## Flexible encoding of mathematics on the computer (2004)

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Venue: | In MKM 2004, volume 3119 of LNCS |

Citations: | 22 - 12 self |

### BibTeX

@INPROCEEDINGS{Kamareddine04flexibleencoding,

author = {Fairouz Kamareddine and Manuel Maarek and J. B. Wells},

title = {Flexible encoding of mathematics on the computer},

booktitle = {In MKM 2004, volume 3119 of LNCS},

year = {2004},

pages = {160--174}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of mathematical texts, including natural language views using the exact text that the mathematician wants to use. Thus, MathLang now supports the ability to capture the essential mathematical structure of mathematics written using natural language text. We show examples of how arbitrary mathematical text can be encoded in MathLang without needing to change any of the words or symbols of the texts or their order. In particular, we show the encoding of a theorem and its proof that has been used by Wiedijk for comparing many theorem prover representations of mathematics, namely the irrationality of √ 2 (originally due to Pythagoras). We encode a 1960 version by Hardy and Wright, and a more recent version by Barendregt. 1 On the way to a mathematical vernacular for computers Mathematicians now use computer software for a variety of tasks: typing mathematical texts, performing calculation, analyzing theories, verifying proofs. Software tools like

### Citations

132 | OMDoc An open markup format for mathematical documents (Version 1.2
- Kohlhase
- 2006
(Show Context)
Citation Context ...a have been refined through many years of development. Research in mathematics, logic, and computer science has led to computer algebra systems (CAS’s) and theorem provers (TPs). Languages like OMDo=-=c [21]-=- show good promise of having a universal way to share CAS and TP data in a mathematical software network [22]. Nevertheless, ordinary mathematicians still write mathematical knowledge (MK) using the t... |

73 | Grammatical framework. a typetheoretical grammar formalism
- Ranta
- 2004
(Show Context)
Citation Context ...) do so for symbolic formulas but not natural language sentences and phrases (e.g., OpenMath or computer algebra systems), (2) handle natural language text via very complicated type systems (e.g., GF =-=[17]),-=- (3) require full formalization (e.g., Mizar [18] or other proof systems), or (4) otherwise restrict mathematicians’ freedom to edit and display their texts in the form that best meets their needs a... |

57 |
The mathematical vernacular, a language for mathematics with typed sets
- Bruijn
- 1994
(Show Context)
Citation Context ...rict mathematicians’ freedom to edit and display their texts in the form that best meets their needs and desires. Theoretical approaches to this issue include De Bruijn’s Mathematical Vernacular (=-=MV) [2] and-=- the recently developed Weak Type Theory (WTT) of Kamareddine andsKamareddine, Maarek, Wells — Encoding Mathematics — MKM 2004 2 Nederpelt [12], both of which strive toward a formalism that captur... |

55 |
The Thirteen Books of Euclid's Elements
- Heath
- 1956
(Show Context)
Citation Context ... the capabilities of MathLang via example encodings. AlthoughsKamareddine, Maarek, Wells — Encoding Mathematics — MKM 2004 3 we are currently testing MathLang by encoding two large mathematical bo=-=oks [14, 10]-=-, this article presents encodings of shorter examples, fully developed in MathLang. Section 3 presents an example (translated earlier into WTT) and shows its encoding in MathLang via both the symbolic... |

36 |
Foundations of Analysis
- Landau
- 1951
(Show Context)
Citation Context ... the capabilities of MathLang via example encodings. AlthoughsKamareddine, Maarek, Wells — Encoding Mathematics — MKM 2004 3 we are currently testing MathLang by encoding two large mathematical bo=-=oks [14, 10]-=-, this article presents encodings of shorter examples, fully developed in MathLang. Section 3 presents an example (translated earlier into WTT) and shows its encoding in MathLang via both the symbolic... |

30 | R.: A Refinement of de Bruijns formal language of mathematics
- Kamareddine, Nederpelt
- 2004
(Show Context)
Citation Context ...ssue include De Bruijn’s Mathematical Vernacular (MV) [2] and the recently developed Weak Type Theory (WTT) of Kamareddine andsKamareddine, Maarek, Wells — Encoding Mathematics — MKM 2004 2 Nede=-=rpelt [12], -=-both of which strive toward a formalism that captures the structure of mathematical text while remaining close to the mathematician’s original text. Carrying the work of MV and WTT forward, we have ... |

23 | Comparing mathematical provers
- Wiedijk
(Show Context)
Citation Context ... view of two versions of Pythagoras’ proof of the irrationality of √ 2. We chose this proof because it has been previously used by Wiedijk as a universal example of proof encoding for theorem prov=-=ers [19, 20]-=-. Both original versions are included in [19]. The first one is an informal version written by G. H. Hardy and E. M. Wright (see figure 7). Section 4.1 is the CML view of our translation into MathLang... |

