Corneli, Joseph

Schubotz, M.

English

It has been argued that the right representation is the key to successful reasoning [Bun13, p. 16]. We would broaden this observation to point out that while both representation and reasoning are crucial, they are not in themselves sufficient for most applied cognitive and computational workflows. In particular, the processes by which representations are created are logically prior. With large-scale projects, human and HCI factors must be considered, and suitable management strategies devised. Below, we consider the representation and workflow issues surrounding the creation of a large machine-readable mathematics knowledge base. Such a knowledge base would differ from a repository of human-readable texts, in that it would support a range of automated reasoning tasks. For example, one such task is automated tutoring. An ongoing master’s thesis [Dud16] on a related simpler task, automated question answering, has helped to motivate the current inquiry.<br/

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     Edinburgh Research Explorer                                      math.wikipedia.org: A vision for a collaborative, semi-formal,language independent math(s) encyclopediaCitation for published version:Corneli, J & Schubotz, M 2017, math.wikipedia.org: A vision for a collaborative, semi-formal, languageindependent math(s) encyclopedia. in 2nd Conference on Artificial Intelligence and Theorem Proving.Obergurgl, Austria, 2nd Conference on Artificial Intelligence and Theorem Proving, Obergurgl, Austria,26/03/17. <http://aitp-conference.org/2017/aitp17-proceedings.pdf>Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:2nd Conference on Artificial Intelligence and Theorem ProvingGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 22. Dec. 2021math.wikipedia.org: A vision for a collaborative,semi-formal, language independent math(s) encyclopediaJ. Corneli and M. Schubotz1 Goldsmiths, University of London, London, UKj.corneli@gold.ac.uk2 Universität Konstanz, Konstanz, Germanymoritz.schubotz@uni.knIntroductionIt has been argued that the right representation is the key to successful reasoning [Bun13, p. 16].We would broaden this observation to point out that while both representation and reasoningare crucial, they are not in themselves sufficient for most applied cognitive and computationalworkflows. In particular, the processes by which representations are created are logically prior.With large-scale projects, human and HCI factors must be considered, and suitable managementstrategies devised. Below, we consider the representation and workflow issues surrounding thecreation of a large machine-readable mathematics knowledge base. Such a knowledge basewould differ from a repository of human-readable texts, in that it would support a range ofautomated reasoning tasks. For example, one such task is automated tutoring. An ongoingmaster’s thesis [Dud16] on a related simpler task, automated question answering, has helpedto motivate the current inquiry.1 What has been done?As far as human-readable mathematics goes, one of the most successful and visible projectsis Wikipedia, the world’s largest encyclopedia. The Mathematics WikiProject indexes 31444articles at the time of writing.1 For some sub-fields of STEM, Wikipedia also contains asignificant amount of related domain knowledge. Until recently, the main – and still mostfamiliar – way of representing this knowledge was in encyclopedic human readable articles indifferent languages, which were to some degree linked to each other. With the implementationof Wikidata this has begun to change. Now, there is a centralized database that has uniqueIDs for each semantic concept. From those concepts, there are links to the individual languageversions of the Wikipedia articles. In addition, Wikidata contains links between concepts, andproperties of the items of various data types, including mathematical formulae [SS16]. In thisway, Wikidata forms the backbone of a knowledge graph. The project, then, exists somewherebetween two domains: digital libraries and artificial intelligence.2 Our plan for the futureWe propose to rely on Wikidata, and build “math.wikipedia.org” as a frontend to that. Thisservice will allow users to conveniently store and retrieve formulas, terms, and other mathe-matical objects. Building a knowledge base with machine-readable formulas and links between1https://en.wikipedia.org/wiki/Wikipedia:WikiProject_Mathematics/Countmath.wikipedia.org Corneli and Schubotzconcepts will produce a richer domain for reasoning than we get from existing projects or othermathematical digital library proposals that lack semantic models.Working with an existing popular platform should help bootstrap the social side, and al-low us to focus our development effort on domain-specific issues. State of the art technologynow exists for mining relevant knowledge from existing knowledge resources. Wikidata anda handful of other projects have proposed knowledge representation formats for mathematics.One current example, Gamma function on Wikidata is pictured in Figure 1. The page is quiteterse. More details would be needed to bring the Wikidata page closer to feature parity withthe Wikipedia pages in English or German. Indeed, better-articulated frameworks would berequired to adequately represent more complex mathematical concepts, e.g., adjunction in acategory.2 Whatever the concept, the definition and any formulas should be accompanied byframing information (e.g., who invented it, where has the concept been discussed?).3 How will the public benefit?Improvements to Wikipedia: If all of this information is available within Wikidata, we andothers can use it to make a better Wikipedia using semi-automated methods. For example,article placeholders, autolinking of entries, automatic provision of references, and so on, couldall be provided. By using the knowledge graph, such services can be delivered with contextualsensitivity; e.g., to point out that “this article is missing a basic explanation of X/Y/Z.”)Reasoning and computational methods available by default: Having a computational“frame” over the material, we would be able to infer, e.g, “Here is an example; here’s how youcompute with it.” An implementation of the key Pólya heuristic “If you can’t solve a problem,there is an easier problem you can solve: find it” would be valuable for tutoring applications.Integration