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Proofassistants using Dependent Type Systems
, 2001
"... this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs ..."
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Cited by 48 (4 self)
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this article we will not attempt to describe all the dierent possible choices of type theories. Instead we want to discuss the main underlying ideas, with a special focus on the use of type theory as the formalism for the description of theories including proofs
A Theorem Prover for a Computational Logic
, 1990
"... We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of line ..."
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Cited by 24 (0 self)
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We briefly review a mechanical theoremprover for a logic of recursive functions over finitely generated objects including the integers, ordered pairs, and symbols. The prover, known both as NQTHM and as the BoyerMoore prover, contains a mechanized principle of induction and implementations of linear resolution, rewriting, and arithmetic decision procedures. We describe some applications of the prover, including a proof of the correct implementation of a higher level language on a microprocessor defined at the gate level. We also describe the ongoing project of recoding the entire prover as an applicative function within its own logic.
Formal proofâ€”theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Cited by 12 (1 self)
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
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 mathematicianâ€™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
Writing PVS proof strategies
 Design and Application of Strategies/Tactics in Higher Order Logics (STRATA 2003), number CP2003212448 in NASA Conference Publication
, 2003
"... Abstract. PVS (Prototype Verification System) is a comprehensive framework for writing formal logical specifications and constructing proofs. An interactive proof checker is a key component of PVS. The capabilities of this proof checker can be extended by defining proof strategies that are similar t ..."
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Abstract. PVS (Prototype Verification System) is a comprehensive framework for writing formal logical specifications and constructing proofs. An interactive proof checker is a key component of PVS. The capabilities of this proof checker can be extended by defining proof strategies that are similar to LCFstyle tactics. Commonly used proof strategies include those for discharging typechecking proof obligations, simplification and rewriting using decision procedures, and various forms of induction. We describe the basic building blocks of PVS proof strategies and provide a pragmatic guide for writing sophisticated strategies. 1
Partial Formalizations And The Lemmings Game
, 1998
"... The computer game Lemmings can serve as a new Drosophila for AI research connecting logical formalizations with information that is incompletely formalizable in practice. In this article we discuss the features of the Lemmings world that make it a challenge to both experimental and theoretical AI an ..."
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The computer game Lemmings can serve as a new Drosophila for AI research connecting logical formalizations with information that is incompletely formalizable in practice. In this article we discuss the features of the Lemmings world that make it a challenge to both experimental and theoretical AI and present some steps toward formalizing the game using situation calculus. Preliminary versions of this paper lack important formulas and references to the literature on reactive AI and to computer vision. The formulas present are not yet integrated into a coherent whole. 1 Work partly supported by ARPA (ONR) grant N000149410775 and partly done while the author was Meyerhoff Visiting Professor at the Weizmann Institute of Science, Rehovot, Israel 1 This document is reachable from the WWW page http://wwwformal.stanford.edu/jmc/home.html. 1 Contents 1 Introduction 3 2 Description of the Lemmings Games 3 3 What Needs to be Formalized? 5 4 An Example of Lemming Play 6 5 Physics of t...
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
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.
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...