OntoMath is an ecosystem of ontologies, text analytics tools, and applications for mathematical knowledge management

1. Alik Kirillovich
(alik.kirillovich@gmail.com)
Higher Institute of Information Technology and Intelligent Systems
Kazan Federal University
02 November, 2018
OntoMath Digital Ecosystem

2. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

3. Traditional libraries
Traditional libraries: documents and bibliographic relationships

4. Traditional libraries
Traditional libraries: documents and bibliographic relationships
Limitation: no direct access to the objects of mathematical knowledge

5. World Digital Mathematical Library (WDML)
WDML: mathematical concepts, objects and logical relationships
Direct access to the objects of mathematical knowledge

6. International Mathematics Union
World Digital Mathematical Library
working group
Workshop 2012
(Washington)
Workshop 2014
(Seoul)
Workshop 2016
(Toronto)
Local working groups: Wolfram Research, Kazan, ….

7. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

8. OntoMath ecosystem
OntoMath is an ecosystem of ontologies, text analytics tools, and applications
for math knowledge management

9. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

10. OntoMath ecosystem
OntoMath is an ecosystem of ontologies, text analytics tools, and applications
for math knowledge management
The core component of the OntoMath digital ecosystem is a Semantic publishing platform

11. Semantic publishing platform (1)
Semantic publishing platform analyzes the underlying semantics in mathematical
scholarly papers and builds their consolidated ontology-based representation
Input: Output:
Collection of math
articles in
Linking Open Data
representation

12. Semantic publishing platform (2)
The generated dataset covers:
• Matadata:
 Title, date, …
 Authors
 Organizations
→ AKT Portal ontology
• Logical structure of documents: Theorem, Proof, Lemma, …
→ Mocassin ontology
• Terminology
→ OntoMathpro ontology
 Formulas, related to terms

13. Input: collection of publications

14. Input: collection of publications

15. AKT Portal ontology
• Journal
• Author
• Publication-Reference
• …
Matadata extraction
Author: V.I. Arnold
Title: On the matricial version…
Journal: Japan. J. Math. 1, 2006
Affiliation: Steklov Math. Institute
Content:
The congruences modulo the
primary numbers n = pa are
studied for the traces of the
matrices An and An−ϕ(n) , where
A is an integer matrix and ϕ(n)
is the number of residues
modulo …

16. AKT Portal ontology
• Journal
• Author
• Publication-Reference
• …
Matadata extraction
Author: V.I. Arnold
Title: On the matricial version…
Journal: Japan. J. Math. 1, 2006
Affiliation: Steklov Math. Institute
Content:
The congruences modulo the
primary numbers n = pa are
studied for the traces of the
matrices An and An−ϕ(n) , where
A is an integer matrix and ϕ(n)
is the number of residues
modulo …

17. Mocassin ontology
Logical structure extraction

18. Mocassin ontology
Logical structure extraction

19. The congruences modulo the primary numbers n
= pa are studied for the traces of the matrices An
and An−ϕ(n) , where A is an integer matrix and
ϕ(n) is the number of residues n, relatively prime
to n.
We present an algorithm to decide whether
these congruences hold for all the integer
matrices A, when the prime number p is fixed.
The algorithm is explicitly applied for many
values of p, …
Ontology-based Terminology Extraction
OntoMathPro ontology

20. Formulas binding
for all integer matrices A, primes p, and natural numbers n. For (1×1)-matrices,
these are just the usual Euler congruences …

21. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

22. Mocassin
Mocassin is an ontology of elements of the logical structure of mathematical papers

23. Outline
1. OntoMath Digital Ecosystem
2. Semantic publishing platform
3. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
4. Applications
1. Semantic search for mathematical formulas
2. Recommender system
5. Conclusion

