OntoMаthPro Ontology:

A Linked Data Hub for Mathematics

Olga Nevzorova, Nikita Zhiltsov,Alexander Kirillovich, Evgeny Lipachev

Kazan Federal University

September 30, 2014

KESW 2014Kazan, Russia

Outline

• OntoMаthPro ontology Goals

Design principles

Concepts

Relations

• Applications: Use case 1: Semantic search for mathematical formulas

Use case 2: Semantic publishing platform

Use case 3: Education

Dimensions

• Expressive formal ontology

• Applied ontologyfor automatic processing mathematical articles

• LOD-compliant dataset,that can be a Linked Data Hub for Mathematics

Concepts

Logics, Set theory, Geometry, Differential Geometry, …

Set, Function, Integral, Lambda matrix, Christoffel Symbol, …

• Taxonomies: Fields of mathematic

Mathematical objects

• Concept description:• English and Russian labels

• Definitions,

• Relations with other concepts

• Links to external terminologies (DBpedia and ScienceWISE)

• Total 3450 concepts

Relations

• Taxonomic relation

• Logical dependency

• Associative relation between objects

• Belongingness of objects to fields of mathematics

• Associative relation between problems and methods

Lambda matrix is a Matrix

Christoffel Symbol is defined by Connectedness

Barycentric Coordinates belongs to Metric Geometry

Matric algebraic linear equation is solved by Gauss method

Chebyshev Iterative Method see also Numerical Solution of Linear Equation Systems

Applications

• Semantic search for mathematical formulas

• Semantic publishing platform and→ RDF dataset PHEIM

• Education

Semantic search for mathematical formulas

• There are currently many math formula search engines out there

• but they are mostly syntactical, and allow only simple search by expression patterns

• Our approach is semantic, and, therefore, can find formulas with respect to a given math concept

• Online demo: http://cll.niimm.ksu.ru/mathsearch

(uni)quation, Springer LaTeX Search, Wikipedia Formula Search, Wolfram Formula Search…

Find formulas, containing “(a + b)2”.

Find formulas that contain variables, expressing angles.

The user enters keywords, filtering suggestions of the system

First column contains a variable standing for the query concept in the relevant formula

Second column contains relevant formula

The third column contains the document fragment (theorem, proof, proposition, …), containing the formula

The user can filter the structural context

Formula details: list of all variables and metadata

Semantic publishing platform

Input: Output:

LOD representationCollection of math articles in

See: [Nevzorova et al. ISWC'13]

RDF-dataset PHEIM

• PHEIM — RDF dataset of the articles of the "Proceeding of Higher Education Institutions: Mathematics" journal

• The dataset covers: Metadata: Title, Authors, Organizations, …

→ AKT Portal ontology

Logical structure of documents: Theorem, Proof, Lemma, …

→ Mocassin ontology

Terminology

→ OntoMathPro ontology

Formulas, related to terms

• Size: 854 284 triples, 1330 articles, about 44 000 variables, 17 000

formulas, 4 000 theorems, 3 000 proofs, 1 000 definitions and other indexed mathematical entities

• SPARQL access point: http://cll.niimm.ksu.ru:8890/sparql

Education (1)

Basic idea:

• A student must know how to conceive a holistic image of the discipline

• We propose to consider an ontology as a close approximation to such image

Education (2)

Task:

• We have taken out a slightly modified small fragment of the ontology on numerical analysis.

• It contains:

• taxonomy of tasks for systems of linear equations

• taxonomy of solving methods

• relationships between them

• We hid all the relations and asked students to reconstruct them

Education (3)

Task:

• We have taken out a slightly modified small fragment of the ontology on numerical analysis. It contains:

• taxonomy of tasks and solving methods for systems of linear equations

• relationships between them

• We hid the relations and asked students to reconstruct them

Education (4)

Evaluation:

• Student's grade = to the similarity between the source graph example and the recovered graph.

• In the experiment, we consider sets of both the graph edges and use precision, recall, and F-score.

• We emphasize that we applied reasoning with respect to ISA relation

Education

Results (F-score):

• ISA Tasks relation: 100%

• ISA Methods relation: 66.7%

• Solves relation: 35%

Conclusion

• We present the ontology of math knowledge

• We describe its applications in information extraction, search and education

http://ontomathpro.org