Abstract not found

. 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...

The accompanying thesis is part of a long-term project to enable computers to do mathematics in the same way that humans do. I will sketch something of the nature of mathematics and the project, and then turn to role of the thesis. Mathematics Mathematics arises from the interaction of two dissimilar modes of reasoning: a ‘soft ’ side, dealing with ideas and analogies, and a ‘hard ’ side, dealing with verification. The ‘hard ’ side is easier to pin down. It consists primarily of formal ‘proofs’, each consisting of a series of assertions. A mathematician can verify that a proof is correct by following it, step by step, checking that each step follows from previous ones via facts already proved to be correct. The ‘soft ’ side is less easily described. It consists of intuitions about the formal objects constructed in mathematical proofs; ideas that one piece of mathematics may analogically correspond to another piece of mathematics; or even analogies between mathematics and objects in the physical world.

This report explains a natural-language interface to the frmilsm of XFST (Xerox Finite State Tool), which is a rich language used for specifying finite state automata and transducers. By using the interface, it is possible to give input to XFST in English aad French, as well as to translate formal XFST code into these languages. It is also possible to edit XFST source files and their natural-laaguage equivalents interac- tively, in parallel.

This paper presents the Durham Mathematical Vernacular (MV) project. The project aims to develop the technology for interactive tools based on proof checking with type theory, which allow mathematicians to develop mathematics in their usual vernacular. Its sub-goals are to develop type theory technology to support MV and to develop the corresponding NL technology. Mathematical language has many significant differences from everyday language, hence a different approach is required to automatically process such language. We discuss some important differences and how they affect implementation. A key requirement is for correctness. Another key feature is that the user defineshis own terminology. We then discuss work in progress, namely the issue of semantic well-formedness in mathematical descriptions, and a prototype being developed to experiment with aspects of the project. 1 Introduction: Defining a Mathematical Vernacular The long term aim of this project is to develop theory and tec...

This paper provides an abstract syntax for a formal language of mathematics. We call our language Weak Type Theory (abbreviated WTT ). WTT will be as faithful as possible to the mathematician 's language yet will be formal and will not allow ambiguities. WTT can be used as an intermediary between the natural language of the mathematician and the formal language of the logician. As far as we know, this is the rst extensive formalization of an abstract syntax of a formal language of mathematics. We compare our work with existing formalizations of languages of mathematics. 1

syntax. The MathLang abstract syntax (that is to say the way we represent MathLang data) is de ned in the following sections. The abstract syntax will only be used in the WTC The Backus-Naur form (BNF) is a metasyntax to formally describe languages.