Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Content

1. Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Context Moritz Schubotz1 , André Greiner-Petter*1 , Philipp Scharpf*1 , Norman Meuschke1 , Howard S. Cohl2 , Bela Gipp1 June 5, 2018 1University of Konstanz, Germany 2National Institute of Standards and Technology, USA *sponsored by SIGIR Student Travel Grant 1/14

2. Motivation

3. Formats of Mathematical Formulae Riemann Zeta Function Rendered Version: ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0 2/14

4. Formats of Mathematical Formulae Riemann Zeta Function Rendered Version: ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0 LATEX: zeta(s) = 0 Rightarrow Re s = frac12 lor Im s=0 2/14

5. Formats of Mathematical Formulae Riemann Zeta Function Rendered Version: ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0 LATEX: zeta(s) = 0 Rightarrow Re s = frac12 lor Im s=0 Mathematica: Implies[ Equal[Zeta[s], 0], Or[ Equal[Re[s], Rational[1, 2]], Equal[Im[s], 0] ] ] 2/14

6. Formats of Mathematical Formulae Riemann Zeta Function Rendered Version: ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0 LATEX: zeta(s) = 0 Rightarrow Re s = frac12 lor Im s=0 Mathematica: Implies[ Equal[Zeta[s], 0], Or[ Equal[Re[s], Rational[1, 2]], Equal[Im[s], 0] ] ] 2/14

7. Formats of Mathematical Formulae Riemann Zeta Function Rendered Version: ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0 LATEX: zeta(s) = 0 Rightarrow Re s = frac12 lor Im s=0 Mathematica: Implies[ Equal[Zeta[s], 0], Or[ Equal[Re[s], Rational[1, 2]], Equal[Im[s], 0] ] ] ← 18 tokens with max depth of 2 ← 16 tokens with max depth of 5 2/14

8. Formats of Mathematical Formulae - MathML A Combined Format Find another format that (1) provides presentation and semantic information, (2) is easy to parse, and (3) is extendible. ⇒ Mathematical Markup Language 3.0. 3/14

13. Formats of Mathematical Formulae - MathML A Combined Format Find another format that (1) provides presentation and semantic information, (2) is easy to parse, and (3) is extendible. ⇒ Mathematical Markup Language 3.0. Part of the MathML for ζ(s) = 0 ⇒ s = 1 2 ∨ s = 0: <math><semantics><mrow>. . . <mo id="5" xref="20">=</mo> <mn id="5" xref="21">0</mn> <mo id="7" xref="19">⇒</mo>. . .</mrow> <annotation−xml encoding="MathML−Content"> <apply><implies id="19" xref="7"/> <apply><eq id="20" xref="5"/> <apply><csymbol id="21" xref="1">ζ</csymbol>. . . </annotation−xml></semantics></math> 3/14

14. Contributions Our Contributions We present the following three main contributions 1. MathMLBen - benchmark for MathML, 2. Evaluate state-of-the art translation tools, 3. Propose a new approach that consider textual context. 4/14

18. MathMLBen - Create a MathML Benchmark Dataset

19. Create MML Benchmark Dataset mathmlben.wmﬂabs.org SwitchSwitch POM-Tagger Dictionaries β β ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ Documents Formula a+b Semantic Formula a+b Manual Refinements Gold Standard MML Multiple Cycles Random Selection Annotated MMLAnnotated MML LaTeXML MML ComparisonMML Comparison VS ® Vecteezy.com & Converter β β Mathematical Language Processor Identifier & Definiens Tree Refinemets MathMLben#422f2d908725a379336f2c6083c5b6edf69157ca β π ζ XML 5/14

25. Create MML Benchmark Dataset mathmlben.wmﬂabs.org Annotated MathML Content MathML do not provide enough semantic information. We overcome this issue by manually annotate content MathML with Wikidata IDs. We used special TEX-macros to 1) annotate identiﬁer with Wikidata IDs, and 2) manipulate the expression tree. 6/14

28. Create MML Benchmark Dataset mathmlben.wmﬂabs.org Annotated MathML Content MathML do not provide enough semantic information. We overcome this issue by manually annotate content MathML with Wikidata IDs. We used special TEX-macros to 1) annotate identiﬁer with Wikidata IDs, and 2) manipulate the expression tree. Original LATEX: W(2, k) 6/14

29. Create MML Benchmark Dataset mathmlben.wmﬂabs.org Annotated MathML Content MathML do not provide enough semantic information. We overcome this issue by manually annotate content MathML with Wikidata IDs. We used special TEX-macros to 1) annotate identiﬁer with Wikidata IDs, and 2) manipulate the expression tree. Original LATEX: W(2, k) LaTeXML Input: wf{Q7913892}{W}(2, w{Q12503}{k}) 6/14

30. Create MML Benchmark Dataset mathmlben.wmﬂabs.org Annotated MathML Content MathML do not provide enough semantic information. We overcome this issue by manually annotate content MathML with Wikidata IDs. We used special TEX-macros to 1) annotate identiﬁer with Wikidata IDs, and 2) manipulate the expression tree. Original LATEX: W(2, k) LaTeXML Input: wf{Q7913892}{W}(2, w{Q12503}{k}) MathML Output: <apply id="p1.1.m1.1.13.1.1.cmml" xref="p1.1.m1.1.13.1.2"> <csymbol cd="wikidata" id=". . ." xref=". . .">Q7913892</csymbol> <cn type="integer" id=". . ." xref=". . .">2</cn> <csymbol cd="wikidata" id=". . ." xref=". . .">Q12503</csymbol> </apply> 6/14

