This paper describes the results of semantic annotating and semantic search in a mathematical collection. We explored semantic models of contexts of a mathematical formula and applied the new knowledge to improving semantic search. The solutions are based on the application of the previously developed OntoMathPro ontology.
1 Introduction
In the field of professional mathematics, considerable
experience has been accumulated in processing and use
of electronic mathematical content in projects for
developing mathematical electronic libraries. Being
well-structured, mathematical texts are favorably
distinguished by presence of standards and markup
languages, availability of electronic mathematical
libraries with highly developed search services, as well
as of software for the automatic processing of individual
components of mathematical texts. Works devoted to
formalizing the level of representations of mathematical
articles are rather widely spread today. For these
purposes, specialized formal languages for representing
mathematical texts, as well as software for converting
these languages [
LaTEX Search (http://latexsearch.com/), and Wolfram Formula Search (http://functions.wolfram.com/).
The mathematical projects carried out by the world
community paved the way for a new idea - creating the
World Digital Mathematical Library (WDML) [
Theorems, axioms, proofs, mathematical definitions, etc are standard classes of mathematical objects. The semantic relationships between elements are important constituents of the object model. To represent the document’s object model, it is proposed to use modern technologies of the semantic web. This way of representing the management of mathematical knowledge will allow creating management tools directly by objects of mathematical knowledge (means of aggregation, semantic search, search by formulas and identification of identical objects). In this paper, we will consider some solutions to the problem of semantic search by formulas that take into account the semantic context of mathematical formulae.
The paper is organized as follows. In Section 2, we present a semantic search service for math formulas. In Section 3, we discuss basic methods for semantic annotation and search, underlying this service. In Section 4 we discuss how these basic methods can be improved by context models of mathematical formulas. In Section 5 we made concluding remarks. 2 Semantic search of mathematical formulas
In this section, we present OntoMath Formula Search
(https://lobachevskii-dml.ru/mathsearch) [
While there are currently many math formula search engines have been developed at present time, they are mostly syntactical, and allow only simple search by expression patterns. For example, to find formulas, containing “(a + b)2”. Unlike them, our service is semantic, and, therefore, can find formulas with respect to a given math concept, regardless of its symbolic representation. For example, to find formulas that contain variables, expressing open sets. The search index stores journal publications in mathematics.
The user enters keywords, filtering suggestions of the
system. The search interface supports the terminology of
OntoMathPro ontology [
Using logical reasoning with respect to the ontology relations, the service finds formulas that contain not only the given concept, but also subclasses of this concepts in the ontology hierarchy. For example, the search results for polygon query contains formulas with variables, denoting such concepts as triangle, parallelogram, trapezoid, and so on.
OntoMath Formula Search is a part of OntoMath
digital ecosystem, an ecosystem of ontologies, text
analytics tools, and applications for math knowledge
management [
The core component of the OntoMath ecosystem is
its semantic publishing platform [
The semantic publishing platform pipeline consists of several steps, one of which is semantic annotation of mathematical texts. In the next two sections, we describe the current implementation of this steep, based on simple context-agnostic models and discuss how this implementation can be improved by context models. 3 Semantic annotation of mathematical texts
The developed model of semantic annotation of
mathematical texts is based on the professional
mathematics ontology OntoMathPro [
Semantic annotation of mathematical texts is carried
out by the system “OntoIntegrator” [
Annotation is performed for mathematical documents in XML format, where the <Math> tag marks mathematical expressions. Mathematical expressions can be included in a noun phrase, and can also be used as prefixes in constructions with hyphens. At the tokenization stage, structures with the <Math> tag are treated as separate objects. In a mathematical text, there are four types of noun phrases (NP) which are marked as TERM1-TERM4 tags (Table 1).
Tag
Noun phrase
TERM1 prepositions, contains no formulas
TERM3 contains no formulas
The construction of nominal groups is described by rules that take into account the internal structure of the nominal group. The noun phrases allocation method works within the bounds of the sentence. When a noun phrase is annotated, the borders of the head word are highlighted with HeadBegin and HeadEnd attributes, its grammatical form is normalized (reduced to the canonical form), and the normal form of NP appears in the Form attribute. Annotated noun phrases are used in the following stages of processing mathematical texts when processing extended formula contexts.
Among all the noun phrases extracted from the texts on the basis of syntactic models, noun groups containing ontological concepts from the OntoMathPro ontology are selected. Such nominal groups will be denoted as NP (Oi) and PP (Oi), where Oi stands for an element of the ontology.
The distribution of nominal groups for the abovementioned types in the experimental collection of 40 mathematical articles is shown in Figure 2. A total of 11865 registered groups are identified in the collection. 4 Improving search based on the models of the formula’s semantic context
Analysis of the semantic context of the mathematical formula is aimed at improving search results, primarily in finding a formula using a textual description of formula variables. To solve the problem, the following models of the semantic context of the formula were proposed. 1)
Model of structural classification of formulas.
According to this model, all mathematical formulas are divided into two classes: the class of “basic” formulas F1 and the class of “auxiliary” formulas F2. Informally, a “basic” formula is a formula, that express significant relationship between mathematical concepts. This informal notion is formalized as follows: mathematical formulas from F1 class have to be longer than 10 characters and include the basic mathematical operators, as well as binary equality and inequality operators.
The distinction between “basic” and “auxiliary” formulas is crucial in the two other context models (see below). Additionally, this distinction is in use in the semantic search service, where search results are restricted to “basic” formulas.
We have developed a tool for automatic classification and have evaluated it on the collection of mathematical papers (50 items) from the “Russian Mathematics (Iz.
