## The Naproche Project Controlled Natural Language Proof Checking of Mathematical Texts

Citations: | 3 - 2 self |

### BibTeX

@MISC{Cramer_thenaproche,

author = {M. Cramer and B. Fisseni and P. Koepke and D. Kühlwein and B. Schröder and J. Veldman},

title = {The Naproche Project Controlled Natural Language Proof Checking of Mathematical Texts},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper discusses the semi-formal language of mathematics and presents the Naproche CNL, a controlled natural language for mathematical authoring. Proof Representation Structures, an adaptation of Discourse Representation Structures, are used to represent the semantics of texts written in the Naproche CNL. We discuss how the Naproche CNL can be used in formal mathematics, and present our prototypical Naproche system, a computer program for parsing texts in the Naproche CNL and checking the proofs in them for logical correctness.

### Citations

460 |
From Discourse to Logic: Introduction to Modeltheoretic
- Kamp, Reyle
- 1993
(Show Context)
Citation Context ...3]. 4 Proof Representation Structures Texts written in the Naproche CNL are translated into Proof Representation Structures or PRSs (see [8], [9]). PRSs are Discourse Representation Structures (DRSs, =-=[7]-=-), which are enriched in such a way as to represent the distinguishing characteristics of SFLM discussed in Section 2. A PRS has five constituents: An identification number, a list of discourse refere... |

51 |
An Introduction to the Theory of Numbers, fourth edition
- Hardy, Wright
- 1960
(Show Context)
Citation Context ...-Formal Language of Mathematics As an example of the semi-formal language of mathematics (SFLM), we cite a proof for the theorem “ √ 2 is irrational” from Hardy-Wright’s introduction to number theory =-=[6]-=-. If √ 2 is rational, then the equation a 2 = 2b 2 is soluble in integers a, b with (a, b) = 1. Hence a 2 is even, and therefore a is even. If a = 2c, then 4c 2 = 2b 2 , 2c 2 = b 2 , and b is also eve... |

49 | Attempto Controlled English: A Knowledge Representation Language Readable by Humans and Machines
- Fuchs, Höfler, et al.
- 2005
(Show Context)
Citation Context ...iff ) and natural language quantification (e.g. for all, there is, every, some, no) works as in natural English, only with some syntactical limitations similar to those in Attempto Controlled English =-=[5]-=-. Sentences in a proof can start with words like then, hence, therefore etc. Mathematical terms can function as noun phrases, and mathematical formulae can function as sentences or sub-clauses. The la... |

21 | Understanding Informal Mathematical Discourse
- Zinn
- 2004
(Show Context)
Citation Context ...d make formal mathematics more feasible to the average mathematician. 7 Related and Future Work Claus Zinn developed a system for processing SFLM texts, which he calls informal mathematical discourse =-=[15]-=-. He used an extended version of DRSs, which he also called Proof Representation Structures, to represent the meaning of a text. The PRS definitions of Naproche and Zinn are different, however. We are... |

19 | Mizar: the first 30 years
- Matuszewski, Rudnicki
(Show Context)
Citation Context ...e limited by the use of formal mathematics system: These are computer systems which facilitate the formalisation of mathematical proofs. In prominent examples of formal mathematics systems like Mizar =-=[11]-=- and Coq [2], this is achieved by allowing the user to use a more readable formal language and to leave out some of the simpler steps of a derivation, which can be filled in by a computer. However, ev... |

15 |
Working with Discourse Representation Theory. An Advanced Course in Computational Semantics
- Blackburn, Bos
(Show Context)
Citation Context ...ains the representation of all claims made inside the scope of that assumption. – Representations of single sentences are produced in a way similar to a standard threading construction algorithm (see =-=[1]-=-). We clarify both the PRS construction algorithm and the functions of the various kinds of PRS conditions by presenting examples which show how the most important PRS conditions are constructed from ... |