with other knowledge bases and AI systems: Thinking of the system asa data consumer, we should be able to use it to index and centrally store concepts from thebroader literature. Thinking of the system as a data provider, if the Wikidata representationsare done right, we should be able to use this data in AI applications, for example, as a resourceto use in more general question-answering systems that make use of mathematical reasoning.4 Related workThe NIST Digital Library of Mathematical Functions [Mil13] and the derived Digital Reposi-tory of Mathematical Formulae [CMS+14, CSM+15] are inspiring examples of formula-centricstorage. Contentmine is an interesting current project that has had success mining knowledgeout of research papers (for now, mostly in chemistry and bioscience). It would be interest-ing to apply the methods to the mathematics and physics content stored in arXiv. Babar isanother project for extracting knowledge out of Wikipedia, specifically [dL12]. The authorsof the current paper have been highly involved with representing mathematics on the web ina format suited for human readers, notably through work with the MathML Association andPlanetMath. Representing mathematical knowledge in a format suitable for both human andmachine interaction requires a somewhat different approach. Michael Kohlhase’s recent workon a Semantic Multilingual Glossary for Mathematics is relevant prior art, as are the earlierOpenMath Content Dictionaries. A range of more formal representations exists in the contextof projects like HOL [GK15], Isabelle, Coq [CFGW04], and Mizar, where contributions to theMizar Mathematical Library correspond to articles in the journal Formalized Mathematics.2https://en.wikipedia.org/wiki/Adjoint_functors2math.wikipedia.org Corneli and Schubotz5 ConclusionsIf we compare the few high-level properties of the gamma function that are expressed in Figure1 with those that are captured in connection with the function’s formalisation in HOL [SH14], itbecomes even more starkly clear that the current Wikidata treatment lacks detail. In addition tolisting more properties, the formal treatment naturally provides the proofs of these properties.At the moment, Wikidata even lacks the property ‘demonstrates’ that could be used to connectthe statement of a theorem with a proof.3 In principle, one could be added – which would thenbeg the question as to how proofs would be represented in the system. However, proofs are justone of many challenging topics to address. A simple example: how should Wikidata answer thequestion “What is the area of a circle?” – given that ‘disk’ would be correct in mathematicalEnglish, but Kreisfläche (lit. circle area) is the relevant concept in German.4More broadly: what role will semantic representation and AI software play in the develop-ment of the online mathematics ecosystem? What workflows will support the use and furtherdevelopment of the constituent online resources? This is the upstream part of the “collabora-tive development of open services based on representations of mathematical knowledge” [Ion16].The fruits of repository- and KB-building efforts depend on due attention to these roots.Acknowledgment Corneli’s work was supported by an EPSRC-funded project studying “TheIntegration and Interaction of Multiple Mathematical Reasoning Processes” (EP/N014758/1).References[Bun13] Alan Bundy. The interaction of representation and reasoning. Proceedings of the RoyalSociety of London A: Mathematical, Physical and Engineering Sciences, 469(2157), 2013.[CFGW04] Lúıs Cruz-Filipe, Herman Geuvers, and Freek Wiedijk. C-CoRN, the constructive Coqrepository at Nijmegen. In International Conference on Mathematical Knowledge Manage-ment, pages 88–103. Springer, 2004.[CMS+14] H. S. Cohl, A. McClain M, B. V. Saunders, M. Schubotz, and J. C. Williams. DigitalRepository of Mathematical Formulae. In CICM, LNCS, 2014.[CSM+15] H. S. Cohl, M. Schubotz, M. A. McClain, B. V. Saunders, C. Y. Zou, A. S. Mohammed,and A. A. Danoff. Growing the DRMF with Generic LATEX Sources. In CICM, 2015.[dL12] Pierre Raymond de Lacaze. BABAR: Wikipedia Knowledge Extraction, 2012.[Dud16] Kaushal Dudhat. Teaching Mathematics to Question Answering System. Master’s thesis,Universität Konstanz, 2016.[GK15] Thibault Gauthier and Cezary Kaliszyk. Sharing HOL4 and HOL Light proof knowledge.In Logic for Programming, Artificial Intelligence, and Reasoning, pages 372–386. Springer,2015.[Ion16] Patrick Ion. The Effort to Realize a Global Digital Mathematics Library. In InternationalCongress on Mathematical Software, pages 458–466. Springer, 2016.[Mil13] Bruce R. Miller. Three years of DLMF: web, math and search. In J. Carette and D. Aspinall,editors, CICM, volume 7961 of LNCS, 2013.[SH14] Umair Siddique and Osman Hasan. On the formalization of gamma function in HOL.Journal of automated reasoning, 53(4):407–429, 2014.[SS16] M. Schubotz and A. P. Sexton. A Smooth Transition to Modern mathoid-based MathRendering in Wikipedia with Automatic Visual Regression Testing. In M. Kohlhase, editor,WiP at CICM, Aachen, 2016.3https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Mathematics#Modelling_proofs_and_properties_of_some_objects4https://www.wikidata.org/wiki/Wikidata_talk:WikiProject_Mathematics#Consistent_naming_convention3math.wikipedia.org Corneli and SchubotzFigure 1: Screenshot of the Wikidata page for Euler’s gamma function https://www.wikidata.org/wiki/Q190573, with German language localization. On the top of the page is the title and theid. Below there is a natural language description and a list of aliases. In the block on theright there are links to the various Wikipedia pages about the gamma function. One of theblocks on the left contains a definition that was manually entered, and originates from theNIST Digital Library of Mathematical Functions. The formula in the blue box, Γ(n) = (n−1)!,was automatically extracted from the English Wikipedia by Yash Nager (a master’s student atUniversität Konstanz), and currently waits for human verification. Links to external identifiersand other related resources are provided.4

math.wikipedia.org: A vision for a collaborative, semi-formal, language independent math(s) encyclopedia

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math.wikipedia.org: A vision for a collaborative, semi-formal, language independent math(s) encyclopedia

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