## Mathematical Vernacular and Conceptual Well-formedness in Mathematical Language (1998)

Venue: | Proceedings of the 2nd Inter. Conf. on Logical Aspects of Computational Linguistics, LNCS/LNAI 1582 |

Citations: | 14 - 9 self |

### BibTeX

@INPROCEEDINGS{Luo98mathematicalvernacular,

author = {Zhaohui Luo and Paul Callaghan},

title = {Mathematical Vernacular and Conceptual Well-formedness in Mathematical Language},

booktitle = {Proceedings of the 2nd Inter. Conf. on Logical Aspects of Computational Linguistics, LNCS/LNAI 1582},

year = {1998}

}

### OpenURL

### Abstract

. This paper investigates the semantics of mathematical concepts in a type theoretic framework with coercive subtyping. The typetheoretic analysis provides a formal semantic basis in the design and implementation of Mathematical Vernacular (MV), a natural language suitable for interactive development of mathematics with the support of the current theorem proving technology. The idea of semantic well-formedness in mathematical language is motivated with examples. A formal system based on a notion of conceptual category is then presented, showing how type checking supports our notion of well-formedness. The power of this system is then extended by incorporating a notion of subcategory, using ideas from a more general theory of coercive subtyping, which provides the mechanisms for modelling conventional abbreviations in mathematics. Finally, we outline how this formal work can be used in an implementation of MV. 1 Introduction By mathematical vernacular (MV), we mean a mathematical and n...

### Citations

921 | The generative lexicon
- Pustejovsky
- 1995
(Show Context)
Citation Context ...s, [CB96]). This situation does not occur in IML: mathematical terms must have been given precise definitions, and the same meaning must result in all contexts. Very recently, the work by Pustejovsky =-=[Pus95]-=- on generative lexicon and the work by Jackendoff on enriched composition [Jac97] have come to our attention. They have studied the conceptual structure and lexical semantics of natural language based... |

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...plies to other related issues. The basic formal theory we use has been studied in several contexts, including the subset theory developed by Nordstrom, Petersson and Smith in Martin-Lof's type theory =-=[NPS90]-=-, the specification calculus by Luo in the Extended Calculus of Constructions and UTT [Luo93], and the related (but different) framework on deliverables [BM92, McK92] and mathematical theories [Luo91a... |

256 |
The Architecture of the Language Faculty
- Jackendoff
- 1997
(Show Context)
Citation Context ...en given precise definitions, and the same meaning must result in all contexts. Very recently, the work by Pustejovsky [Pus95] on generative lexicon and the work by Jackendoff on enriched composition =-=[Jac97]-=- have come to our attention. They have studied the conceptual structure and lexical semantics of natural language based on an idea of coercion. It would be interesting to study the connections of thei... |

190 |
Linguistic Semantics
- LYONS
- 1995
(Show Context)
Citation Context ...can be considered to be defined from some basic concepts such as those in a foundational language. For these reasons, the traditional arguments against componential analysis in lexical semantics (cf, =-=[Lyo95]-=-) do not apply to mathematical language. 6 Implementing MV: Discussion and Related Work As stated in Section 2, one of our research objectives is to develop the implementation technology of MV based o... |

147 |
LEGO Proof Development System: User's Manual
- Luo, Pollack
- 1992
(Show Context)
Citation Context ...ciated techniques to implement MV based on existing theorem proving technology. Such technology is represented by type theory based proof development systems such as ALF [MN94], Coq [Coq96], and Lego =-=[LP92]-=-. One of the motivations of our work reported here is to study the linguistic structure of mathematical language based on which MV is to be designed and implemented. In particular, we are interested i... |

108 |
Computation and Reasoning, A Type Theory for Computer Science
- Luo
- 1994
(Show Context)
Citation Context ...ctive types including type constructors for functional types A ! B, dependent functional types \Pi(A; B), types of dependent pairs \Sigma (A; B), and types of natural numbers, lists, trees, etc. (See =-=[Luo94]-=- for details.) UTT is implemented in the proof system Lego [LP92]. 4.1 Conceptual categories A (conceptual) category represents a mathematical concept. We shall use the judgement form C : Cat to denot... |

104 |
Type-Theoretical Grammar
- Ranta
- 1995
(Show Context)
Citation Context ...more closely related to our research in their objectives and methods than others; among the more related are de Bruijn's work on mathematical vernacular [dB94], Ranta's work on type-theoretic grammar =-=[Ran94]-=- and his research on informal mathematical language based on type theory [Ran95] and its implementation [Ran97] 10 , Coscoy's work on proof explanation in Coq [CKT95], and the Mizar project [Miz], whe... |

