=Paper=
{{Paper
|id=Vol-2414/paper11
|storemode=property
|title=Towards Formula Concept Discovery and Recognition
|pdfUrl=https://ceur-ws.org/Vol-2414/paper11.pdf
|volume=Vol-2414
|authors=Philipp Scharpf,Moritz Schubotz,Howard S. Cohl,Bela Gipp
|dblpUrl=https://dblp.org/rec/conf/sigir/ScharpfSCG19
}}
==Towards Formula Concept Discovery and Recognition==
https://ceur-ws.org/Vol-2414/paper11.pdf
Towards Formula Concept Discovery and
Recognition
Philipp Scharpf1 , Moritz Schubotz2 , Howard S. Cohl3 , and Bela Gipp2
1
Department of Computer and Information Science
University of Konstanz, Germany
first.last@uni-konstanz.de
2
Department of Information Technology
University of Wuppertal, Germany
last@uni-wuppertal.de
3
National Institute of Standards and Technology, United States
first.last@nist.gov
Abstract. Citation-based Information Retrieval (IR) methods for sci-
entific documents have proven to be effective in academic disciplines that
use many references. In science, technology, engineering, and mathemat-
ics (STEM), researchers cite less often but employ mathematical concepts
to refer to prior knowledge (Moed et al.). Our long-term goal is to gen-
eralize citation-based IR-methods and apply the generalized method to
both classical references and mathematical concepts. In this paper, we
suggest how mathematical formulae could be cited and define a Formula
Concept Retrieval challenge with two subtasks: Formula Concept Discov-
ery (FCD) and Formula Concept Recognition (FCR). While the former
aims at the definition and exploration of a Formula Concept that names
bundled equivalent representations of a formula, the latter is designed
to match a given formula to a prior assigned concept ID. Moreover, we
present first Machine Learning based approaches to tackle the FCD and
FCR tasks, which we apply to a standardized test-collection (NTCIR
arXiv dataset). Our FCD approach yields a recall of 68% for retrieving
equivalent representations of frequent formulae, and 72% for extracting
the formula name from the surrounding text. FCD and FCR will enable
citing formulae within mathematical documents and facilitate seman-
tic search as well as similarity computations for plagiarism detection or
document recommender systems.
Keywords: Natural Language Processing · Mathematical Language Pro-
cessing · Mathematical Information Retrieval · Feature Analysis · Ma-
chine Learning
2 Philipp Scharpf, Moritz Schubotz, Howard S. Cohl, and Bela Gipp
1 Introduction
Documents from Science, Technology, Engineering, and Mathematics (STEM)
often contain a significant amount of mathematical formulae. Since they are vi-
tal to understanding the content of these documents, semantic search engines or
recommender systems need to process and analyze them alongside the text. In
information science and technology, the semantics of natural language is typi-
cally grasped via conceptualization [25]. In the case of mathematical language,
we argue for the introduction of a definition for a mathematical Formula Con-
cept as a collection of equivalent formulae with different representations (see [15]
for a discussion of the definition difficulties). Once defined, the technical imple-
mentation of a Formula Concept can be Formula Concept Discovery (FCD) and
Formula Concept Recognition (FCR). The first term (FCD) refers to the explo-
ration of formula concepts by examining a multitude of formula examples from
various sources and occurrences. Figure 1 illustrates how the same equation, in
this case, the Klein-Gordon equation from Quantum Physics, can be represented
in different formats that seem very diverse at first glance but actually represent
the same mathematical concept. We will present first implementations of FCD
and FCR in the following.
Fig. 1: Various representations of the Klein-Gordon equation extracted from
physics papers [2], [22], [7], [21], [6], [12], [11], [4], [20].
Towards Formula Concept Discovery and Recognition 3
2 Related Work
Mathematical Information Retrieval (MathIR) addresses the information need
in STEM fields by retrieving, processing and analyzing mathematical formulae.
