## Flyspeck in a Semantic Wiki Collaborating on a Large Scale Formalization of the Kepler Conjecture

### Cached

### Download Links

Citations: | 2 - 0 self |

### BibTeX

@MISC{Lange_flyspeckin,

author = {Christoph Lange and Sean Mclaughlin and Florian Rabe},

title = {Flyspeck in a Semantic Wiki Collaborating on a Large Scale Formalization of the Kepler Conjecture},

year = {}

}

### OpenURL

### Abstract

Abstract. Semantic wikis have been successfully applied to many problems in knowledge management and collaborative authoring. They are particularly appropriate for scientific and mathematical collaboration. In previous work we described an ontology for mathematical knowledge based on the semantic markup language OMDoc and a semantic wiki using both. We are now evaluating these technologies in concrete application scenarios. In this paper we evaluate the applicability of our infrastructure to mathematical knowledge management by focusing on the Flyspeck project, a formalization of Thomas Hales ’ proof of the Kepler Conjecture. After describing the Flyspeck project and its requirements in detail, we evaluate the applicability of two wiki prototypes to Flyspeck, one based on Semantic MediaWiki and another on our mathematics-specific semantic wiki SWiM. 1 Scientific Communication and the Flyspeck Project Scientific communication consists mainly of exchanging documents, and a great deal of scientific work consists of collaboratively authoring them. Common instances are writing down first hypotheses, commenting on results of experiments or project steps, and structuring, annotating, or re-organizing existing items of knowledge, as depicted in Buchberger’s figure on the right.

### Citations

468 |
Interactive Theorem Proving and Program Development, Coq’Art:the Calculus of Inductive Constructions
- Bertot, Castéran
- 2004
(Show Context)
Citation Context ...osely”. It was found by Hales using a regular expression search of an English dictionary for the expression “F.*P.*K”, for “Formal Proof of Kepler” 2sExamples of proof assistants are Isabelle[29], Coq=-=[3]-=-, and Twelf[30]. (The word “formalize” is used in many contexts in this field. In the remainder of this paper, we use “formal” and “formalize” loosely, possibly referring to any degree of colloquial o... |

313 | System description: Twelf — a meta-logical framework for deductive systems
- Pfenning, Schürmann
- 1999
(Show Context)
Citation Context ... found by Hales using a regular expression search of an English dictionary for the expression “F.*P.*K”, for “Formal Proof of Kepler” 2sExamples of proof assistants are Isabelle[29], Coq[3], and Twelf=-=[30]-=-. (The word “formalize” is used in many contexts in this field. In the remainder of this paper, we use “formal” and “formalize” loosely, possibly referring to any degree of colloquial or scientific fo... |

198 |
SPARQL Query Language for RDF
- Prud’hommeaux, Seaborne
- 2006
(Show Context)
Citation Context ... Another powerful feature of SWiM is that authors can embed inline SPARQL queries into wiki pages. Query 1 can be posed without explicitly annotating “unprovenness”, making use of negation as failure =-=[33]-=-: SELECT ?l WHERE { ?l rdf:type odo:Lemma . ?l swrc:isAbout <Composite_Regions> . OPTIONAL { ?p rdf:type odo:Proof . ?p odo:proves ?l . } FILTER ( ! bound(?p) ) } As OMDoc supports all degrees of form... |

164 |
Isabelle: A generic theorem prover, volume 828
- Paulson
- 1994
(Show Context)
Citation Context ...xamine closely”. It was found by Hales using a regular expression search of an English dictionary for the expression “F.*P.*K”, for “Formal Proof of Kepler” 2sExamples of proof assistants are Isabelle=-=[29]-=-, Coq[3], and Twelf[30]. (The word “formalize” is used in many contexts in this field. In the remainder of this paper, we use “formal” and “formalize” loosely, possibly referring to any degree of coll... |

159 |
Harmelen, “OWL web ontology language overview,” W3C recommendation
- McGuinness, Van
- 2004
(Show Context)
Citation Context ...sing terms of the OMDoc document ontology[17]. This ontology models those aspects of the three layers of mathematical knowledge supported by OMDoc to the extent supported by the expressivity of OWL-DL=-=[23]-=-, including a limited inference of dependencies. Modeling all modules of the OMDoc specification in this ontology is not totally complete, though most mathematical statements as well as key aspects of... |

114 | OMDoc: An Open Markup Format for Mathematical Documents (Version
- Kohlhase
(Show Context)
Citation Context ...ics. This is natural. Mathematical logic, depending as it does on symbols and relationships between symbols, lends itself well to formal exposition. Languages like MathML[22], OpenMath[27], and OMDoc=-=[13]-=- were developed to represent the clearly defined and hierarchical structures of mathematics in a way that preserves the intricate relationships.Inthistradition,OMDocisalanguagethatemploysContentMathML... |

