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Natural mathematical language for the computer

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If you have a question about this talk, please contact Laura Rimell.

This is joint work with Peter Schodl and Kevin Kofler, also from Vienna. We are currently working towards the creation of an automatic mathematical research system that can support mathematicians in their daily work, providing services for abstract mathematics as easily as Latex provides typesetting services, the arXiv provides access to preprints, Google provides web services, Matlab provides numerical services, or Mathematica provides symbolic services.

The ultimate goal is to be able to read, understand, and process automatically ordinary mathematical text of the kind found in scholarly articles and books, as far as they do not involve historical, anecdotal, or other content that requires cultural knowledge from outside core mathematics. This restriction reduces the difficult problems of automatic natural language processing to a manageable level.

A limited part of our vision—expected to take 50 man years to bring a system far enough that it will grow by itself in a wikipedia-like fashion—is being realized through the project ``A modeling system for mathematics’’ (MoSMath), currently supported by a grant of the Austrian Science Foundation FWF . Within this project, we attempt to create a modeling and documentation language for conceptual and numerical mathematics called FMathL (formal mathematical language), suited to the habits of mathematicians.

FMathL allows to specify problems in their natural mathematical form, with functions, sets, operators, measures, quantifiers, tables, cases, etc. Formal models are specified close to how they would be communicated informally when describing them in a lecture or paper, except that no relevant details are suppressed.

A faithful representation of the semantics in terms of a so-called semantic matrix is the heart of our approach. FMathL enables users to express arbitrary mathematics in a form that is faithfully translated into the semantic matrix. Application modules can therefore be fed by algorithms that extract from the semantic matrix the relevant information.

At present we have a fragment of FMathL designed to encode parts of mathematics (mainly related to optimization problems) in the semantic matrix, checking it for semantic adequacy (currently on the level of types only, ignoring many more subtle issues), and preparing it for automatic re-rendering in natural language.

An interface to the controlled natural language of Naproche (developped in Germany for representing human-readable formal proofs) enables us to read and represent texts written in this language, and to recreate Naproche-texts from texts represented in the semantic matrix.

We are currently working on an interface to the Grammatical Framework, which has multiple language support with correct inflection.

This talk is part of the NLIP Seminar Series series.

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