sachsen, Germany

    Markus Krötzsch

  2. Bosch Center for Artificial Intelligence, Renningen, Germany

    Daria Stepanova

A Computation of Support and Confidence

A Computation of Support and Confidence

Notation. Given a logical formula \(\phi \) with some free variables \(x_1, \dots , x_n\), all other variables being by default existentially quantified, we define:

$$ \#(x_1, \dots , x_n): \phi \quad :=\quad |\{ (x_1, \dots , x_n)\ :\ \phi (x_1, \dots , x_n) \text { is true }\}|$$

We remind the reader of the two following definitions:

Definition

14 (Prediction of a rule): The predictions P of a rule \(\varvec{B} \Rightarrow h\) in a KB \(\mathcal {K}\) are the head atoms of all instantiations of the rule where the body atoms appear in \(\mathcal {K}\). We write \(\mathcal {K}\wedge (\varvec{B} \Rightarrow h) \models P\).

Definition

19 (Support): The support of a rule in a KB is the number of positive examples predicted by the rule.

A prediction of a rule is a positive example if and only if it is in the KB. This observation gives rise to the following property:

Proposition 34

(Support in practice): The support of a rule \(\varvec{B}\Rightarrow h\) is the number of instantiations of the head variables that satisfy the query \(\varvec{B} \wedge h\). This value can be written as:

$$\text {support}(\varvec{B} \Rightarrow h(x, y)) = \#(x,y): \varvec{B} \wedge h(x,y)$$

Definition

20 (Confidence): The confidence of a rule is the number of positive examples predicted by the rule (the support of the rule), divided by the number of examples predicted by the rule.

Under the CWA, all the predicted examples are either positive examples or negative examples. Thus, the standard confidence of a rule is the support of the rule divided by the number of prediction of the rule, written:

$$ \textit{std-conf}(\varvec{B} \Rightarrow h(x, y)) = \frac{\#(x,y): \varvec{B} \wedge h(x,y)}{\#(x,y): \varvec{B}}$$

Assume h is more functional than inverse functional. Under the PCA, a predicted negative example is a prediction h(x, y) that is not in the KB, such that, for this x there exists another entity \(y'\) such that \(h(x,y')\) is in the KB. When we add the predicted positive examples, the denominator of the PCA confidence becomes:

$$\#(x,y): (\varvec{B} \wedge h(x,y)) \vee (\varvec{B} \wedge \lnot h(x,y) \wedge \exists y'. h(x,y'))$$

We can simplify this logical formula to deduce the following formula for computing the PCA confidence:

$$\textit{pca-conf}(\varvec{B} \Rightarrow h(x,y)) = \frac{\#(x,y): \varvec{B} \wedge h(x,y)}{\#(x,y): \varvec{B} \wedge \exists y'. h(x,y')}$$

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Suchanek, F.M., Lajus, J., Boschin, A., Weikum, G. (2019). Knowledge Representation and Rule Mining in Entity-Centric Knowledge Bases. In: Krötzsch, M., Stepanova, D. (eds) Reasoning Web. Explainable Artificial Intelligence. Lecture Notes in Computer Science(), vol 11810. Springer, Cham. https://doi.org/10.1007/978-3-030-31423-1_4

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