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Understanding Mathematical Discourse
 DIALOGUE. AMSTERDAM UNIVERSITY
, 1999
"... Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers ..."
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Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers a welldefined set of discourse relations and forces/allows us to apply mathematical reasoning. We give a brief discussion on selected linguistic phenomena of mathematical discourse, and an analysis from the mathematician’s point of view. Requirements for a theory of discourse representation are given, followed by a discussion of proofs plans that provide necessary context and structure. A large part of semantics construction is defined in terms of proof plan recognition and instantiation by matching and attaching.
Verifying textbook proofs
 Technische Universität
, 1998
"... In the first half of the 1960s, Paul Abrahams implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, “was primarily directed towards the verification of textbook proofs, i.e., proofs resembling those that normally appear in mathematical t ..."
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In the first half of the 1960s, Paul Abrahams implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, “was primarily directed towards the verification of textbook proofs, i.e., proofs resembling those that normally appear in mathematical textbooks and journals”. Abrahams did not succeed. If, so Abrahams, “a computer were to check a textbook proof verbatim, it would require far more intelligence than is possible with the current state of the programming art”. Therefore, so Abrahams, “the user must create a rigorous, i.e., completely formalised, proof that he believes represents the intent of the author of the textbook proof, and use the computer to check this rigorous proof”. Abrahams points further out that “it it a trivial task to program a computer to check a rigorous proof; however, it it not a trivial task to create such a proof from a textbook proof”. Abrahams was right. In all later projects, proofs had to be written in a formal language in order to verify them. One wellknown example is the the Automath project: van BenthemJutting formalised a whole textbook of Landau, the ‘Grundlagen der Analysis’, into a formal language, autqe [8]. A second example is the Mizar project [5]: proofs have
Towards the Mechanical Verification of Textbook Proofs
 In Proceedings of the 7th. Workshop on Logic, Language, Information and Computation (WOLLIC2000
, 2000
"... Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an indepth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends ..."
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Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an indepth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends and integrates stateoftheart technologies from Natural Language Processing (Discourse Representation Theory) and Automated Reasoning (Proof Planning) in a novel and promising way, having the potential to initiate progress in both of these disciplines.
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...
Checking Textbook Proofs
 Int. Workshop on FirstOrder Theorem Proving (FTP'98), Technical Report E1852GS981
, 1998
"... . Our longrange goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and deduction perspective and give an indepth analysis for a sample textbook proof. A three phase model for proof understanding is developed: parsing, str ..."
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. Our longrange goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and deduction perspective and give an indepth analysis for a sample textbook proof. A three phase model for proof understanding is developed: parsing, structuring and refining. It shows that the combined application of techniques from both NLP and AR is quite successful. Moreover, it allows to uncover interesting insights that might initiate progress in both AI disciplines. Keywords: automated reasoning, natural language processing, discourse analysis 1 Introduction In [12], John McCarthy notes that "Checking mathematical proofs is potentially one of the most interesting and useful applications of automatic computers". In the first half of the 1960s, one of his students, namely Paul Abrahams, implemented a Lisp program for the machine verification of mathematical proofs [1]. The program, named Proofchecker, "was primarily directed towar...
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...
COMPUTER UNDERSTANDING OF MATHEMATICAL PROOFS
, 1977
"... Mathematical proofs constitute a mixture of formulas with a subset of natural language. They can be represented as a sequence of lines expressible in the symbolism of predicate calculus. The transition from step to step may depend on a series of logical manipulations and/or on intricate mathematical ..."
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Mathematical proofs constitute a mixture of formulas with a subset of natural language. They can be represented as a sequence of lines expressible in the symbolism of predicate calculus. The transition from step to step may depend on a series of logical manipulations and/or on intricate mathematical knowledge associated with the domain of the proof. The organization of the proof may depend on different conventions adopted by mathematicians in communication with each other. This paper deals with problems involved in following the mathematical argument along those lines. Some of the ideas were implemented as a part of a system for teaching axiomatic set theory to college students. The most powerful and frequently used rules of inference utilize a resolution theorem prover. To the best of our knowledge this is the only resolution theorem prover, perhaps the only general purpose theorem prover used in actual production.