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145
Tabular Representation of Relations
, 1992
"... Multidimensional mathematical expressions, called tables, have proven to be useful for documenting digital systems. This paper describes 10 classes of tables, giving their syntax and semantics. Several abbreviations that can be useful in tables are introduced. Simple examples are provided. 1 Int ..."
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Cited by 51 (11 self)
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Multidimensional mathematical expressions, called tables, have proven to be useful for documenting digital systems. This paper describes 10 classes of tables, giving their syntax and semantics. Several abbreviations that can be useful in tables are introduced. Simple examples are provided. 1 Introduction In earlier papers, [1,2], we have shown how the documentation required for the professional construction and use of computing systems can consist of descriptions of a set of mathematical relations. Those papers discuss the documents very abstractly; the contents of the documents are specified without restricting the notations or formats to be used. This paper complements the earlier papers by defining multidimensional notations (which we call tabular expressions) that have proven useful for describing the specified mathematical functions in practical applications [3, 4, 5, 6, 7, 8]. A companion paper [9], presents an interpretation of logical expressions that is designed for these...
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 45 (17 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
The Power of Vacillation in Language Learning
, 1992
"... Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there ..."
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Cited by 44 (11 self)
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Some extensions are considered of Gold's influential model of language learning by machine from positive data. Studied are criteria of successful learning featuring convergence in the limit to vacillation between several alternative correct grammars. The main theorem of this paper is that there are classes of languages that can be learned if convergence in the limit to up to (n+1) exactly correct grammars is allowed but which cannot be learned if convergence in the limit is to no more than n grammars, where the no more than n grammars can each make finitely many mistakes. This contrasts sharply with results of Barzdin and Podnieks and, later, Case and Smith, for learnability from both positive and negative data. A subset principle from a 1980 paper of Angluin is extended to the vacillatory and other criteria of this paper. This principle, provides a necessary condition for circumventing overgeneralization in learning from positive data. It is applied to prove another theorem to the eff...
Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
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Cited by 42 (20 self)
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
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Cited by 33 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
Chaincomplete posets and directed sets with applications. Algebra univers
, 1976
"... Let a poset P be called chaincomplete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets atisfying some sort of chaincompleteness condition (see [,11, [31, [6], I71, [17], [,181, ['191, [,211, [,221), and used them in a variety o ..."
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Cited by 27 (0 self)
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Let a poset P be called chaincomplete when every chain, including the empty chain, has a sup in P. Many authors have investigated properties of posets atisfying some sort of chaincompleteness condition (see [,11, [31, [6], I71, [17], [,181, ['191, [,211, [,221), and used them in a variety of applications. In this paper we study the
Lipschitz maps and nets in Euclidean space
 Geom. Funct. Anal
, 1998
"... In this paper we discuss the following three questions. 1. Given a realvalued function f ∈ L ∞ (R n) with inf f(x)> 0, is there a biLipschitz homeomorphism φ: R n → R n such that the Jacobian determinant det Dφ = f? ..."
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Cited by 24 (0 self)
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In this paper we discuss the following three questions. 1. Given a realvalued function f ∈ L ∞ (R n) with inf f(x)> 0, is there a biLipschitz homeomorphism φ: R n → R n such that the Jacobian determinant det Dφ = f?