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**11 - 20**of**20**### Under supervision of:

, 2006

"... MathLang is a language for mathematics on computers. It allows computerisation of existing and new mathematical texts written in the Common Mathematical Language, and checking the grammatical correctness of this computerisation. The framework also allows the user to take incremental steps towards th ..."

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MathLang is a language for mathematics on computers. It allows computerisation of existing and new mathematical texts written in the Common Mathematical Language, and checking the grammatical correctness of this computerisation. The framework also allows the user to take incremental steps towards the generation of a fully formalised document in such a way that the result can be checked by a proof checker. This report describes the language itself, its grammar, elements, characteristics and one of the concrete syntaxes: the plain syntax. An encoding of a large example is presented and explained. The main focus of the report lies on the implementation of the heart of the framework: MathLang-Core. While the architecture has changed to a more XML-centred design during the implementation, both the old and proposed new architectures are discussed. Four components can be distinguished in the framework: the parser, the abstract syntax tree, the checker and the printer. The latter has many different instances for many formats.

### Computerising Mathematical Text with MathLang

"... Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."

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Mathematical texts can be computerised in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerising mathematical texts which is flexible enough to connect the different approaches to computerisation, which allows various degrees of formalisation, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Four Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), Robert Lamar (since 2006)), and Christoph Zengler (since 2008) and over a dozen master’s degree and undergraduate

### Systems related to the FMathL vision

, 2010

"... There are already many automatic mathematical assistants that provide expert help in specialized domains. Known classes include computer algebra systems, automated deduction systems, modeling systems, matrix packages, numerical prototyping languages, decision trees for scientific computing software, ..."

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There are already many automatic mathematical assistants that provide expert help in specialized domains. Known classes include computer algebra systems, automated deduction systems, modeling systems, matrix packages, numerical prototyping languages, decision trees for scientific computing software, etc.. Such existing tools already provide partial functionality of the kind to be created in the project but only tied to specific applications, or with a limited scope. This document describes a number of current systems related to the FMathL vision, and some of their limitations when viewed in the light of this vision. The PI’s website (www.mat.univie.ac. at/~neum/FMathL.html) contains a large selection of additional resources and references to existing related systems. L ATEX

### INRIA Grenoble Rhône-Alpes

, 2010

"... LISE is a multidisciplinary project involving lawyers and computer scientists with the aim to put forward a set of methods and tools to (1) define software liability in a precise and unambiguous way and (2) establish such liability in case of incident. This paper provides an overview of the overall ..."

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LISE is a multidisciplinary project involving lawyers and computer scientists with the aim to put forward a set of methods and tools to (1) define software liability in a precise and unambiguous way and (2) establish such liability in case of incident. This paper provides an overview of the overall approach taken in the project based on a case study. The case study illustrates a situation where, in order to reduce legal uncertainties, the parties to a contract wish to include in the agreement specific clauses to define as precisely as possible the share of liabilities between them for the main types of failures of the system.

### Heriot-Watt University Edinburgh, Scotland

, 2005

"... The evolution of types and logic in the 20th century ∗ ..."

### MathLang, a framework for computerising

, 2007

"... Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected) ..."

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Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) University of West of England, Bristol 1A brief history • There are two influencing questions:

### MathLang, a framework for computerising

, 2007

"... Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the cont ..."

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Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) Saarbruecken, Germany 1A brief history • There are two influencing questions:

### MathLang, a framework for computerising

, 2006

"... – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ ..."

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– If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs and much of computer science in the early part of the 20th century was built by mathematicians and logicians. • There were also important inventions in computer science made by physicists (e.g., von Neumann) and others, but we ignore these in this talk. Brasilia University 1An example of a computable function/solvable problem • E.g., 1.5 chicken lay down 1.5 eggs in 1.5 days. • How many eggs does 1 chicken lay in 1 day? • 1.5 chicken lay 1.5 eggs in 1.5 days. • Hence, 1 chicken lay 1 egg in 1.5 days. • Hence, 1 chicken lay 2/3 egg in 1 day. Brasilia University 2Unsolvability of the Barber problem • which man barber in the village shaves all and only those men who do not shave themselves? • If John was the barber then – John shaves Bill ⇐ ⇒ Bill does not shave Bill – John shaves x ⇐ ⇒ x does not shave x – John shaves John ⇐ ⇒ John does not shave John • Contradiction. Brasilia University 3Unsolvability of the Russell set problem