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Little Theories
 Automated DeductionCADE11, volume 607 of Lecture Notes in Computer Science
, 1992
"... In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to forma ..."
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Cited by 48 (15 self)
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In the "little theories" version of the axiomatic method, different portions of mathematics are developed in various different formal axiomatic theories. Axiomatic theories may be related by inclusion or by theory interpretation. We argue that the little theories approach is a desirable way to formalize mathematics, and we describe how imps, an Interactive Mathematical Proof System, supports it.
Theory Interpretation in Simple Type Theory
 HIGHERORDER ALGEBRA, LOGIC, AND TERM REWRITING, VOLUME 816 OF LECTURE NOTES IN COMPUTER SCIENCE
, 1993
"... Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admit ..."
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Cited by 36 (17 self)
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Theory interpretation is a logical technique for relating one axiomatic theory to another with important applications in mathematics and computer science as well as in logic itself. This paper presents a method for theory interpretation in a version of simple type theory, called lutins, which admits partial functions and subtypes. The method is patterned on the standard approach to theory interpretation in rstorder logic. Although the method is based on a nonclassical version of simple type theory, it is intended as a guide for theory interpretation in classical simple type theories as well as in predicate logics with partial functions.
On Solving Presburger and Linear Arithmetic with SAT
 In Proc. of Formal Methods in ComputerAided Design (FMCAD 2002), LNCS
, 2002
"... We show a reduction to propositional logic from quantifierfree Presburger arithmetic, and disjunctive linear arithmetic, based on FourierMotzkin elimination. While the complexity of this procedure is not better than competing techniques, it has practical advantages in solving verification problems ..."
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Cited by 26 (2 self)
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We show a reduction to propositional logic from quantifierfree Presburger arithmetic, and disjunctive linear arithmetic, based on FourierMotzkin elimination. While the complexity of this procedure is not better than competing techniques, it has practical advantages in solving verification problems. It also promotes the option of deciding a combination of theories by reducing them to this logic.
Proof Representations in Theorem Provers
, 1998
"... s and compressed postscript files are available via http://svrc.it.uq.edu.au Proof Representations in Theorem Provers Geoffrey Norman Watson Abstract This is a survey of some of the proof representations used by current theorem provers. The aim of the survey is to ascertain the range of mechani ..."
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Cited by 4 (0 self)
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s and compressed postscript files are available via http://svrc.it.uq.edu.au Proof Representations in Theorem Provers Geoffrey Norman Watson Abstract This is a survey of some of the proof representations used by current theorem provers. The aim of the survey is to ascertain the range of mechanisms used to represent proofs and the purposes to which these representations are put. This is done within a simple framework. It examines both internal and external representations, although the focus is on representations that could be exported to an external proof checker. A number of examples from various provers are given in a series of appendices. 1 Contents 1 Introduction 3 2 Aim of the Survey 3 2.1 Why Construct Proofs . . . . . . . . . 3 2.2 Levels of Representation . . . . . . . . 4 3 Scope of the Survey 5 3.1 Ergo . . . . . . . . . . . . . . . . . . . 5 3.2 HOL . . . . . . . . . . . . . . . . . . 6 3.3 Isabelle . . . . . . . . . . . . . . . . . 7 3.4 Nuprl . . . . . . . . . . . ...
Proof Script Pragmatics in IMPS
 Automated Deduction CADE12, volume 814 of Lecture Notes in Computer Science
, 1994
"... . This paper introduces the imps proof script mechanism and some practical methods for exploiting it. 1 Introduction imps, an Interactive Mathematical Proof System [4, 2], is intended to serve three ultimate purposes: { To provide mathematics education with a mathematics laboratory for students ..."
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Cited by 4 (2 self)
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. This paper introduces the imps proof script mechanism and some practical methods for exploiting it. 1 Introduction imps, an Interactive Mathematical Proof System [4, 2], is intended to serve three ultimate purposes: { To provide mathematics education with a mathematics laboratory for students to develop axiomatic theories, proofs, and rigorous methods of symbolic computation. { To provide mathematical research with mechanized support covering a range of concrete and abstract mathematics, eventually with the help of a large theory library of formal mathematics. { To allow applied formal methods to use exible approaches to formalizing problem domains and proof techniques, in showing software or hardware correctness. Thus, the goal of imps is to provide mechanical support for traditional methods and activities of mathematics, and for traditional styles of mathematical proof. Other automated theorem provers may be intended for quite dierent sorts of problems, and they can theref...