24. OntoMаthPro (1)
OntoMathPro (http://ontomathpro.org) is an ontology of mathematical knowledge

25. OntoMаthPro (2)
Concepts:
• Taxonomies:
 Fields of mathematics
 Mathematical objects
• Concept description:
 English and Russian labels
 Definitions,
 Relations with other concepts
 Links to external terminologies (DBpedia and ScienceWISE)
• Total: 3450 concepts
Logics, Set theory, Geometry, Differential Geometry, …
Set, Function, Integral, Lambda matrix, Christoffel Symbol, …

26. OntoMаthPro (3)
Relations:
• Taxonomic relation (ISA)
• Belongingness of objects to fields of mathematics
• Logical dependency
• Associative relation between objects
• Associative relation between problems and methods
Lambda matrix is a Matrix
Barycentric Coordinates belongs to Metric Geometry
Christoffel Symbol is defined by Connectedness
Chebyshev Iterative Method see also Numerical Solution of Linear Equation Systems
Matric algebraic linear equation is solved by Gauss method

27. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

28. OntoMath ecosystem
OntoMath is an ecosystem of ontologies, text analytics tools, and applications
for math knowledge management

29. • 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://lobachevskii-dml/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.
Semantic search for mathematical formulas

30. The user enters keywords, filtering suggestions of the system

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

32. Second column contains relevant formula

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

34. The user can filter the structural context

35. Formula details: list of all variables and metadata

36. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

37. OntoMath ecosystem
OntoMath is an ecosystem of ontologies, text analytics tools, and applications
for math knowledge management

38. Recommender system (1)

39. Extracted keywords and Related pages are added
Recommender system (2)

40. Links to Term pages
Recommender system (3)

41. Links to other article pages
Recommender system (4)

42. Term page
Recommender system (5)

43. Outline
1. World Digital Mathematical Library
2. OntoMath Digital Ecosystem
3. Semantic publishing platform
4. Ontologies
1. Mocassin ontology
2. OntoMathPro ontology
5. Applications
1. Semantic search for mathematical formulas
2. Recommender system
6. Publications

44. Main publications
• Alexander Elizarov, Alexander Kirillovich, Evgeny Lipachev, Olga Nevzorova. Digital Ecosystem
OntoMath: Mathematical Knowledge Analytics and Management // Kalinichenko L., Kuznetsov S.,
Manolopoulos Y. (eds). XVIII International Conference on Data Analytics and Management in Data
Intensive Domains (DAMDID/RCDL 2016). Communications in Computer and Information Science,
vol 706. Springer, Cham, 2017. Pp. 33-46
• Olga Nevzorova, Nikita Zhiltsov, Alexander Kirillovich, Evgeny Lipachev. OntoMathPRO Ontology: A
Linked Data Hub for Mathematics // Pavel Klinov, Dmitry Mouromstev (eds.) Proceedings of the 5th
International Conference on Knowledge Engineering and Semantic Web (KESW 2014).
Communications in Computer and Information Science, vol. 468. Springer, Cham, 2014. Pp. 105–
119
• Olga Nevzorova, Nikita Zhiltsov, Danila Zaikin, Olga Zhibrik, Alexander Kirillovich, Vladimir Nevzorov,
Evgeniy Birialtsev. Bringing Math to LOD: A Semantic Publishing Platform Prototype for Scientific
Collections in Mathematics // Harith Alani, et al. (eds.) Proceedings of the 12th International
Semantic Web Conference (ISWC 2013). Lecture Notes in Computer Science, vol. 8218. Springer
Berlin Heidelberg, 2013. Pp. 379-394
• Alexander Elizarov, Alexander Kirillovich, Evenvgeny Lipachev, Olga Nevzorova. Semantic Formula
Search in Digital Mathematical Libraries // Proceedings of the 2nd Russia and Pacific Conference on
Computer Technology and Applications (RPC 2017). IEEE, 2017
• A.M. Elizarov, A.V. Kirillovich, E.K. Lipachev, A.B. Zhizhchenko, N.G. Zhil'tsovc. Mathematical
Knowledge Ontologies and Recommender Systems for Collections of Documents in Physics and
Mathematics // Doklady Mathematics, 2016, Vol. 93, Num. 2. Pp. 1–3