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Parsing Formulae in Textbook Proofs
 Tilburg University
, 1999
"... bes the factorization of objects a; b; c; : : : ; l. Albeit the missing c, the phrase a; b; : : : l (line 9) enumerates also the factors a; b; c; : : : ; l. Equally, q b 1 1 : : : q b j j (line 13) refers to the same mathematical entity as q b 1 1 q b 2 2 : : : q b j (line 11). Clearly, th ..."
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bes the factorization of objects a; b; c; : : : ; l. Albeit the missing c, the phrase a; b; : : : l (line 9) enumerates also the factors a; b; c; : : : ; l. Equally, q b 1 1 : : : q b j j (line 13) refers to the same mathematical entity as q b 1 1 q b 2 2 : : : q b j (line 11). Clearly, the terms p a 1 1 : : : p a \Gammab i : : : p a k k and p a 1 1 p a 2 2 : : : p a k k do not describe the same object. The proper name the fu
MathNat Mathematical Text in a Controlled Natural Language
"... Abstract. The MathNat 1 project aims at being a first step towards automatic formalisation and verification of textbook mathematical text. First, we develop a controlled language for mathematics (CLM) which is a precisely defined subset of English with restricted grammar and dictionary. To make CLM ..."
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Abstract. The MathNat 1 project aims at being a first step towards automatic formalisation and verification of textbook mathematical text. First, we develop a controlled language for mathematics (CLM) which is a precisely defined subset of English with restricted grammar and dictionary. To make CLM natural and expressive, we support some complex linguistic features such as anaphoric pronouns and references, rephrasing of a sentence in multiple ways producing canonical forms and the proper handling of distributive and collective readings. Second, we automatically translate CLM into a system independent formal language (MathAbs), with a hope to make MathNat accessible to any proof checking system. Currently, we translate MathAbs into equivalent first order formulas for verification. In this paper, we give an overview of MathNat, describe the linguistic features of CLM, demonstrate its expressive power and validate our work with a few examples.
Verifying Textbook Proofs
 INT. WORKSHOP ON FIRSTORDER THEOREM PROVING (FTP'98), TECHNICAL REPORT E1852GS981
, 1998
"... ..."
The Creation and Use of a Knowledge Base of Mathematical Theorems and Definitions
, 1995
"... IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or ..."
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IPR is an automatic theoremproving system intended particularly for use in higherlevel mathematics. It discovers the proofs of theorems in mathematics applying known theorems and definitions. Theorems and definitions are stored in the knowledge base in the form of sequents rather than formulas or rewrite rules. Because there is more easilyaccessible information in a sequent than there is in the formula it represents, a simple algorithm can be used to search the knowledge base for the most useful theorem or definition to be used in the theoremproving process. This paper describes how the sequents in the knowledge base are formed from theorems stated by the user and how the knowledge base is used in the theoremproving process. An example of a theorem proved and the English proof output are also given. 1 Introduction The motivating goal behind this work is to develop a theoremproving system which will be useful to both an expert and a nonexpert in the attempt to prove theorems in ...
A DRTbased approach for formula parsing in textbook proofs
 IN THIRD INTERNATIONAL WORKSHOP ON COMPUTATIONAL SEMANTICS (IWCS3
, 1999
"... Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs, however, only a limited quantity of knowledge is necessary. In addition, we have gained something very essent ..."
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Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs, however, only a limited quantity of knowledge is necessary. In addition, we have gained something very essential: inference. A prerequisite for parsing textbook proofs is to being able to parse formulae that occur in these proofs. Parsing formulae alone in the empty context is trivial. But within the context of textbook proofs the task soon gets complex. Several kinds of references from the text to parts or sets of terms and formulae have to be handled. We describe some of the linguistic phenomena that occur in mathematical texts. The focus is on our treatment of term reference which is embedded in the DRT.
On the Relationship Between Structure and Reference in Mathematical Discourse
"... We study the relationship between the structure of discourse and the use of referring expressions in the mathematical domain. We address linguistic, algorithmic as well as representation issues. We show how referential expressions refer to mathematical statements and how a knowledge intensive approa ..."
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We study the relationship between the structure of discourse and the use of referring expressions in the mathematical domain. We address linguistic, algorithmic as well as representation issues. We show how referential expressions refer to mathematical statements and how a knowledge intensive approach, domain reasoning with the use of proof plans, are used for discourse understanding. We propose to represent discourse plans as underspecified discourse representation structures being selected and instantiated during discourse processing. Our main emphasis is on the handling of abstract discourse entities. 1 Motivation We have the following practical application in mind: the automatic verification of mathematical textbook proofs. Imagine a program that understands mathematical discourse. Such a device reads proofs, say mathematical arguments taken from textbooks on elementary mathematics, and is then able to communicate its knowledge about what it has read and analyzed. It answers ques...
Structuring Textbook Proofs
, 1999
"... We propose a promising research problem, the machine verification of textbook proofs. It shows that textbook proofs are a sufficiently complex and highly structured form of discourse, embedded in a welldefined and wellunderstood domain, thus offering an ideal domain for discourse analysis. Because ..."
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We propose a promising research problem, the machine verification of textbook proofs. It shows that textbook proofs are a sufficiently complex and highly structured form of discourse, embedded in a welldefined and wellunderstood domain, thus offering an ideal domain for discourse analysis. Because recognizing the structure of a proof is a prerequisite for verifying the correctness of a given mathematical argument, we define a four component model of discourse segmentation. 1 Introduction In order to advance our knowledge of discourse understanding, we have to 1. tackle realworld problems, that is study discourse that is sufficiently complex; 2. build ontologies and formalize knowledge about the domain of discourse; 3. seriously address representation issues; 4. apply reasoning techniques. This is nothing new. But did you ever see a natural language system where each of these four issues has been successfully addressed? Contrarily, many research resources has been spent on a family...