Up until now, various formula search engines have been developed, and transla-
tions between different markups (LaTeX, Presentation, and Content MathML)
and standards elaborated [5]. Since Wikipedia is only semi-structured, Wikidata4
was launched to provide direct access to specific interlingual facts (RDF5 triples)
and retrieve information systematically. Wikidata is a free and open semantic
knowledge-base that can be read and edited by humans and machines [23]. Wiki-
data stores items with statements and their references. In the case of mathemat-
ical knowledge, this includes formulae, e.g., pressure (Q39552) with a defining
formula property (P2534) p = FS . To scalably seed information into Wikidata, a
Primary Sources tool6 was introduced, allowing active users to quickly browse
through new claims and their references to approve or reject them. The arXiv.org
e-Print server [10] makes available free preprints for a large collection of publica-
tions from Physics, Mathematics, Computer Science, Economics and more. Many
authors provide their LaTeX source code. Both Wikipedia and arXiv articles
were extracted as part of the NTCIR MathIR Task [1]. In 2017, the Special In-
terest group for Math Linguistics (SIGMathLing)7 was initiated as a forum and
resource cooperative for the linguistics of mathematical/technical documents.
For Mathematical Language Processing (MLP), the formula parts (operators,
identifiers, numbers) have to be annotated using the Mathematical Markup Lan-
guage (MathML). There are several tools available, most prominently the La-
TeXML converter8 . Furthermore, the occurring symbols (variables, constants)
need to be disambiguated, i.e., their meaning inferred from the context and se-
mantically annotated. There have been attempts to automatically retrieve the
semantics of identifiers from the surrounding text [18]. While Wikipedia articles
more commonly contain variable definitions in the text, in general, many paper
articles often omit them. This leaves the task of manual annotation inevitable
for building machine-interpretable datasets. The NIST Digital Repository of
Mathematical Formulae (DRMF) [3] and NIST Digital Library of Mathematical
Functions (DLMF) [9] are two examples of maintained high-quality semantic
datasets. At this moment, Wikidata contains approximately 3600 items with
a "defining formula" property. Moreover, the benchmark MathMLben [17] was
created to evaluate tools for mathematical format conversion (from LaTeX to
MathML to Computer Algebra Systems), containing approximately 300 formulae
from Wikipedia, the arXiV and the DLMF, which were augmented by Wikidata
macros [16].
4
http://www.wikidata.org
5
https://www.w3.org/RDF
6
https://www.wikidata.org/wiki/Wikidata:Primary_sources_tool
7
https://sigmathling.kwarc.info
8
https://dlmf.nist.gov/LaTeXML
4 Philipp Scharpf, Moritz Schubotz, Howard S. Cohl, and Bela Gipp
3 Formula Concept Retrieval Challenge
We define as the goal to be eventually able to map all of the various representa-
tions of a formula to a unique and open concept ID, e.g., linking all occurrences
of the Klein-Gordon equation shown in Figure 1 to the Wikidata item Q868967 9 .
We define two subtasks of the Formula Concept Retrieval challenge:
– Formula Concept Discovery (FCD) as a method to find common equivalent
representations and a name candidate for a given formula, and
– Formula Concept Recognition (FCR) as the approach to recognize formulae
in documents as being instances of prior defined formula concept.
4 Our Approach
In the following, we present our first efforts to implement and evaluate a Formula
Concept Discovery (FCD). We approach FCD by retrieving equivalent formu-
lations with different representations (see Figure 2) as well as name candidates
from the surrounding text. The initial step is to identify formula candidates
which occur most often within a given dataset, assuming that they are potential
seeds of popular formula concepts. We first tried formula clustering but discov-
ered that it was not a suitable method for FCD since the number of clusters
is a priori unclear and the tested algorithms were not able to group equivalent
formulae. Subsequently, we decided to start with a ranking of formula duplicates
(with the same LaTeX string), which yielded reasonable results. We employed
the NTCIR arXiv dataset [1] which is comprised of 104062 document sections
containing over 60 million formulae. We confined our computations to the sub-
ject class of astrophysics (680 astro-ph documents), employing a domain expert
to semantically evaluate the results. From the duplicate ranking, we selected a
formula length range between 10 and 30 characters and restricted our selection
to duplicates occurring in at least two different documents. This yielded 3495
formulae. We then manually selected all equations, and discarded all stubs with-
out a right-hand-side, as well as simple variable dependence definitions, such as
x = x (t) and x = y or x = const. For the first 50 samples from the duplicate
ranking, we retrieved the operators and identifiers from the provided MathML
and tags, as well as the surrounding text (words within a window
of ±500 characters around the formula). We encoded both tags using the Tfid-
fVectorizer from the Python package Scikit-learn [13] and Doc2Vec model [8]
from the Python package Gensim [14]. We then compared the performance of a
k-nearest neighbor classifier (Scikit-learn) on the four resulting vector encodings
(math2vec [24] and math tf-idf for the formulae, semantics2vec and semantics
tf-idf for the surrounding text) to retrieve equivalent representations.
