The Numerate Web: Mathematical Formulas and Computations on the Web of Data

SWJ Review

In this article, the author presents the Numerate Web, an approach that leverages and extends earlier work to advance the support for the representation of mathematical models in RDF. This work has a significant potential impact, is well-motivated, and is supported through the demonstration of examples. The syntax and incorporated shorthand notations for incorporating mathematical equations are well explained and several algorithms for calculation execution are provided. Nevertheless, despite the numerous strengths of this article, the major shortcoming is the lack of a rigorous quantitative evaluation of the approach. Instead, how this work can be leveraged in the context of two case studies is provided. Additionally, the mathematics in the examples included were relatively straightforward. Could this approach be used for calculus or solving differential equations? There is a mention regarding the incorporation of time-varying behavior as future work, but the discussion on the limitation of this approach should be extended. In terms of mathematics, it should be made very clear what this approach can and cannot do.

Listed below are many of the grammatical issues found within the article. Several issues were likely missed, so it is highly recommended that the author addresses the following and also carefully proofreads the article afterward. For example, I didn't comment on the use of Oxford commas, but you mostly use them but in some places do not. Whether or not to use Oxford commas is debatable, but whatever you decide, it should be consistent throughout the paper.

Section 1
Page 1
Line 42-43 - Single sentence paragraph, should be combined with the following paragraph.
Line 48-49 - Single sentence paragraph, should be combined with the previous paragraph.
Line 49 - footnote should go after the punctuation: "...that both have RDF serializations^1." -> "...that both have RDF serializations.^1"
Page 2
Line 12-14 - Single sentence paragraph, should be combined with the following paragraph or the thought should be expanded upon.
Line 37-38 - Single sentence paragraph, should be combined with the previous paragraph.
Line 39-40 - Single sentence paragraph, should be combined with the following paragraph, which is also a single sentence paragraph.
Section 2
Line 50 - Missing comma: "In 2003 Marchiori..." -> "In 2003, Marchiori..."
Page 3
Line 22 - Missing comma: "In 2011 Lange..." -> "In 2011, Lange..."
Line 25-26 - phrasing and missing comma: "Additional to OMDoc the work introduces..." -> "In addition to OMDoc, the work introduces..."
Line 29 - Missing comma: "In 2012 Ferre..." -> "In 2012, Ferre..."
Line 45-46 - Unnecessary comma: "For example, constants, and variables are only..." -> "For example, constants and variables are only..."
Line 49 - Missing comma: "In 2014 Munoz..." -> "In 2014, Munoz..."
Section 3
Page 4
Line 15-16 - Single sentence paragraph, should be combined with the following paragraph.
Line 45-46 - Single sentence paragraph, should be combined with the previous paragraph.

As noted for these first 4 pages, many single-sentence paragraphs are included and continue to be included in the remainder of the paper. The use of single-sentence paragraphs is not technically grammatically incorrect. It can serve a stylistic purpose typically for emphasis in story-telling, but that is not the case here so we recommend that such occurrences should be corrected. The remainder of this review will not continue to include comments for single-sentence paragraphs, but that is not because they went unnoticed. We leave it to the authors to remedy this issue.

Section 4
Page 6
Line 25 - Figure 5 caption, typo and missing article: "Example for representig a gear motor as RDF model" -> "Example for representing a gear motor as an RDF model"

Section 5
Line 45 - missing comma and article: "As mentioned in Section 1 these objects may be represented using Content MathML as markup language." -> "As mentioned in Section 1, these objects may be represented using Content MathML as a markup language."
Page 7
Line 16 - missing comma: "Therefore an OWL ontology for OpenMath..." -> "Therefore, an OWL ontology for OpenMath..."
Page 8
Line 46 - footnote should go after the punctuation: "...within the POPCORN definition^2." -> "...within the POPCORN definition.^2"