31. Create MML Benchmark Dataset mathmlben.wmﬂabs.org MathMLBen Collection We annotated in total 305 formulae. • 1 to 100: randomly sampled from Wikipedia. Used for ’National Institute of Informatics Testbeds and Community for Information access Research Project’ (NTCIR) 11 • 101 to 200: randomly sampled from the sources of NIST Digital Library of Mathematical Functions (contains 9,897 labeled formulae). • 201 to 305: 70% from NTCIR arXiv and 30% from NTCIR-12 Wikipedia datasets. All data is available at https://mathmlben.wmflabs.org/. 7/14

35. Create MML Benchmark Dataset mathmlben.wmﬂabs.org 8/14

36. Evaluate State-Of-The-Art LATEX to MathML Conversion Tools

37. Benchmarking Conversion Tools SwitchSwitch POM-Tagger Dictionaries β β ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ Documents Formula a+b Semantic Formula a+b Manual Refinements Gold Standard MML Multiple Cycles Random Selection Annotated MMLAnnotated MML LaTeXML MML ComparisonMML Comparison VS ® Vecteezy.com & Converter β β Mathematical Language Processor Identifier & Definiens Tree Refinemets MathMLben#422f2d908725a379336f2c6083c5b6edf69157ca β π ζ XML 9/14

41. Benchmarking Conversion Tools - Accuracy Tested Conversion Tools 1. LaTeXML: Perl tool used to create DLMF 2. LaTeX2MathML: small Python project 3. Mathoid: service that allows to generate also SVG and PNG 4. SnuggleTeX: Java library developed at University of Edinburgh 5. MathToWeb: Java web application 6. TeXZilla: Javascript web application 7. Mathematical: Ruby application that can generate SVG/PNG 8. CAS: Computer Algebra System that is capable to parse LATEX 9. Part-Of-Math (POM) Tagger: grammar-based LATEX parser that was used to perform translations from LATEX to CAS. 10/14

42. Benchmarking Conversion Tools - Accuracy Tested Conversion Tools 1. LaTeXML: Perl tool used to create DLMF 2. LaTeX2MathML: small Python project 3. Mathoid: service that allows to generate also SVG and PNG 4. SnuggleTeX: Java library developed at University of Edinburgh 5. MathToWeb: Java web application 6. TeXZilla: Javascript web application 7. Mathematical: Ruby application that can generate SVG/PNG 8. CAS: Computer Algebra System that is capable to parse LATEX 9. Part-Of-Math (POM) Tagger: grammar-based LATEX parser that was used to perform translations from LATEX to CAS. 10/14

43. Benchmarking Conversion Tools - Accuracy 305 305 295 305 288 229 290 305 305 0 50 100 150 200 250 300 0 10 20 30 40 50 60 70 80 SuccessfullyParsedExpressions TreeEditDistance Average Distance of Presentation Subtree Average Distance of Content Subtree Successfully Parsed LaTeX Expressions Average of Structural Distances & Successfully Parsed Expressions 11/14

44. Benchmarking Conversion Tools - Runtime 372,76 29,65 20,77 9,57 4,17 3,69 1,79 1,41 1,00 10,00 100,00 1000,00 LatexML Mathoid Mathema�cal Latex2MML MathToWeb POM SnuggleTeX TeXZilla Performance of Tools Dura�on in Seconds 12/14

45. Improvements for Conversion Tools by Considering Textual Context

46. Approach to Improve Conversion Tools SwitchSwitch POM-Tagger Dictionaries β β ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ Documents Formula a+b Semantic Formula a+b Manual Refinements Gold Standard MML Multiple Cycles Random Selection Annotated MMLAnnotated MML LaTeXML MML ComparisonMML Comparison VS ® Vecteezy.com & Converter β β Mathematical Language Processor Identifier & Definiens Tree Refinemets MathMLben#422f2d908725a379336f2c6083c5b6edf69157ca β π ζ XML 13/14

49. Approach to Improve Conversion Tools POM-Tagger Dictionaries ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ ~~~~~~~~~~ ~~~~~~~ ~~~~~~~~ Documents Formula a+b Semantic Formula a+b Manual Refinements Gold Standard MML Multiple Cycles Random Selection Annotated MMLAnnotated MML LaTeXML MML ComparisonMML Comparison VS ® Vecteezy.com & Converter β β Mathematical Language Processor Identifier & Definiens Tree Refinemets MathMLben#422f2d908725a379336f2c6083c5b6edf69157ca β π ζ XML SwitchSwitch β β 13/14

50. Approach to Improve Conversion Tools Example for item 101 mathmlben.wmﬂabs.org/101 A function f(x, y) is continuous at a point (a, b) if lim (x,y)→(a,b) f(x, y) = f(a, b), (1) that is, for every arbitrarily small positive constant there exists δ(> 0) such that |f(a + α, b + β) − f(a, b)| < , (2) for all α and β that satisfy |α|, |β| < δ. 14/14

51. Approach to Improve Conversion Tools Example for item 101 mathmlben.wmﬂabs.org/101 A function f(x, y) is continuous at a point (a, b) if lim (x,y)→(a,b) f(x, y) = f(a, b), (1) that is, for every arbitrarily small positive constant there exists δ(> 0) such that |f(a + α, b + β) − f(a, b)| < , (2) for all α and β that satisfy |α|, |β| < δ. 14/14

52. Thanks for you attention!

Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Content

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Improving the Representation and Conversion of Mathematical Formulae by Considering their Textual Content