9
https://www.wikidata.org/wiki/Q868967
Towards Formula Concept Discovery and Recognition 5
5 Our Results
Table 1 shows the results of our approach for discovering Formula Concepts.
We rank the fetched formulae by the number of duplicates d and also list the
number of documents dˆ they appear in. The main investigation was to compare
the performance of four different encodings in terms of the retrieved number of
equivalent representations using the kNN recommendation algorithm provided
by Scikit-learn. Calculating the overall success distribution, we discovered that
the math2vec (em ) encoding clearly outperforms the others by yielding 71% of the
retrieved instances, followed by semantics tf-idf (ês ) with 15%, semantics2vec
(es ) with 11%, and math tf-idf (êm ) with 4%. On average, there were 3 matches
per formula from 3 different documents. Overall, for 34/50 = 68% of the sample
formulae, we could retrieve equivalent representations. Finally, we listed the five
top name candidates from the surrounding text and evaluated whether they
contain a suitable name for the Formula Concept to be seeded as a Wikidata
item. For our 50 examples, we achieve a recall of 36/50 = 72% for the formula
name. Furthermore, for 41/50 = 82% of the retrieved name candidates, there
was a Wikidata QID available to tag the formula concept.
6 Future Work
Having launched FCD as a method for tagging formulae with Wikidata QIDs, we
can now employ FCR to identify formulae within STEM documents using their
constituting parts (operators and identifiers) in a SPARQL query10 . However,
since at the moment only less than 4000 formulae are seeded into Wikidata [19]
and storing multiple representations as "defining formula" of the same formula
concept item is not endorsed, we argue for the creation of a specific Wikidata-
attached Formula Concept Database. It should include formalized augmentation
to generate equivalent forms using, e.g., commutations, additional sub- and su-
perscripts, unit and reference frame variations, etc. Most importantly, a method
for inferring substitutions or implicit terms needs to be developed.
Hubble’s law (Q179916) equation of state (Q214967)
p = ωρ ȧ = aH
p = κρ Hi = Ṙ/R
ω = p/ρ H = ȧ/a
pd = ωρd H(t) = ȧ/a
Fig. 2: Clustering equivalent representations of formulae in the semantic space
as named Formula Concept Wikidata items.
This work was supported by the German Research Foundation (DFG grant
GI-1259-1).
10
W3C Recommendation: https://www.w3.org/TR/rdf-sparql-query
6 Philipp Scharpf, Moritz Schubotz, Howard S. Cohl, and Bela Gipp
Table 1: Formula Concept Discovery (FCD). Top-50 results of a cross-document
duplicate search in the subject class astro-ph of the NTCIR arXiv dataset.
Equivalent formulae are retrieved to bundle concept candidates using a k-nearest
neighbor (kNN) recommendation, while comparing the relative success s of dif-
ferent encodings (math2vec: em , math tf-idf : êm , semantics2vec: es , semantics
tf-idf : ês ). The number of duplicates d and originating distinct documents dˆ
are shown as well as a retrieved sample formula. Furthermore, it is evaluated
whether the first five words of the surrounding text are candidates for the name
of the formula and whether a Wikidata QID is available.
