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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.
Parsing formulae in textbook proofs
 Tilburg University
, 1999
"... Our longrange goal is to implement a program for the machine verification of textbook proofs. The problem was first tackled by Simon [Sim88]. However, Simon does not claim to have used or developed an adequate theory for semantics construction. A prerequisite for parsing textbook proofs is to being ..."
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Our longrange goal is to implement a program for the machine verification of textbook proofs. The problem was first tackled by Simon [Sim88]. However, Simon does not claim to have used or developed an adequate theory for semantics construction. A prerequisite for parsing textbook proofs is to being able to parse terms and formulae that occur in these proofs. We analyze a typical textbook proof and describe some of the linguistic phenomena one has to cope with, focusing on problems that involve the handling of terms and formulae. We propose, contrarily to Simon, an adequate theory for semantics construction, namely DRT, which, however, must be adapted to fit our needs. 2 Linguistic analysis Fig. 1 depicts a typical proof taken from a standard textbook on elementary number theory. We analyze it from the linguistics perspective focusing on potential difficulties: Constants and variables have a domain and scope which extends across text and formulae. Obviously, the occurrences of p in p  ab, p  a and p  b (Fig. 1, line 4) refer to their first occurrence in if p is prime (same line). However, it is not required that the occurrences of p, a and b in line 8 refer to the occurrences of p, a and b in line 4. The term abc...l (line 8) describes 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 b1 1... qbj j the terms p a1 (line 13) refers to the same mathematical entity as qb1 1 qb2 2...qbj j
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
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.
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...
A Framework for the Creation and Use of a Knowledge Base of Mathematical Theorems and Definitions
, 1995
"... A mathematician knows thousands of theorems and definitions and is able to choose just the needed results and use them at just the right time in the theoremproving process. The problem of codifying some bit of this process is the topic of this paper. IPR is an automatic theoremproving system i ..."
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A mathematician knows thousands of theorems and definitions and is able to choose just the needed results and use them at just the right time in the theoremproving process. The problem of codifying some bit of this process is the topic of this paper. IPR is an automatic theoremproving system intended particularly for use in mathematics. It discovers the proofs of theorems in mathematics by applying known theorems and definitions from a knowledge base. Theorems and definitions are stored in the knowledge base in the form of "sequents" rather than formulas or rewrite rules. The sequentsinto which a theorem is reduced before being put into the knowledge base consistently mirror the ways a human might use the theorem. Because the data are in this form, natural fetching algorithms can be used to search the knowledge base for the most useful theorem to be used in the theoremproving process. 1 Introduction The theory presented here, implemented in IPR, provides a natur...