Mathematical Vernacular and Conceptual Well-Formedness in Mathematical Language

@inproceedings{Luo1997MathematicalVA,
  title={Mathematical Vernacular and Conceptual Well-Formedness in Mathematical Language},
  author={Zhaohui Luo and Paul Callaghan},
  booktitle={LACL},
  year={1997}
}
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… 

Object languages in a type-theoreticmeta-frameworkPaul

TLDR
Techniques for providing a convenient syntax for object languages implemented via a type-theoretic Logical Framework, and reports on work in progress are presented.

Coercive Subtyping

TLDR
This approach, subtyping with speciied implicit coercions is treated as a feature at the level of the logical framework; in particular, the meaning of an object being in a supertype is given by coercive deenition rules for the deenitional equality.

Mathematical Vernacular in Type Theory-based Proof Assistants

TLDR
The Durham Mathematical Vernacular (MV) project is presented, the general design of a prototype to support experimentation with issues of MV is discussed, current work on the prototype is explained, and a prototype is demonstrated at the conference.

Computer-Assisted Reasoning with Natural Language: Implementing a Mathematical Vernacular

TLDR
The Durham Mathematical Vernacular project is presented, namely the issue of semantic well-formedness in mathematical descriptions, and a prototype being developed to experiment with aspects of the project.

Weak Type Theory : a formal language for mathematics

TLDR
A formal language for expressing mathematics, called Weak Type Theory (‘WTT’), which is close to the usual way in which mathematicians express themselves in writing and has a high degree of reliability in the sense that it gives a true image of what the mathematician intends to say.

Vernacular Mathematics, Discourse Representation, and Arbitrary Objects

This paper discusses some of the issues that have to be addressed for a systematic treatment of vernacular mathematics. In vernacular proofs, variable letters are introduced and used in a way that

An Implementation of LF with Coercive Subtyping & Universes

TLDR
It is claimed that the combination of Tarski-style universes together with coercive subtyping provides an ideal formulation of universes that is both semantically clear and practical to use.

Understanding informal mathematical discourse

TLDR
This thesis aims to provide a general framework as well as an implementation for such a proof engine that is capable of processing mathematical proofs automatically, at least with regard to translating the mathematicians’ expert language into the system's artificial formal language.

Flexible Encoding of Mathematics on the Computer

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

References

SHOWING 1-10 OF 32 REFERENCES

Context-Relative Syntactic Categories and the Formalization of Mathematical Text

TLDR
The rules presented are a fragment of a grammar implemented in the proof editor ALF, which can be used as an interactive proof text editor, so that the user chooses proof steps, wordings, and grammatical structures, and the system checks that the text is both mathematically and grammatically correct.

Coercive Subtyping in Type Theory

TLDR
In this approach, subtyping with specified implicit coercions is treated as a feature at the level of the logical framework; in particular, subsumption and coercion are combined in such a way that the meaning of an object being in a supertype is given by coercive definition rules for the definitional equality.

Coercive Subtyping

TLDR
This approach, subtyping with speciied implicit coercions is treated as a feature at the level of the logical framework; in particular, the meaning of an object being in a supertype is given by coercive deenition rules for the deenitional equality.

Mathematical Vernacular in Type Theory-based Proof Assistants

TLDR
The Durham Mathematical Vernacular (MV) project is presented, the general design of a prototype to support experimentation with issues of MV is discussed, current work on the prototype is explained, and a prototype is demonstrated at the conference.

Deliverables: A Categorial Approach to Program Development in Type Theory

TLDR
A method for formally developing functional programs using the “propositions as types” paradigm, where a function together with its proof of correctness forms a morphism in a category whose objects are input/output specifications.

Program Speciication and Data Reenement in Type Theory

TLDR
It is shown that a type theory with a strong logical power and nice structural mechanisms provides an adequate formalism for modular development of programs and speciications and can be developed by means of the existing proof development systems based on type theories.

Type-Theoretical Interpretation and Generalization of Phrase Structure Grammar

TLDR
A generalization of phrase structure grammar, in which all functional categories (such as verbs and adjectives) have type restrictions, that is, their argument types are speciic domains.

Computation and reasoning - a type theory for computer science

  • Zhaohui Luo
  • Computer Science
    International series of monographs on computer science
  • 1994
TLDR
A set-theoretic model for the specification and development of programs and a unifying theory of dependent types is proposed.