# Formula Name (QID) d / dˆ sem , sêm , ses , sês Encoding: sample formula Name candidates from surrounding text
1 H = ȧ/a hubble parameter (Q179916) 32 / 32 0.0, 0.1, 0.0, 0.9 ês : Hi = Ṙ/R hubble, parameter, time, factor, equations
2 p = ωρ equation of state (Q214967) 6/5 0.3, 0.0, 0.1, 0.6 es : pd = wρd equation, state, quintessence, expansion, pressure
3 ω = p/ρ accelerating universe (Q1049613) 4/3 0.7, 0.0, 0.0, 0.3 em : p = ωρ universe, accelerating, indefinitely, strain, values
4 p = −A/ρα dark fluid (Q5223514) 4/4 0.7, 0.0, 0.3, 0.0 em : p = − ρAα chaplygin, gas, dark, generalized, fluid
5 pd = wρd dark energy (Q18343) 4/3 0.3, 0.0, 0.3, 0.3 es : pX = ωX ρX energy, dark, equation, represent, pressure
6 H = ȧ/a N/A (Q179916) 4/4 0.4, 0.1, 0.2, 0.3 êm : H = a0 /a scale, factor, usual, equation, state
7 k = |k| wavenumber (Q192510) 3/3 0.8, 0.0, 0.2, 0.0 em : k = |k| oscillatory, behavior, depend, time, wavenumber
8 f = e−φ R N/A (N/A) 3/2 1.0, 0.0, 0.0, 0.0 em : f (φ) = e−φ R string, lowenergy, effective, action, theory
9 p = κρ equation of state (Q214967) 3/2 0.3, 0.0, 0.7, 0.0 es : pD = w(z)ρD equation, state, ary, patch, exceeds
10 w = pX /ρX equation of state (Q214967) 3/3 0.6, 0.0, 0.1, 0.3 em : pX = wX ρX equation, state, dark, energy, wmap
11 µ = mp /me proton-to-electron mass ratio (Q2912520) 3 / 3 1.0, 0.0, 0.0, 0.0 em : mi = µmp ratio, proton, electron, masses, technique
12 φc = M/g critical value (Q2189464) 3/3 0.0, 0.0, 0.0, 0.0 N/A field, critical, value, takes
13 p = − ρAα chaplygin gas (Q5073250) 3/3 0.8, 0.0, 0.0, 0.2 em : p = −Aρ−α state, generalized, chaplygin, gas, equation
14 p = αρ polytropic gas (Q831024) 3/2 0.7, 0.0, 0.2, 0.2 ês : wα = pα /ρα constant, gas, cosmological, matter, polytropic
15 M = M
e /Γ connected manifold (Q2721559) 3/3 0.0, 0.0, 0.0, 0.0 N/A multiply, connected, equally, quotient, manifolds
16 g(a) = 4(a)/a dark energy (Q18343) 3/2 1.0, 0.0, 0.0, 0.0 em : g(a) = ∆(a)/a models, dark, energy, growth, history
17 α = dns /d ln k N/A (Q192510) 3/3 1.0, 0.0, 0.0, 0.0 em : dns /d ln k = αs introduced, customary, notation, comoving, wavenumber
18 ψ = −iθ N/A (N/A) 3/2 0.0, 0.0, 0.0, 0.0 N/A real imaginary universe
R
19 dt = a(η)dη N/A (Q11471) 2/2 0.5, 0.0, 0.3, 0.3 ês : t =a(η)dη time, related, cosmic, relation, overdot
√ √
20 ∆xmin = β lower bound (Q21067468) 2/2 1.0, 0.0, 0.0, 0.0 em : ∆xmin = h̄ β positive, constant, lower, bound, implies, dimensional
i i
21 k = ap modes (N/A) 2/2 0.0, 0.0, 0.0, 0.0 N/A modes, comoving, obtained, scaling, coincide
22 ϕ = δAµ perturbations (Q911364) 2/2 0.0, 0.0, 0.0, 0.0 N/A note, valid, perturbations, gauge, theories
23 hab = gab − na nb metric (Q865746) 2/2 0.0, 0.0, 0.0, 0.0 N/A bulk, scalar, curvature, induced, metric
24 K = Kab hab brane (Q385601) 2/2 1.0, 0.0, 0.0, 0.0 em : K = Kαβ hαβ vector, field, unit, normal, brane
p p