13 | System description: SystemOnTPTP
- Sutcliffe
- 2000
(Show Context)
Citation Context ...scharged in 80 seconds. All tests were carried out on a 2 GHz Intel Pentium M with 2 GB Ram. 6 We access automated theorem provers, currently Otter and E Prover, through Geoff Sutcliff’s SystemOnTPTP =-=[12]-=-.16 M. Cramer, B. Fisseni, P. Koepke, D. Kühlwein, B. Schröder, J. Veldman 6 Conclusion We discussed the particular properties of the Semi-Formal Language of Mathematics and presented the Naproche CN... |

3 |
Representation and Inference for Natural Language - Volume II: Working with Discourse Representation Structures
- Blackburn, Bos
- 1999
(Show Context)
Citation Context ...Ss, have a canonical translation into first order logic and can be processed further by logical tools. This translation is performed in a way similar to the DRS to firstorder translation described in =-=[1]-=-. For example, the discourse referents normally trigger existential quantifiers, but in a PRS condition of the form A ⇒ B the discourse referents of A trigger universal quantifiers. There are two main... |

2 |
Mathematisch-logische Aspekte von Beweisreprsentationsstrukturen
- Cramer
- 2009
(Show Context)
Citation Context ...ed. For a PRS to be satisfied in a certain structure A, it must be possible to sequentially modify the context [A, ∅, ∅] by each condition in the PRS. The details of the PRS semantics can be found in =-=[4]-=-. Note that textual referents are not included in the PRS semantics. They are used in the Logic module of the Naproche system (See 5.3). 4.5 Translating PRSs into First-Order Logic PRSs, like DRSs, ha... |

2 |
Generating Proof Representation Structures for the Project NAPROCHE
- Kolev
- 2008
(Show Context)
Citation Context ...alternative of using the mathematical WYSIWYG editor TeXmacs [13]. 4 Proof Representation Structures Texts written in the Naproche CNL are translated into Proof Representation Structures or PRSs (see =-=[8]-=-, [9]). PRSs are Discourse Representation Structures (DRSs, [7]), which are enriched in such a way as to represent the distinguishing characteristics of SFLM discussed in Section 2. A PRS has five con... |

2 |
Grundlagen der Analysis (Third Edition
- Landau
- 1960
(Show Context)
Citation Context ...tion symbols, relation symbols, verbs, adjectives and nouns. They can be marked by the words definition or define. Here is an extract from our reformulation of Edmund Landau’s Grundlagen der Analysis =-=[10]-=-, with all structure markers marked in bold font: Axiom 3: For every x, x ′ ̸= 1. Axiom 4: If x ′ = y ′ , then x = y. Theorem 1: If x ̸= y then x ′ ̸= y ′ . Proof: Assume that x ̸= y and x ′ = y ′ . T... |

1 |
The Controlled Natural Language of Naproche in a nutshell, http://virt054.zim.uni-duisburg-essen.de/NAPROCHE-WIKI/doku.php?id= dokumentation:language. Naproche Project 17
- Cramer
(Show Context)
Citation Context ...istics of SFLM. We first discuss the syntax of the Naproche CNL, and finally present a CNL version of the example proof from Section 2. A more detailed specification of the CNL syntax can be found in =-=[3]-=-. In the Naproche CNL we distinguish between macrostructure (general text structure) and microstructure (grammatical structure in a sentence). 3.1 Macrostructure A Naproche text is structured by struc... |

1 |
A calculus for Proof Representation Structures, Diploma thesis
- Kühlwein
(Show Context)
Citation Context ...native of using the mathematical WYSIWYG editor TeXmacs [13]. 4 Proof Representation Structures Texts written in the Naproche CNL are translated into Proof Representation Structures or PRSs (see [8], =-=[9]-=-). PRSs are Discourse Representation Structures (DRSs, [7]), which are enriched in such a way as to represent the distinguishing characteristics of SFLM discussed in Section 2. A PRS has five constitu... |

1 | A short introduction to Naproche v0.1 - Kühlwein |