107 | A proof of the Kepler conjecture
- Hales
(Show Context)
Citation Context ...es about a week to formalize a single page of mathematical text. Based on such estimates, Hales expects that it will take around 20 manyears to complete the Flyspeck project. Hales is compiling a book=-=[8]-=- of lemmas from different areas of mathematics that are needed in his proof. That book is currently 450 pages, and contains a significant percentage of the mathematical results used in the proof. It c... |

81 | Isar — a generic interpretative approach to readable formal proof documents
- Wenzel
- 1999
(Show Context)
Citation Context ...powerful (OMDoc markup) Fig. 6. Summary of the evaluation of the features 4 Related Work Outside of wikis, the combination of computerized proofs and human-readable text has been investigated in Isar =-=[38]-=-, an alternative literate programming language for Isabelle, and in Mizar [26], whose language of Mizar is close to mathematical vernacular. In contrast to Isar, there is a large web-based library of ... |

63 |
A computer-checked proof of the four-colour theorem. Available at http://research.microsoft.com/~gonthier/4colproof.pdf
- Gonthier
- 2011
(Show Context)
Citation Context ...knowledge, we anticipate that the definitions will undergo refactoring in order to facilitate the actual development of the proofs.(Historically,thishasbeenthecasewithmanylargecomputerizedproofs, cf. =-=[5]-=-.) Refactoring support by the wiki would thus be advantageous. 4 Case Studies and Evaluation So far, the Flyspeck project has four core members who collaborate via GoogleCode[9]. While the services of... |

52 |
From SPARQL to rules (and back
- Polleres
- 2007
(Show Context)
Citation Context ...feature of SWiM is that authors can embed inline SPARQL queries into wiki pages. Query 1 can be posed without explicitly annotating “unprovenness”, assuming a SPARQL semantics with negation as failure=-=[31]-=-: 9 Types of OMDoc links are automatically extracted from the markup; see above. 10sSELECT ?l WHERE { ?l rdf:type odo:Lemma . ?l swrc:isAbout <Composite_Regions> . OPTIONAL { ?p rdf:type odo:Proof . ?... |

46 |
The Wiki Way: Collaboration and Sharing on the Internet
- Leuf, Cunningham
- 2001
(Show Context)
Citation Context ...workflow of incremental formalization inherent to scientific writing. Wikis have not only shown to be appropriate for writing, but are also effective for project management, e.g. in corporate settings=-=[21,33]-=-. We are therefore interested in applying our technologies to scientific knowledge engineering projects. 2 The Flyspeck Project The target of our case study is the Flyspeck Project, which seeks to for... |

33 | Reasoning in semantic wikis
- Krötzsch, Schaffert, et al.
- 2007
(Show Context)
Citation Context ...ed by one wiki page and relations between resources by links between pages. Both pages and links can be typed with terms from ontologies[28], which are either preloaded into the wiki or modeled ad hoc=-=[16]-=-. This is the prevalent approach of adding semantics to wikis, although other ways have been investigated[34]. Semantic wikis offer enhanced navigation capabilities. For example, they can usually disp... |

23 | a semantic wiki for mathematical knowledge management
- Lange
(Show Context)
Citation Context ...rations,definitions,axioms, theorems, proofs, examples, etc.), and collections of interrelated statements can be grouped into theories. With SWiM, a semantic wiki for mathematical knowledge management=-=[18]-=-, we have investigated collaborative editing of OMDoc documents. Additionally, we host a public knowledge base and experimental ground about mathematical knowledge management on the web, which is powe... |

18 | I.: A Search Engine for Mathematical Formulae
- Kohlhase, Sucan
(Show Context)
Citation Context ... structures[14]. We are going to cover them with the document ontology and utilize them for browsing. Querying Proof search will be greatly simplified if the semantic-aware search engine MathWebSearch=-=[15]-=- is used. It applies substitution tree indexing to mathematical formulae. That means, for example, that a query for � f(x ? z)dx would also find � f(y + z)dy. Equivalence up to α-conversion is obvious... |

15 | HANDSCHUH S.: Annotation and navigation in semantic wikis
- OREN, DELBRU, et al.
(Show Context)
Citation Context ... where one resource (e.g. one mathematical theorem) is represented by one wiki page and relations between resources by links between pages. Both pages and links can be typed with terms from ontologies=-=[28]-=-, which are either preloaded into the wiki or modeled ad hoc[16]. This is the prevalent approach of adding semantics to wikis, although other ways have been investigated[34]. Semantic wikis offer enha... |

