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29
On a Formal Semantics of Tabular Expressions
 Science of Computer Programming
, 1997
"... In [15, 22, 25, 26] Parnas et al. advocate the use of relational model for documenting the intended behaviour of programs. In this method, tabular expressions (or tables) are used to improve readability so that formal documentation can replace conventional documentation. Parnas [23] describes sever ..."
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Cited by 21 (5 self)
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In [15, 22, 25, 26] Parnas et al. advocate the use of relational model for documenting the intended behaviour of programs. In this method, tabular expressions (or tables) are used to improve readability so that formal documentation can replace conventional documentation. Parnas [23] describes several classes of tables and provides their formal syntax and semantics. In this paper, an alternative, more general and more homogeneous semantics is proposed. The model covers all known types of tables used in Software Engineering. Contents 1 Introduction 2 2 Introductory examples 4 3 Relations 9 3.1 Cartesian Products, Functions, Relations . . . . . . . . . . . . . . . 9 3.2 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 InputOutput Relations . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Raw Table Skeleton 14 5 Cell Connection Graph and Medium Table Skeleton 15 6 Raw and Medium Table Elements 19 Supported by NSERC of Canada Grant 7 Well Do...
Syntactical Analysis of Total Termination
 In Proceedings of the 4th International Conference on Algebraic and Logic Programming
, 1994
"... Termination is an important issue in the theory of term rewriting. In general termination is undecidable. There are nevertheless several methods successful in special cases. In [5] we introduced the notion of total termination: basically terms are interpreted compositionally in a total wellfounded ..."
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Cited by 15 (8 self)
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Termination is an important issue in the theory of term rewriting. In general termination is undecidable. There are nevertheless several methods successful in special cases. In [5] we introduced the notion of total termination: basically terms are interpreted compositionally in a total wellfounded order, in such a way that rewriting chains map to descending chains. Total termination is thus a semantic notion. It turns out that most of the usual techniques for proving termination fall within the scope of total termination. This paper consists of two parts. In the first part we introduce a generalization of recursive path order presenting a new proof of its wellfoundedness without using Kruskal's theorem. We also show that the notion of total termination covers this generalization. In the second part we present some syntactical characterizations of total termination that can be used to prove that many term rewriting systems are not totally terminating and hence outside the scope of the...
Termination of Term Rewriting By Interpretation
, 1992
"... We investigate how to prove termination of term rewriting systems by interpretation of terms. This can be considered as a generalization of polynomial interpretations. A classification of types of termination is proposed built on properties in the semantic level. A transformation on term rewritin ..."
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Cited by 13 (3 self)
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We investigate how to prove termination of term rewriting systems by interpretation of terms. This can be considered as a generalization of polynomial interpretations. A classification of types of termination is proposed built on properties in the semantic level. A transformation on term rewriting systems eliminating distributive rules is introduced. Using this distribution elimination a new termination proof of SUBST from [9] is given.
Interpolating hereditarily indecomposable Banach spaces
"... Abstract. It is shown that every Banach space either contains ℓ 1 or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, L p (λ), 1 < p <∞, is a quotient of a H.I Banach space. Introduction.A Banach space X is said to be Hereditarily Indecomposable ( ..."
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Cited by 12 (3 self)
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Abstract. It is shown that every Banach space either contains ℓ 1 or it has an infinite dimensional closed subspace which is a quotient of a H.I. Banach space.Further on, L p (λ), 1 < p <∞, is a quotient of a H.I Banach space. Introduction.A Banach space X is said to be Hereditarily Indecomposable (H.I.) if for every pair of closed subspaces Y, Z of X with Y ∩ Z = {0}, Y + Z is not a closed subspace.(By “subspace”, in the sequel, we mean closed infinite dimensional subspace of X). The H.I spaces are a new and, as we believe, a fundamental class of Banach spaces. The celebrated example of a Banach space with no unconditional basic sequence, due to W. Gowers and B. Maurey ([GM]),
Maintenance Of Geometric Representations Through Space Decompositions
, 1997
"... The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of ..."
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Cited by 10 (4 self)
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The ability to transform between distinct geometric representations is the key to success of multiplerepresentation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG $ brep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations. 1. Motivation 1.1. Modern theory of representations The modern field of solid modeling owes much of its success to the theoretical foundations laid by members of the Production Automation Project at the University of Rochester in the 1970's. The history of these development...
Computational Adequacy for Recursive Types in Models of Intuitionistic Set Theory
 In Proc. 17th IEEE Symposium on Logic in Computer Science
, 2003
"... This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown ..."
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Cited by 9 (2 self)
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This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domaintheoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic ZermeloFraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambdacalculus with callbyvalue operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domaintheoretic models.
Constructive set theories and their categorytheoretic models
 IN: FROM SETS AND TYPES TO TOPOLOGY AND ANALYSIS
, 2005
"... We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicat ..."
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Cited by 9 (0 self)
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We advocate a pragmatic approach to constructive set theory, using axioms based solely on settheoretic principles that are directly relevant to (constructive) mathematical practice. Following this approach, we present theories ranging in power from weaker predicative theories to stronger impredicative ones. The theories we consider all have sound and complete classes of categorytheoretic models, obtained by axiomatizing the structure of an ambient category of classes together with its subcategory of sets. In certain special cases, the categories of sets have independent characterizations in familiar categorytheoretic terms, and one thereby obtains a rich source of naturally occurring mathematical models for (both predicative and impredicative) constructive set theories.
Solid Modeling
, 2001
"... This article revisits the main ideas and foundations of solid modeling in engineering, summarizes recent progress and bottlenecks, and speculates on possible future directions ..."
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Cited by 6 (1 self)
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This article revisits the main ideas and foundations of solid modeling in engineering, summarizes recent progress and bottlenecks, and speculates on possible future directions
Contributions to the Theory of Rough Sets
, 1999
"... . We study properties of rough sets, that is, approximations to sets of records in a database or, more formally, to subsets of the universe of an information system. A rough set is a pair hL; Ui such that L; U are definable in the information system and L ` U . In the paper, we introduce a langua ..."
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Cited by 6 (1 self)
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. We study properties of rough sets, that is, approximations to sets of records in a database or, more formally, to subsets of the universe of an information system. A rough set is a pair hL; Ui such that L; U are definable in the information system and L ` U . In the paper, we introduce a language, called the language of inclusionexclusion, to describe incomplete specifications of (unknown) sets. We use rough sets in order to define a semantics for theories in the inclusionexclusion language. We argue that our concept of a rough set is closely related to that introduced by Pawlak. We show that rough sets can be ordered by the knowledge ordering (denoted kn ). We prove that Pawlak's rough sets are characterized as kn greatest approximations. We show that for any consistent (that is, satisfiable) theory T in the language of inclusionexclusion there exists a kn greatest rough set approximating all sets X that satisfy T . For some classes of theories in the language of i...
Modal Logics For Products Of Topologies
 Studia Logica
, 2004
"... We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 S4. We axiomatize t ..."
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Cited by 4 (0 self)
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We introduce the horizontal and vertical topologies on the product of topological spaces, and study their relationship with the standard product topology. We show that the modal logic of products of topological spaces with horizontal and vertical topologies is the fusion S4 S4. We axiomatize the modal logic of products of topological spaces with horizontal, vertical, and standard product topologies. We prove that both of these logics are complete for the product of rational numbers Q Q with the appropriate topologies.