Section 6
Page 9
Line 40 - missing comma: "Analogous to connecting programming languages to SPARQL endpoints via APIs a hypothetical Content" -> "Analogous to connecting programming languages to SPARQL endpoints via APIs, a hypothetical Content"
Page 10
Line 14 - missing comma: "In [30] we already proposed..." -> "In [30], we already proposed..."
Line 16 - footnote should go after the punctuation: "...is reviewed and available on the OpenMath website^3." -> "...is reviewed and available on the OpenMath website.^3"
Line 42 - missing comma: "With rdf:resource and rdf:resourceset it is possible to select..." -> "With rdf:resource and rdf:resourceset, it is possible to select..."
Line 43 - missing comma: "However, for traversing the edges further operators are necessary." -> "However, for traversing the edges further, operators are necessary."
Line 43-44 - phrasing can be improved and it is not clear what is meant here. Why does it say "with one" when it seems from the examples that both operators expect multiple values? It should be clarified that "one and multiple" is referring to the output of the functions rather than the input: "For this purpose, two additional operators for RDF properties with one and multiple values are defined: rdf:value and rdf:valueset." -> For this purpose, two additional operators for RDF properties with the ability to return a single value or multiple values, respectively, are defined: rdf:value and rdf:valueset."
Page 11
Line 7 - missing comma: "Complementary to the operator rdf:value the operator rdf:valueset is able..." -> "Complementary to the operator rdf:value, the operator rdf:valueset is able..."
Line 41 - the quotes don't match up: 'A literal with the content "‘This is an English text."’ and the language label "‘en"’ is representable...' -> 'A literal with the content "‘This is an English text.’" and the language label "‘en’" is representable...'
Line 48 - footnote should go after the punctuation: "...and reduce the amount of data required for encoding^4." -> "...and reduce the amount of data required for encoding.^4"
Page 12
Line 1 - missing comma: "For the RDF operators defined in the previous sections short forms for URIs are not necessary for the functionality." -> "For the RDF operators defined in the previous sections, short forms for URIs are not necessary for the functionality."
Line 3 - typo: "...to assign parts of of URIs to..." -> "...to assign parts of URIs to..."
Line 4-5 - incompletes sentence: "In this case, the prefixes...ontology about persons." -> "In this case, the prefixes...ontology about persons are used."
Line 5 - typo and phrasing: "As can be can be seen,..." -> "As shown,..."
Line 17 - missing comma: "In order to support prefix declarations in OpenMath semantic attributions could be used, comparable to..." -> "In order to support prefix declarations in OpenMath, semantic attributions could be used, comparable to..."
Line 25-26 - redundancy: "It is possible to overwrite a prefix within a child object is possible." -> "It is possible to overwrite a prefix within a child object."
Line 35 - tense agreement: "...the inheritance of the prefixes to child objects itself." -> "...the inheritance of the prefixes to child objects themselves."
Line 45 - spelling: "...elements fulfil a certain..." -> "...elements fulfill a certain..."
Page 13
Line 1 - missing word: "...the example shown the efficiency..." -> "...the example shown of the efficiency..."
Line 2 - typo: "...has to be loaded from the from the RDF database." -> "...has to be loaded from the RDF database."
Line 3 - missing comma: "If the filter condition could be pushed down to the database then this would allow..." -> "If the filter condition could be pushed down to the database, then this would allow..."
Line 35-36 - missing comma and unnecessary comma: "Therefore it can be checked for consistency by OWL reasoners, and it can be..." -> "Therefore, it can be checked for consistency by OWL reasoners and it can be..."
Line 41 - incorrect pluralization: "In order to improve the usability of mathematical expressions input and output when..." -> "In order to improve the usability of mathematical expression inputs and outputs when..."
Section 7
Page 14
Line 33 - typo: "...their linkage with with RDF resources..." -> "...their linkage with RDF resources..."
Line 33 - missing comma: "On this basis the creation..." -> "On this basis, the creation..."
Page 17
Line 26 - unnecessary article: "The Algorithm 1..." -> "Algorithm 1..."
Page 18
Line 44 - unnecessary article: "The algorithm 2..." -> "Algorithm 2..."
Page 19
Line 20-21 - unnecessary article: "...(line 12 of the Algorithm 2)." -> "...(line 12 of Algorithm 2)."
Page 20
Line 1-2 - unnecessary article: "To support this, the algorithms 1 and 3 must be adapted..." -> "To support this, Algorithms 1 and 3 must be adapted..."
Line 4 - phrasing: "An example depicts Figure 7, which shows..." -> "An example is depicted in Figure 7, which shows..."
Line 25 - footnote goes after the punctuation: "...Ontology^9 (MUO)." -> "...Ontology (MUO).^9"
Line 29-30 - phrasing: "...with QUDT contains [56, pp. 294]." -> "...with QUDT is contained in [56, pp. 294]."
Line 42 - unnecessary article: "...into the algorithm 3..." -> "...into Algorithm 3..."
Line 46 - footnote goes after the punctuation: "An example is shown in Listing 13^11, where..." -> "An example is shown in Listing 13,^11 where..."
Page 21
Line 40 - missing comma: "For this purpose the conversion..." -> "For this purpose, the conversion..."
Line 42 - missing commas: "For the given example therefore the conversion..." -> "For the given example, therefore, the conversion..."
Page 22
Line 8 - missing comma: "...via OWL restrictions as shown in Listing 14." -> "...via OWL restrictions, as shown in Listing 14."
Section 8
Line 29 - missing commas: "The first case study OpenMath Content Dictionaries (Section 8.2) investigates..." -> "The first case study, OpenMath Content Dictionaries (Section 8.2), investigates..."
Line 33 - missing commas: "The second case study process chain planning and evaluation (Section 8.3) investigates..." -> "The second case study, process chain planning and evaluation (Section 8.3), investigates..."
Line 39 - typo: "...described insection 8.1 was..." -> "...described in Section 8.1 was..."
Line 49 - footnote goes after the punctuation: "...representation of mathematical objects and the execution of calculations^12." -> "...representation of mathematical objects and the execution of calculations.^12"
Page 23
Line 37 - redundancy: "For example, the KOMMA ontology editor, for example, supports textual..." -> "For example, the KOMMA ontology editor supports textual..."
Line 42 - capitalization of proper noun: "As already described in section 5, OpenMath..." -> "As already described in Section 5, OpenMath..."
Page 24
Line 4 - footnote goes after the punctuation: "...platform eniLINK^14, an extension..." -> "...platform eniLINK,^14 an extension..."
Page 26
Line 2 - typo: "...sums or products in in any..." -> "...sums or products in any..."
Line 5 - footnote goes after the punctuation: "...SPARQL query^19." -> "...SPARQL query.^19"
Line 44 - typo: "...calculations wer developed with..." -> "...calculations were developed with..."
Page 31
Line 46 - footnote goes after the punctuation: "...into the Schema.org vocabulary^21." -> "...into the Schema.org vocabulary.^21"
Line 46-47 - phrasing: "An example of the use of the GoodRelations ontology for the domain mountain sports equipment gives [67]." -> "An example of the use of the GoodRelations ontology for the domain mountain sports equipment is given in [67]."
Page 32
Line 36 - unnecessary comma: "...the integration of external data in mathematical models is possible, if it is available in an RDF..." -> "...the integration of external data in mathematical models is possible if it is available in an RDF..."
Line 41 - capitalization: "...in section 8.1 extended..." -> "...in Section 8.1 extended..."
Line 47 - unnecessary article: "The figure 18..." -> "Figure 18..."
Page 34
Line 18 - phrasing: "Questions are here the embedding..." -> "Questions include the embedding..."