25 v = |dp/dρ| equation of state (Q214967) 2/2 1.0, 0.0, 0.0, 0.0 em : vc = dpc /dρc equation, state, suggests, effective, velocity
√
26 Q = GM limit (Q246639) 2/2 0.0, 0.0, 0.0, 0.0 N/A limit, rhoades, value, write
27 ζ = Hδφ/φ̇ N/A (Q10886678) 2/2 1.0, 0.0, 0.0, 0.0 em : R = (H/φ̇)δφψ curvature, perturbation, uniform, density, valid
√
28 mγ = e/ π photon mass (Q3198) 2/2 0.0, 0.0, 0.0, 0.0 N/A photon, mass, gauge, mechanism, schwinger
R
29 dη = dt/a(t) conformal time (Q2482717) 2/2 0.6, 0.0, 0.1, 0.3 ês : t = a(η)dη conformal, time, ase, figure, fig
30 Tg = Ho tg N/A (Q126818) 2/2 0.0, 0.0, 0.0, 0.0 N/A dimensionless, factor, eq, extragalactic, object
31 H = a0 /a N/A (Q179916) 2/2 0.7, 0.0, 0.1, 0.2 ês : H = ȧ/a conformal, time, background, scale, factor
32 θ = A exp(−ζt) exponential decrease (Q574576) 2/2 0.0, 1.0, 0.0, 0.0 êm : ψ(t, r) = ψ(r) exp(−iωt) decreases, exponentially, slowly
33 pi = ωi ρi N/A (N/A) 2/2 0.7, 0.0, 0.1, 0.1 es : wX = pX /ρX case, expected, current, observations, restrict
34 i∂t Φ = HΦ schrödinger evolution (Q165498) 2/2 0.0, 0.0, 0.0, 0.0 N/A evolution, shrödinger
35 H(t) = ȧ/a N/A (Q179916) 2/2 0.8, 0.1, 0.0, 0.1 em : ȧ = aH data, scale, function, combined, sn
36 pΛ = −ρΛ dark energy (Q18343) 2/2 1.0, 0.0, 0.0, 0.0 em : pD = −ρD dark, contributions, matter, energy, matterdominated
37 PM = wρM equation of state (Q214967) 2/2 0.6, 0.0, 0.3, 0.1 es : px = wρx pressure, write, related, equation, state
38 fν = ρν /ρd neutrino (Q2126) 2/2 0.0, 0.0, 0.0, 0.0 N/A matter, neutrino
39 At = rAs fluctuation (Q5462624) 2/2 0.0, 0.0, 0.0, 0.0 N/A fluctuation
40 pm = γρm nonrelativistic matter (Q55921784) 2/2 1.0, 0.0, 0.0, 0.0 em : γ = p/ρ matter, components, universe, nonrelativistic, ordinary
41 Ωi = ρi /ρc expansion rate (N/A) 2/2 1.0, 0.0, 0.0, 0.0 em : Ω = ρ/ρcrit universe, constant, rate, expansion, variables
n
42 P (k) = Ak inflation (Q273508) 2/2 0.0, 0.0, 0.0, 0.0 N/A fluctuations, field, inflation, universe, inflationary
43 LI = M (τ )φ[x(τ )] N/A (N/A) 2/2 0.0, 0.0, 0.0, 0.0 N/A idea, quantitative, viewpoint, arises, study
ab
44 L = κhab T N/A (N/A) 2/2 0.0, 0.0, 0.0, 0.0 N/A standard coupling
45 wi = Pi /ρi equation of state (Q214967) 2/2 0.7, 0.0, 0.2, 0.1 ês : wα = pα /ρα relative, contributions, components, equations, state
46 M̄ = B/C N/A (N/A) 2/2 0.3, 0.0, 0.3, 0.3 es : M̄ = B
C
minimum
47 Ψ = Ψ` + Ψs N/A (N/A) 2/2 0.0, 0.0, 0.0, 0.0 N/A split, dropped, note, long, short
48 z = aφ̇/H equation (Q21086835) 2/2 0.7, 0.0, 0.0, 0.3 ês : zq = aφ̇/H quantity, equation
49 uµ = dxµ /dτ comoving fluid (Q5462744) 2/2 1.0, 0.0, 0.0, 0.0 em : kµ = dxµ /dv cosmological, fundamental, observer, comoving, fluid
50 φ̇ = −Wφ firstorder differential equation (Q11214) 2/2 1.0, 0.0, 0.0, 0.0 em : χ̇ = −Wχ equation, firstorder, differential, scale, factor
Towards Formula Concept Discovery and Recognition 7
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