11 | Flyspeck I: Tame graphs
- Nipkow, Bauer, et al.
(Show Context)
Citation Context ...cal geometry, graph theory and hypermaps, single and multivariable calculus, and plane and spherical trigonometry. The first steps toward a computerized proof have already been taken. Nipkow and Bauer=-=[24]-=- proved the correctness of a fundamental algorithm in Isabelle. The other two main parts of the computer code, linear programming and global optimization, are currently being investigated in doctoral ... |

9 | Proving bounds for real linear programs in isabelle/hol
- Obua
- 2005
(Show Context)
Citation Context ...orrectness of a fundamental algorithm in Isabelle. The other two main parts of the computer code, linear programming and global optimization, are currently being investigated in doctoral dissertations=-=[35,25]-=-. There is a project page[9] for Flyspeck that documents some of this progress. That page has a source repository containing the book of lemmas, as well as the formalized definitions of some important... |

7 |
Formal Global Optimisation with Taylor Models,” in IJCAR, ser
- Zumkeller
- 2006
(Show Context)
Citation Context ...orrectness of a fundamental algorithm in Isabelle. The other two main parts of the computer code, linear programming and global optimization, are currently being investigated in doctoral dissertations=-=[35,25]-=-. There is a project page[9] for Flyspeck that documents some of this progress. That page has a source repository containing the book of lemmas, as well as the formalized definitions of some important... |

5 |
C.: Cooperative repositories for formal proofs
- Corbineau, Kaliszyk
- 2007
(Show Context)
Citation Context ...ofsisnotcompatiblewithothertheoremprovers,asallcalculi and proof tactics need to be defined in the Logiweb system itself. ProofWiki is an integration of ProofWeb, a web frontend to Coq, into MediaWiki=-=[4]-=-. Coq’s conversion tools are used to generate a human-readable and browsable HTML or L ATEX presentation with linked symbols from the proof scripts. Generating index pages, such as lists of all defini... |

5 | The Layers of Logiweb
- Grue
- 2007
(Show Context)
Citation Context ...s early stage, but it may be required later if the central maintainer approach does not turn out to work. Logiweb is a distributed system for publishing machine checked mathematics in high-quality PDF=-=[7]-=-. While the author does not call it a “wiki”, it shares part of the key wiki principles: anybody can contribute to a Logiweb site and edit new pages in a simple text syntax with a browser. On the othe... |

4 |
Semantically Annotated L ATEX for Scientific Publications
- SALT
- 2007
(Show Context)
Citation Context ...antic markup languages for representing structures of scientific knowledge, and editing tools understanding them, are a promising approach to supporting this work.Besides generic approaches like SALT =-=[6]-=-, the most extensive work in semantic markup has been in the domain of mathematics. Mathematical logic, depending on symbols and relationships between symbols, naturally lends itself well to formal ex... |

3 |
Flyspeck project. http://code.google.com/p/flyspeck/. The purpose of the flyspeck project is to produce a formal proof of the Kepler Conjecture
- Hales
(Show Context)
Citation Context ...rithm in Isabelle. The other two main parts of the computer code, linear programming and global optimization, are currently being investigated in doctoral dissertations[35,25]. There is a project page=-=[9]-=- for Flyspeck that documents some of this progress. That page has a source repository containing the book of lemmas, as well as the formalized definitions of some important functions and inequalities.... |

3 | Documents with flexible notation contexts as interfaces to mathematical knowledge
- Kohlhase, Müller, et al.
- 2007
(Show Context)
Citation Context ...totype we realized that the narrative structure of the book is not adequately represented by a simple hierarchy of categories. OMDoc has more powerful ways of putting content into narrative structures=-=[14]-=-. We are going to cover them with the document ontology and utilize them for browsing. Querying Proof search will be greatly simplified if the semantic-aware search engine MathWebSearch[15] is used. I... |

3 | Towards scientific collaboration in a semantic wiki
- Lange
- 2007
(Show Context)
Citation Context ...reloaded, it should be possible to add other annotations ad hoc. We have not focusedondocumentediting indetailhere,butadditionaleditingservicesrelying on the document ontology are planned for SWiM 0.3=-=[19,20]-=-. Finally, using the module system of OMDoc and refactoring the knowledge into more smaller theories could help to simplify the structure of Flyspeck for browsing and to explicate the dependencies bet... |

