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14
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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Cited by 7 (6 self)
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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. 1
Bidirectional Natural Deduction
 AI*IA Notizie
, 1993
"... The goal of this paper is to present a theorem prover able to perform both forward and backward reasoning supported by a well defined formal system. This system for bidirectional reasoning has been proved equivalent to Gentzen's classical system of propositional natural deduction. This pap ..."
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Cited by 4 (2 self)
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The goal of this paper is to present a theorem prover able to perform both forward and backward reasoning supported by a well defined formal system. This system for bidirectional reasoning has been proved equivalent to Gentzen's classical system of propositional natural deduction. This paper, primarily aimed at developing a deeper theoretical understanding of bidirectional reasoning, provides basic concepts to be incorporated into an innovative theorem prover to support interactive proofs construction in general domains. 1
Verifying Textbook Proofs
 INT. WORKSHOP ON FIRSTORDER THEOREM PROVING (FTP'98), TECHNICAL REPORT E1852GS981
, 1998
"... ..."
Computing Presuppositions and Implicatures in Mathematical Discourse
"... In any wellwritten mathematical discourse a certain amount of mathematical and metamathematical knowledge is presupposed and implied. We give an account on presuppositions and implicatures in mathematical discourse and describe an architecture that allows to e ectively interpret them. Our approach ..."
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Cited by 1 (1 self)
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In any wellwritten mathematical discourse a certain amount of mathematical and metamathematical knowledge is presupposed and implied. We give an account on presuppositions and implicatures in mathematical discourse and describe an architecture that allows to e ectively interpret them. Our approach heavily relies on proof methods that capture common patterns of argumentation in mathematical discourse. This pragmatic information provides a highlevel strategic discourse understanding and allows to compute the presupposed and implied information.
Towards the Mechanical Verification of Textbook Proofs
"... 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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Cited by 1 (1 self)
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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...
An Intelligent Tutoring System for Induction Proofs
"... interfaces. We will specify and implement abstract interfaces for the student model, the dialogue history and the problem state. 4 Diagnosis and Therapy. We will view the diagnosis task as a plan recognition problem. We will explore the possibilities (i) of using proof plans and Oyster/Clam's proof ..."
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interfaces. We will specify and implement abstract interfaces for the student model, the dialogue history and the problem state. 4 Diagnosis and Therapy. We will view the diagnosis task as a plan recognition problem. We will explore the possibilities (i) of using proof plans and Oyster/Clam's proof planning facility to support the diagnosis and therapy task; (ii) of adapting a probabilistic plan recognition approach using Bayes's belief networks. This is the approach taken in Andes, a physics tutoring system [GCV98] with similar domain properties. Since specifying and implementing the diagnosis and therapy module of Intuition will be a significant project in itself, we will recruit a PhD student who will concentrate his research efforts solely on these components: 4.1 Knowledge acquisition (6pm), literature survey (3pm), and summarisation of intermediate results (1pm). 4.2 Diagnosis module (9pm), and summarisation of intermediate results (1pm). 4.3 Therapy module (9pm), and summar...