97 | Semi-productive polysemy and sense extension
- Copestake, Briscoe
- 1995
(Show Context)
Citation Context ...ypes fast 6= someone who is a typist and who is fast However, the NL example is defeasible -- further information may affect which interpretation is chosen (eg a race between typists and accountants, =-=[CB96]-=-). This situation does not occur in IML: mathematical terms must have been given precise definitions, and the same meaning must result in all contexts. Very recently, the work by Pustejovsky [Pus95] o... |

93 |
The ALF proof editor and its proof engine
- Magnusson, Nordström
- 1994
(Show Context)
Citation Context ...eoretic semantics and the associated techniques to implement MV based on existing theorem proving technology. Such technology is represented by type theory based proof development systems such as ALF =-=[MN94]-=-, Coq [Coq96], and Lego [LP92]. One of the motivations of our work reported here is to study the linguistic structure of mathematical language based on which MV is to be designed and implemented. In p... |

90 |
A survey of the project AUTOMATH
- Bruijn
- 1980
(Show Context)
Citation Context ...r has been substantially influenced and improved by general discussions in the works above, and by communications with Aarne Ranta. In particular, Prof de Bruijn's pioneering work on MV (and Automath =-=[dB80]-=-) has significantly influenced the research field and our work. His work on MV offers many insights in designing mathematical vernaculars, which we very much believe are deserving of further investiga... |

46 | Coercive subtyping
- Luo
- 1997
(Show Context)
Citation Context ...hen \Sigma (A; B)s\Sigma (A 0 ; B 0 ). The expected composite coercions with respect to these generalisations can be easily constructed from the component coercions, but we omit the details here (see =-=[Luo98]-=-). 5.2 The subcategory relation We can use the notion of coercion directly in the calculus of section 4 by developing a notion of subcategory relation as a lifted version of the subtyping relation. We... |

40 | Typing algorithm in type theory with inheritance
- Saibi
- 1997
(Show Context)
Citation Context ...ing (eg, between types of natural numbers and integers) and that of inheritancebased subtyping (eg, between record types). Anthony Bailey has implemented coercion mechanisms in Lego (and Saibi in Coq =-=[Sai97]-=-), and considered its applications to formal development of mathematics (Galois theory) based on type theory [Bai98]. For some meta-theoretic results of coercive subtyping, see [JLS97, SL98]. The mech... |

37 | Théry, Extracting Text from Proof
- Coscoy, Kahn, et al.
- 1995
(Show Context)
Citation Context ...'s work on type-theoretic grammar [Ran94] and his research on informal mathematical language based on type theory [Ran95] and its implementation [Ran97] 10 , Coscoy's work on proof explanation in Coq =-=[CKT95]-=-, and the Mizar project [Miz], where a mathematical vernacular has been defined and implemented in a batch system whose logic is based on set theory, and subsequently used to formalise an impressive a... |

30 |
The machine-checked literate formalisation of algebra in type theory
- Bailey
- 1998
(Show Context)
Citation Context ...ormalism of type checking of certain type systems [vBJMP93], or Alex Jones' work on the Decidable Dependency Theorem in Linear Algebra [Jon95]. A good source of further information is Bailey's thesis =-=[Bai98]-=-, which contains much discussion on how to formalise maths with type theory and how to make the formalised version more understandable, followed by the formalisation of a substantial example (parts of... |

28 | Program specification and data refinement in type theory - Luo - 1993 |

25 | Coercive subtyping in type theory
- Luo
- 1996
(Show Context)
Citation Context ... set" and "finite 2 sequence ". Coercive subtyping allows a satisfactory treatment of this phenomenon through overloading of unit types (types containing only one object). This idea fir=-=st appeared in [Luo97]-=-, where overloading pairs of \Sigma -types and product types is considered, and is further developed in Bailey's thesis [Bai98], where he makes extensive use of coercions and overloading. The techniqu... |

24 | A higher-order calculus and theory abstraction
- Luo
- 1991
(Show Context)
Citation Context ...[NPS90], the specification calculus by Luo in the Extended Calculus of Constructions and UTT [Luo93], and the related (but different) framework on deliverables [BM92, McK92] and mathematical theories =-=[Luo91a]-=-. Here, we apply this theory to mathematical concepts and the related well-formedness issues. In the next section, we shall extend this to introduce a notion of subcategory based on the theory of coer... |