3 |
Flyspeck: A blueprint for the formal proof of the Kepler conjecture. source files at http://code.google.com/p/flyspeck/source/browse/trunk
- Hales
- 2008
(Show Context)
Citation Context ...We use “computerized” to mean that a theorem, proof or definition has been expressed in a proof assistant. Note that we consider computerized definitions and proofs formal “documents” as well. 2book =-=[10]-=- of lemmas from different areas of mathematics that are needed in his proof. Its 450 pages contain a significant percentage of the mathematical results used in the proof, covering such disparate topic... |

2 |
The Connexions Markup Language (CNXML). http: //cnx.org/aboutus/technology/cnxml
- Hendricks, Galvan
- 2007
(Show Context)
Citation Context ...ems, but in practice it does not: formulæ are written in Content MathML[22], and the CNXML markup language used for larger structures allows for annotating texts as mathematical statements like lemmas=-=[10]-=-, but this structural information is not yet used by the system. Recently, there is a growing interest in integrating proof assistants with wikis 11 . Logiweb and ProofWiki are wiki-like systems that ... |

2 |
using Semantic MediaWiki – ontoworld.org. http://ontoworld.org/index
- Sites
- 2007
(Show Context)
Citation Context ...ing of typed links (called properties). External ontologies can be referenced from the wiki, but at most sites powered by Semantic MediaWiki, site-specific ontologies are developed in an ad hoc manner=-=[26]-=-. Prototype In Semantic MediaWiki, we imported the Twelf master source of Flyspeck via a custom upload page. The Twelf file was first enhanced by special comment lines marking the beginning and end of... |

1 |
2000 mathematics subject classification. http: //www.ams.org/msc
- Society
- 2000
(Show Context)
Citation Context ...ctionof“primitivevolumes”, which in turn is a section of the chapter “volume calculations”). Standardized ways of classifying mathematical topics, such as the Mathematical Subject Classification (MSC)=-=[1]-=-, could be added later. Project-organization metadata such as whether the proof of a lemma has already been computerized, or if someone is currently attempting a proof. This is essential so that two p... |

1 |
SALT–SemanticallyAnnotated L ATEX for scientific publications
- Groza, Möller
- 2007
(Show Context)
Citation Context ...g such documents. For example, we can write scientific documents in a semantic markup language with an editor that knows the structures available in this language. Besides generic approaches like SALT=-=[6]-=-, the most extensive work in semantic markup has been in the domain of mathematics. This is natural. Mathematical logic, depending as it does on symbols and relationships between symbols, lends itsel... |

1 |
sTEX: A L ATEX-based workflow for OMDoc. In OMDoc – An open markup format for mathematical documents [Version 1.2] [13], chapter 26.15
- Kohlhase
(Show Context)
Citation Context ... markup step by step, or one could start the formalization on the TEX side. There, one would formalize the book to sTEX, a content-oriented TEX notation for OMDoc, which can then be converted to OMDoc=-=[12]-=-. Either way involves a TEX-to-XML transformation, which has been tested in large scale in our group[2]. Annotating ThecasestudyshowedthattheeditingofontologiesinSWiMshould become more flexible. While... |

1 |
The OMDoc document ontology. http://kwarc.info/projects/ docOnto/omdoc.html, seen
- Lange
- 2007
(Show Context)
Citation Context ... its (typed) relations to other items of mathematical knowledge in the wiki are extracted from the OMDoc XML markup and explicitly represented as RDF triples using terms of the OMDoc document ontology=-=[17]-=-. This ontology models those aspects of the three layers of mathematical knowledge supported by OMDoc to the extent supported by the expressivity of OWL-DL[23], including a limited inference of depend... |

1 |
SWiM development roadmap. https://trac.kwarc.info/swim/ roadmap/, seen
- Lange
- 2007
(Show Context)
Citation Context ...reloaded, it should be possible to add other annotations ad hoc. We have not focusedondocumentediting indetailhere,butadditionaleditingservicesrelying on the document ontology are planned for SWiM 0.3=-=[19,20]-=-. Finally, using the module system of OMDoc and refactoring the knowledge into more smaller theories could help to simplify the structure of Flyspeck for browsing and to explicate the dependencies bet... |

1 |
using Semantic MediaWiki. http://www.semantic-mediawiki.org/w/index
- Sites
- 2008
(Show Context)
Citation Context ...ng of typed links (called properties). External ontologies can be referenced from the wiki, but at most sites powered by Semantic MediaWiki, site-specific ontologies are developed in an ad hoc manner =-=[34]-=-. Prototype In Semantic MediaWiki, we imported the Twelf master source of Flyspeck via a custom upload page. The Twelf file was first enhanced by special comment lines marking the beginning and end of... |