ontomаthpro ontology: a linked data hub for mathematics
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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 third column contains the document fragment (theorem, proof, proposition, …), containing the formula
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
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