Abstract section need to discuss more about the proposed work. The major findings should be discussed with the significance of Numerate Web.
The major contribution of this paper: proposed an approach for structured semantic models of systems as an ontology for mathematical expressions and design of a textual syntax and methods for accessing RDF data within mathematical formulas. Finally, the proposed work has been evaluated. Introduction section should be more descriptive with the proposed work and its findings.
Authors have discussed the limitations in the related work section but it should be concisely discussed the need to propose a new web as Numerate Web. Community and readers should be satisfied with the proposal. It is also noticed that related work section only discussed till 2014 papers so is there no any research has been done after 2014 in this area or authors missing to include please check it carefully.
Numerate Web Applications has included as a new layer of Semantic Web Layer Cake, it is challenging as there are already several mathematical ontologies and vocabularies are available so is it really needed to include a new layer. Authors should justify this.
Algorithms are organized well and the proposed approach can provide an essential basis for future applications in digital system models and mathematical knowledge management.
There are some typo mistakes that need to handle carefully such as, “Fig. 5. Example for representig representing a gear motor as RDF model”

The author presented a formal approach aiming to define mathematical models using RDF-based techniques and overcome limitations of current semantic-based languages (e.g., SWRL and SPARQL). The proposal also includes a specific rule language and a textual syntax (named POPCORN-LD) for modeling mathematical objects and properties following the linked data guidelines. Two simple case studies are evaluated using a KOMMA-based application framework extending the basic software with additional functionalities.

The proposed approach is interesting and coherent with Semantic Web journal aims. The idea is quite original but it is difficult to understand the real benefits of the work with respect to state of the art frameworks in real-world scenarios. This aspect should be emphasized in the case study section. The manuscript is well written and easy to follow. The background section provides a brief but satisfactory overview of the domain and related work. The software is publicly available on GitHub and the user interface is minimalist and simple to use. Source code is well organized and the README file details all steps required to run the software.

General remarks:
- Section 2, include a comparison table to summarize and highlight the main features of all cited works. I also suggest to identify, if existing, further recent approaches proposed in the last 2 years;
- Section 7.6 is very interesting and should be extended with more details and examples about inheritance and overwriting of mathematical rules;
- Section 8.1, KOMMA is a very useful framework for this work but a possible integration in Protégé should be taken into consideration to facilitate the usage of the system in the Semantic Web community;
- is it possible to use the proposed system without KOMMA? Are there any specific APIs? The case study described is Section 8.3 can be easily extended to Industry 4.0 scenarios for data management/processing. An implementation running on embedded platforms could be very useful;
- a performance evaluation section is suggested to compare the proposed approach with existing works also in terms of processing time.

Minor remarks and suggestions:
- Section 1, introduction should end with some details about all sections of the paper. Move here the paragraph reported in Section 3, p.5, lines 18-29;
- Sections 4-5-6-7 can be organized as subsections of Section 3, representing the fundamental elements of the whole Numerate Web vision;
- Section 8.4 is very concise. Aggregate with Section 9;
- p.12, line 3, "of of" --> "of";
- p.25, figure 10 is not cited in the text;
- rename the Github repository, "numerateweb-swj-2022" --> "numerateweb";
- include a docker (or vagrant) configuration file to simplify building and running the web application.