24 | Deliverables: A categorical approach to program development in type theory - McKinna, Burstall - 1993 |

17 | Deliverables: a categorical approach to program development in type theory - Burstall, McKinna - 1992 |

12 |
The coq proof assistant reference manual version 8.3. INRIA
- Coq
- 2010
(Show Context)
Citation Context ...ntics and the associated techniques to implement MV based on existing theorem proving technology. Such technology is represented by type theory based proof development systems such as ALF [MN94], Coq =-=[Coq96]-=-, and Lego [LP92]. One of the motivations of our work reported here is to study the linguistic structure of mathematical language based on which MV is to be designed and implemented. In particular, we... |

12 |
de Bruijn. The Mathematical Vernacular, a language for mathematics with typed sets
- G
(Show Context)
Citation Context ... in MV. Focussing on the conceptual structure reflected in the use of substantives (or common noun phrases) and adjectives -- the key linguistic entities used to represent mathematical concepts (cf., =-=[dB94]-=-), we consider fine-structured categories that allow advanced treatment of deciding whether an expression is well-formed. With our type-theoretic semantics, checking semantic well-formedness is suppor... |

12 | Some proof-theoretic and algorithmic aspects of coercive subtyping. Types for proofs and programs - Jones, Luo, et al. - 1998 |

8 | Context-relative syntactic categories and the formalization of mathematical text - Ranta - 1996 |

3 | Mathematical vernacular in type theory-based proof assistants
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- 1998
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Citation Context ... a set, the if-clause would have been trivially false (ie, ill-typed and not derivable) 7 . We omit further specific discussion of how objects and proof terms will be treated under this strategy (see =-=[CL98]-=- for more information). 4.2 Category Constructors Categories can either be base categories or the result of applying a category constructor. A category or a category constructor is defined by giving o... |

2 |
The formalization of linear algebra in LEGO: The decidable dependency theorem
- Jones
- 1995
(Show Context)
Citation Context ... Ruys' work on the Fundamental Theorem of Algebra [Ruy], or the formalism of type checking of certain type systems [vBJMP93], or Alex Jones' work on the Decidable Dependency Theorem in Linear Algebra =-=[Jon95]-=-. A good source of further information is Bailey's thesis [Bai98], which contains much discussion on how to formalise maths with type theory and how to make the formalised version more understandable,... |

2 | Type-theoretical interpretation and generalization of phrase structure grammar
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- 1995
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Citation Context ...; among the more related are de Bruijn's work on mathematical vernacular [dB94], Ranta's work on type-theoretic grammar [Ran94] and his research on informal mathematical language based on type theory =-=[Ran95]-=- and its implementation [Ran97] 10 , Coscoy's work on proof explanation in Coq [CKT95], and the Mizar project [Miz], where a mathematical vernacular has been defined and implemented in a batch system ... |

2 |
A grammatical framework (some notes on the source files
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- 1997
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Citation Context ... Bruijn's work on mathematical vernacular [dB94], Ranta's work on type-theoretic grammar [Ran94] and his research on informal mathematical language based on type theory [Ran95] and its implementation =-=[Ran97]-=- 10 , Coscoy's work on proof explanation in Coq [CKT95], and the Mizar project [Miz], where a mathematical vernacular has been defined and implemented in a batch system whose logic is based on set the... |

1 |
Formalizing Mathematics in Type Theory
- Ruys
(Show Context)
Citation Context ... studied: equality is a particular issue that needs careful consideration. A lot of serious mathematics has already been done based on type theory, eg Ruys' work on the Fundamental Theorem of Algebra =-=[Ruy]-=-, or the formalism of type checking of certain type systems [vBJMP93], or Alex Jones' work on the Decidable Dependency Theorem in Linear Algebra [Jon95]. A good source of further information is Bailey... |

1 | Coercive subtyping: coherence and conservativity - Soloviev, Luo - 1998 |

1 |
Typechecking in pure type systems. submitted for publication
- Jutting, McKinna, et al.
- 1993
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Citation Context ...deration. A lot of serious mathematics has already been done based on type theory, eg Ruys' work on the Fundamental Theorem of Algebra [Ruy], or the formalism of type checking of certain type systems =-=[vBJMP93]-=-, or Alex Jones' work on the Decidable Dependency Theorem in Linear Algebra [Jon95]. A good source of further information is Bailey's thesis [Bai98], which contains much discussion on how to formalise... |