VUZ)” journal. To evaluate this tool, we just compare a count of formulas classified as F1 automatically by this tool, with a count of formulas classified as F1 manually by a human assessor. The result of evaluation for ten papers is presented in Table 2. 2) The expanded syntactic context of the variable formula description, allowing to take into account the compatibility (grouping) of the types of noun groups shown above. Various combinations of noun phrases were studied. The most informative was found to be the construction of the form NP1(O1) + PP2(O2), which makes it possible to form a hypothesis about the connection between the two classes of ontology and determine the stable relation between noun phrases.
If NP and PP are found in the neighborhood of a formula from class F1, and they follow each other directly and contain ontological concepts in their composition, that is, they satisfy the pattern NP12 (O1, O2) = NP1(O1) + PP2(O2), where symbols O1 and O2 are
OntoMathPro ontology concepts, then the compound 69 50 42 69 70 73 81 73 80 62 noun phrase NP12 is allocated. A comparison of text strings was carried out using the Jaccard measure (the threshold value was set to 0.7).
The selected pairs of O1 and O2 concepts will undergo an expert analysis to confirm the existence of a significant link between these concepts. These are examples of pairs found by the NP + PP model associated with ontological concepts (Table 3). множество точек пересечения прямых x5 = x567 the set of points of intersection of lines x5 = x567
We evaluated the experiments on extracting expanded syntactic contexts of variable formulas which were performed using the collection of 50 papers under analysis.
To compare the labels of ontological concepts and the textual description of formula variables, we used the Jaccard metric in the form k = (k1 + k2) / 2, where с нормой with the norm с границей γ with the bound γ в представлении U(ξ) = U.(ξ) + U*(ξ), ξ ∈ D, ; in the view U(ξ) = U.(ξ) + U*(ξ), ξ ∈ D, ; symbols k1 and k2 are the Jaccard coefficients obtained by comparing pairs (NP, O1), (PP, O2). The results are represented in Table 4. The analysis of the results allows us to conclude that the selected pairs demonstrate stable links between concepts in the ontology, which thus makes it possible to improve the OntoMathPro ontology. (Кокасательное пространство F1 + к (кокасательное проективному пространству F2) / The пространство F1 + проективное 1 cotangent space F1 + to the projective пространство F2) / cotangent space F2 space F1 + projective space F2 (базис F1 + с условием F2) / basis F1 + with condition F2 (базис F1 + условие F2) / 1 basis F1 + condition F2 (оператор из F1 + в подпространство про(сотпреарнасттовро F2) / opFer1ator F1 ++ 1 F2) / operator from F1 + to a subspace F2 space F2
(предел F1 + в норме F2) / the limit F1 + in the norm F2 (предел F1 + норма F2) / the 1 limit F1 + norm F2 (Векторное произведение F1 + в новом базисе F2) / A vector product F1 + in a new basis F2
(расстояние от точки F1 + до множества F2) / distance F1 + from point to set F2 (векторное произведение F1 p+rгoоdлuоcнt омFн1ы+й бinазиaс Fh2o)l/ovneocmtoicr 1 basis F2
(расстояние F1 + множество F2) / distance F1 + set F2 (Оператору F1 + на сетке F2) / To the operator F1 + on the net F2 (оператор F1 + сеть F2) / operator F1 + net F2 0,6 0,88 0,88 0,6 0,74 0,74
Models of stable context patterns. We automatically extracted stable context patterns containing mathematical formulas from the collection of mathematical papers, and then manually selected patterns with fixed semantics among them. Examples of such patterns are mathematical concept definitions, formula descriptions, or patterns with auxiliary words such as “в виде” (“in the form of”). Some of these patterns are presented in Table 5.
This approach was inspired by corpus-driven and
empirically well-founded methodologies, such as Corpus
Pattern Analysis (CPA) [
We have implemented a customized semantic annotation tool for the pattern with auxiliary words “в виде” (“in the form of”), and we are going to implement such customized tools for the other frequent patterns. Множество точек пересечения прямых : = :;<
lines x5 = x567 Производная (/')@() ≡ ,@() в представлении U() = A() + @(), ∈ , Derivative (/')@() ≡ ,@() in the view U(ξ) = U.(ξ) + U*(ξ), ξ ∈ D, C = ∩ C' – множество внутренних узлов. C = ∩ C' is set of internal nodes Метод механических квадратур линейных интегродифференциальных уравнений вида + , H ℎH , + , H ℎ, , K = , 0 ≤ ≤ 1, где () и ℎH(, ), ℎ, (, ) – известные функции.
quadratures of linear integrodifferential equations + , H ℎH , + H, ℎ, , K = , 0 ≤ ≤ 1, where () и ℎH(, ), ℎ, (, ) are known.
This paper describes the results of semantic annotation and semantic search in a mathematical collection, as well as new solutions for improving semantic search related to building semantic models of contexts of a mathematical formula. These solutions are based on the OntoMathPro ontology and allow enriching the ontology with new concept links.
These studies are in line with the project of developing the World Digital Mathematical Library, the purpose of which is to unite digitized versions of the entire body of mathematical scientific literature in a distributed system of interlinked repositories.
The developed technologies have been tested on digital math collections of Kazan Federal University. Acknowledgments. A. Kirillovich would like to thank Sergio Pinillos Prado and Mikhail Malakhaltsev from the University of Los Andes, as well as Zaira Alexandra Gil Vargas for their assistance. This work was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, grant agreement no. 1.2368.2017, and Russian Foundation for Basic Research and the government of the region of the Russian Federation, grant № 18-47-160007.