15 | Mathlang: Experience-driven development of a new mathematical language
- Kamareddine, Maarek, et al.
(Show Context)
Citation Context ...good formation of the text at a grammatical level. It is very important that this checking does not require full formalization or committing to any specific foundation of mathematics. In earlier work =-=[13], -=-we introduced MathLang, defined its XML-based concrete syntax, implemented a weak type checker, and tested the encoding of substantial real mathematical documents. MathLang’s CML layer is new and is... |

15 | The Seventeen Provers of the World
- Wiedijk
- 2006
(Show Context)
Citation Context ...esents a bigger example: Pythagoras’ proof of the irrationality of √ 2. We chose this proof because it has been previously used by Wiedijk as a universal example of proof encoding for theorem prov=-=ers [19]-=-. We encode two informal versions of this proof in MathLang: an earlier one due to Hardy and Wright as well as a more explicit (but still informal) one due to Barendregt. For all 3 example encodings p... |

14 | P.: Mathematical vernacular and conceptual well-formedness in mathematical language
- Luo, Callaghan
- 1999
(Show Context)
Citation Context ...nformal definitions (<CMP> tag) are text with embedded OpenMath formulas. WTT The overall document is a formal encoding. Like N.G. de Bruijn’s MV and Z. Luo and P. Callaghan’s Mathematical Vernacu=-=lar [15], -=-WTT is a theoretical language and does not have any natural language representation. MathLang’s approach A computerized structure covers the entire documents. Pieces of CML transformation procedures... |

14 |
On equivalents of well-foundedness
- Rudnicki, Trybulec
- 1999
(Show Context)
Citation Context ...age sentences and phrases (e.g., OpenMath or computer algebra systems), (2) handle natural language text via very complicated type systems (e.g., GF [17]), (3) require full formalization (e.g., Mizar =-=[18] o-=-r other proof systems), or (4) otherwise restrict mathematicians’ freedom to edit and display their texts in the form that best meets their needs and desires. Theoretical approaches to this issue in... |

6 |
Towards an Interactive Mathematical Proof Mode
- Barendregt
- 2003
(Show Context)
Citation Context ...s expressiveness as a replacement of XSL. A possible intermediate stage from CML-texts to fully formalized texts (the “later computations” in figure 2) could be to use the Mathematical Proof Langu=-=age [1]-=-, a language between informal and formalized mathematics. CML OMDoc's approach WTT's approach MathLang's approach TPs' approach Fig. 1. Approaches. Informal data is represented by blobby shapes ( ). C... |

6 | MKM from Book to Computer: A Case Study
- Davenport
- 2003
(Show Context)
Citation Context ...elatively unambiguous, so that automated processing will often be possible without human interaction, (5) sensibly organized, so that most desired ways of browsing the data will not be difficult, and =-=(6)-=- automation-friendly, to facilitate further more complex computations. The language formalism of MathLang has three main features. (1) A symbolic structure. MathLang encodes mathematical texts via a s... |

5 |
TeXmacs as authoring tool for publication and dissemination of formal developments
- Audebaud, Rideau
- 2003
(Show Context)
Citation Context ...rogrammer. (We consider here the generation of natural language texts from Coq [3], the design of an electronic library of mathematics [4], the MoWGLI project [5], work to interface Coq with TEX MACS =-=[8]-=-, and the documentation system of FoC [16].) Figure 2 shows that the original CML text is first divided into a symbolic part and a natural language part and that afterward, the full original text can ... |

3 | Extensible Stylesheet Language (XSL) Recommendation - Deach - 1999 |

2 |
A Natural Language Explanation for Formal Proofs. LACL
- Coscoy
- 1996
(Show Context)
Citation Context ...obby shapes). The natural language is produced with generic computations according to some structured text given by the programmer. (We consider here the generation of natural language texts from Coq =-=[3]-=-, the design of an electronic library of mathematics [4], the MoWGLI project [5], work to interface Coq with TEX MACS [8], and the documentation system of FoC [16].) Figure 2 shows that the original C... |

1 |
Formal Proof Authoring: an Experiment. UITP
- Théry
- 2003
(Show Context)
Citation Context ... believe this is largely because existing computer MK representations either fail to capture the mathematical structure of natural language as used in mathematics or require writing in a rigid format =-=[7]-=-. Mathematical typesetting systems like L ATEX fail to capture the mathematical structure of both natural language text and symbolic formulas. Existing computer mathematics systems that capture mathem... |

1 |
V.: FoCDoc: The documentation system of FoC. Calculemus
- Maarek, Prevosto
- 2003
(Show Context)
Citation Context ...on of natural language texts from Coq [3], the design of an electronic library of mathematics [4], the MoWGLI project [5], work to interface Coq with TEX MACS [8], and the documentation system of FoC =-=[16]-=-.) Figure 2 shows that the original CML text is first divided into a symbolic part and a natural language part and that afterward, the full original text can be retrieved, as well as a fully symbolic ... |