Interpretation and Instantiation of Theories for Reasoning about Formal Specifications
 QUEENSLAND UNIVERSITY OF TECHNOLOGY
, 1997
"... In this paper an outline is given of an approach to formally reasoning about importation, parameterisation and instantiation of specifications written in a modular extension of the Z language (called Sum). Interpretation and instantiation of theories in first order logic are well understood. We illu ..."
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Cited by 4 (1 self)
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In this paper an outline is given of an approach to formally reasoning about importation, parameterisation and instantiation of specifications written in a modular extension of the Z language (called Sum). Interpretation and instantiation of theories in first order logic are well understood. We illustrate how to use these results directly to provide a framework within which we can soundly and efficiently reason about modular specifications. A reasoning environment within the Ergo 4:1 theorem prover has been constructed that provides the theory management, construction and extension facilities needed to support such a reasoning process. Sum specifications are mapped to the appropriate Ergo structures by a straightforward translation process. A simple example in Sum is presented to demonstrate the use of these theory extension mechanisms. As far as the authors are aware, no other system offers interpreted automated support for reasoning about parameterisation and instantiation of modular...
Exploiting Parallelism in Interactive Theorem Provers
 Proceedings of TPHOLs, volume 1479 of LNCS
, 1998
"... . This paper reports on the implementation and analysis of the MP refiner, the first parallel interactive theorem prover. The MP refiner is a shared memory multiprocessor implementation of the inference engine of Nuprl. The inference engine of Nuprl is called the refiner. The MP refiner is a co ..."
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Cited by 4 (1 self)
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. This paper reports on the implementation and analysis of the MP refiner, the first parallel interactive theorem prover. The MP refiner is a shared memory multiprocessor implementation of the inference engine of Nuprl. The inference engine of Nuprl is called the refiner. The MP refiner is a collection of threads operating as sequential refiners running on separate processors. Concurrent tactics exploit parallelism by spawning tactics to be evaluated by other refiner threads simultaneously. Tests conducted with the MP refiner running on a four processor Sparc shared memory multiprocessor reveal that parallelism at the inference rule level can significantly decrease the elapsed time of constructing proofs interactively. 1 Introduction An interactive theorem prover is a computer program that employs automated deduction to construct proofs with the aid of a user. Many interactive theorem provers require users to supply programs, called tactics, to carry out inference. Tacti...
A General Method for Safely Overwriting Theories in Mechanized Mathematics Systems
 Lecture Notes in Computer Science (Proc. Intl. Zurich Sem. Digital Comm.). Spinger
, 1994
"... We propose a general method for overwriting theories with model conservative extensions in mechanized mathematics systems. Model conservative extensions, which include the denition of new constants and the introduction of new abstract datatypes, are \safe" because they preserve models as well as ..."
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Cited by 2 (1 self)
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We propose a general method for overwriting theories with model conservative extensions in mechanized mathematics systems. Model conservative extensions, which include the denition of new constants and the introduction of new abstract datatypes, are \safe" because they preserve models as well as consistency. The method employs the notions of theory interpretation and theory instantiation. It is illustrated using manysorted rstorder logic, but it works for a variety of underlying logics. Supported by the MITRESponsored Research program. 1 1 Introduction Mathematical reasoning is always performed in some mathematical context consisting of vocabulary and assumptions. The formal counterpart of a context is a theory consisting of a formal language plus a set of sentences of the language called axioms. (We will denote a theory T by the pair (L; ) where L is the formal language of T and is the set of axioms of T .) An extension of a theory T is any theory T 0 obtained by ...
Comparison of IMPS, PVS and Larch with respect to theory treatment and modularization
, 1996
"... This paper serves as a report of the literature study I performed between November '95 and February '96 concerning concepts for Isabelle Modules. It compares three proof systems which are recent and successful enough to serve as exemplaries for a study of their theory handling. ..."
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Cited by 1 (0 self)
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This paper serves as a report of the literature study I performed between November '95 and February '96 concerning concepts for Isabelle Modules. It compares three proof systems which are recent and successful enough to serve as exemplaries for a study of their theory handling.
Discoveries and Experiments in the Automation of Mathematical Reasoning
, 2002
"... vii List of Figures xii Chapter 1. ..."