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Notes on Refinement, Interpolation and Uniformity.
"... The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The abs ..."
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The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The absence of the desirable interpolation properties from many logics that have been used in refinement, motivates a thorough investigation of methods to expand a specification formalism orthogonally, so that the critical uniform interpolants become available. A potential breakthrough is outlined in this paper. 1. A refinement paradigm Let us consider program development by means of stepwise refinements. One postulates some abstract data typelike specification 1 (ADT), suitable for the problem at hand, which has to be implemented on the available system. The end product consists of (the text of) an abstract program manipulating the postulated ADT, together with a suite of (texts of) modules implementin...
Compositional Structuring in the BMethod: A Logical Viewpoint of the Static Context
 in Proceedings of the International Conference of B and Z Users ZB2000
, 2000
"... The BMethod provides a collection of structuring mechanisms which support information hiding, modularisation and compositionality of module operations, although, in order to achieve compositionality and independent (parallel) refinement, sharing is restricted in B. In this paper we elaborate some n ..."
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Cited by 4 (3 self)
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The BMethod provides a collection of structuring mechanisms which support information hiding, modularisation and compositionality of module operations, although, in order to achieve compositionality and independent (parallel) refinement, sharing is restricted in B. In this paper we elaborate some noninterference and compositionality assumptions that underlie structuring mechanisms such as uses, sees and imports and show how they may be violated by inducing emerging properties which alter the context of the used, seen or imported machine. We discuss how such situations can be avoided by considering necessary and sufficient conditions for logical conservativeness and modularisation. As proof obligations, these conditions ensure that the properties of the context of the seen, used or imported component are conserved, i.e. that they are preserved but not enriched. From a logical viewpoint, these proof obligations require that the uniform interpolant of the contextual extension axioms is implied by the base context.
Integrating Heterogeneous Formalisms: Framework and Application
, 1999
"... In this paper we study the problem of integrating heterogeneous formal notations, emphasising the integration of the axiomatic (logical) semantics. We present a general method for integrating the consequence systems of interrelated heterogeneous formal notations and explain why and when the resultin ..."
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Cited by 1 (1 self)
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In this paper we study the problem of integrating heterogeneous formal notations, emphasising the integration of the axiomatic (logical) semantics. We present a general method for integrating the consequence systems of interrelated heterogeneous formal notations and explain why and when the resulting compound muiltilogical consequence system may need the additional support of structural (&quot;locality&quot;) axioms in order to incorporate the interrelations between the components. We illustrate the use of this method as a basis for the integration of VDM and B formalisms based on LPF and Classical logic.
Uniformity, Interpolation and Module specification in a Development Workspace
 the proceedings of the TFM'98 workshop
, 1997
"... . Interpolation and Schematic Reasoning are shown to underlie critical and somewhat complementary aspects of designing and (syntactically) manipulating specification modules. In addition, the presence of a Uniform presentation of interpolants facilitates the specification of modules. Also, the a ..."
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. Interpolation and Schematic Reasoning are shown to underlie critical and somewhat complementary aspects of designing and (syntactically) manipulating specification modules. In addition, the presence of a Uniform presentation of interpolants facilitates the specification of modules. Also, the ability to encapsulate and manipulate Uniform Schemata may assist us in reasoning with (abstractions of) hidden data. Unfortunately, most formalisms that have been used in fundamental approaches to software engineering lack uniform interpolation and do not directly support schematic reasoning. This paper reveals the critical role of uniform interpolants and uniform schemata from the perspective of modularity, and quotes a general construction indicating that a potentially large class of calculi can be extended conservatively so that a uniform presentation of the critical interpolants becomes available and the manipulation of uniform schemata is supported. 1 Introduction There is a ...
Modularity and Interpolation in a Development Workspace.
, 1997
"... The potential benefits of a uniform version of interpolation are hindered by its absence from many expressive logics. This motivates a thorough investigation of appropriate expansions of known entailments so that an easytoderive, uniform presentation of the interpolants is supported. This paper p ..."
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The potential benefits of a uniform version of interpolation are hindered by its absence from many expressive logics. This motivates a thorough investigation of appropriate expansions of known entailments so that an easytoderive, uniform presentation of the interpolants is supported. This paper presents the skeleton of a general construction and indicates that a potentially large class of entailments can be extended so that a uniform presentation of the interpolants is available. 1 Introduction There is a well established relation between interpolation [8] and modularity properties of refinements [23, 24, 29, 5, 32, 40, 39, 12, 11] and databases [25]. On the other hand, many logics that have been used in refinement or databases lack the desirable interpolation properties. To compensate for this inadequacy, several groups of researchers have proposed techniques to restrict these logics to fragments that have the desirable modularity properties. Some of these enterprises have focuse...
Semantics of First Order Parametric Specifications
"... Parametricity is one of the most effective ways to achieve compositionality and reuse in software development. Parametric specifications have been thoroughly analyzed in the algebraic setting and are by now a standard part of most software development toolkits. However, an effort towards classifying ..."
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Parametricity is one of the most effective ways to achieve compositionality and reuse in software development. Parametric specifications have been thoroughly analyzed in the algebraic setting and are by now a standard part of most software development toolkits. However, an effort towards classifying, specifying and refining algorithmic theories, rather than mere datatypes, quickly leads beyond the realm of algebra, and often to full first order theories. We extend the standard semantics of parametric specifications to this more general setting. The familiar semantic characterization of parametricity in the algebraic case is expressed in terms of the free functor, i.e. using the initial models. In the general case, initial models may not exist, and the free functor is not available. Various syntactic, semantic, and abstract definitions of parametricity have been offered, but their exact relationships are often unclear. Using the methods of categorical model theory, we establish the eq...
Borrowing Interpolation
, 2011
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.
Interpolation for Predefined Types
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2008
"... ... model theoretic framework of the theory of institutions. For this semantics we develop a generic interpolation result which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of i ..."
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... model theoretic framework of the theory of institutions. For this semantics we develop a generic interpolation result which can be easily applied to various concrete situations from the theory and practice of specification and programming. Our study of interpolation is motivated by a number of important applications to computing science especially in the area of structured specifications.
Development Workspaces: an introduction
"... Interpolation and schematic reasoning underlie critical and somewhat complementary aspects of the design and manipulation of specification modules. Unfortunately, many formalisms that have been used in formal approaches to software engineering lack uniform interpolation and do not directly suppor ..."
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Interpolation and schematic reasoning underlie critical and somewhat complementary aspects of the design and manipulation of specification modules. Unfortunately, many formalisms that have been used in formal approaches to software engineering lack uniform interpolation and do not directly support schematic reasoning. This paper presents the skeleton of a novel general construction for extending conservatively a base entailment system Espec to an entailment system Edev such that Edev provides a uniform presentation of interpolants for entailments between Especsentences and supports the precise logical encapsulation of uniform schemata on Espec . This conservative extension is presented by means of a Subentailment system hEspec ; J : Espec!Edevi called a Development Workspace on Espec . 1 Introduction There is a well established relation between interpolation [6, 21, 31, 2] and modularity properties of refinement via implementations (eg., [12, 11, 10] and [22, 23, 41, 38, 4...
The VDM+B project: Objectives and Progress
"... Abstract. The VDM+B project is developing the formal underpinnings for an integration of VDM and B enabling their couse within one development. In this paper, we describe the objectives for the project, the approach being undertaken and the current status of the work. 1 ..."
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Abstract. The VDM+B project is developing the formal underpinnings for an integration of VDM and B enabling their couse within one development. In this paper, we describe the objectives for the project, the approach being undertaken and the current status of the work. 1