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15
On Free Type Definitions in Z
, 1992
"... Recent discussions in the Z community have considered the issue of the consistency of the free type construct in Z. A key question is whether free type definitions which met the criterion for consistency given in the Z Reference Manual, are conservative over Zermelo set theory (i.e. ZF without the a ..."
Abstract

Cited by 14 (3 self)
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Recent discussions in the Z community have considered the issue of the consistency of the free type construct in Z. A key question is whether free type definitions which met the criterion for consistency given in the Z Reference Manual, are conservative over Zermelo set theory (i.e. ZF without the axiom of replacement). The main purpose of this paper is to give an introduction to the issues and to show that the answer to this question is "yes" (given the axiom of choice). A byproduct of the arguments we give here is that the criterion given in the Z reference manual may be replaced by an intuitively simpler one without loss of expressive power from the theoretical or practical point of view.
Consistency and refinement for partial specification in Z
 FME'96: Industrial Benefit of Formal Methods, Third International Symposium of Formal Methods Europe, volume 1051 of Lecture Notes in Computer Science
, 1996
"... . This paper discusses theoretical background for the use of Z as a language for partial specification, in particular techniques for checking consistency between viewpoint specifications. The main technique used is unification, i.e. finding a (candidate) least common refinement. The corresponding no ..."
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Cited by 13 (10 self)
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. This paper discusses theoretical background for the use of Z as a language for partial specification, in particular techniques for checking consistency between viewpoint specifications. The main technique used is unification, i.e. finding a (candidate) least common refinement. The corresponding notion of consistency between specifications turns out to be different from the known notions of consistency for single Z specifications. A key role is played by correspondence relations between the data types used in the various viewpoints. 1 Partial specification It is generally agreed that systems of a realistic size cannot be specified in single linear specifications, but rather should be decomposed into manageable chunks which can be specified separately. The traditional method for doing this is by hierarchical and functional decomposition. Nowadays, it is often claimed [11] that this is not the most natural or convenient (in relation to "perceived complexity") method  rather systems s...
Constructive consistency checking for partial specification in Z
 in Z. Science of Computer Programming
, 1999
"... This paper describes how to check consistency between partial specifications in Z, i.e. how to establish that different partial specifications of one system do not impose contradictory requirements. Using the traditional refinement relation in Z, we present techniques for constructing unifications ( ..."
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Cited by 10 (2 self)
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This paper describes how to check consistency between partial specifications in Z, i.e. how to establish that different partial specifications of one system do not impose contradictory requirements. Using the traditional refinement relation in Z, we present techniques for constructing unifications (least common refinements) of partial specifications, which represent their combined requirements. Three relatively simple conditions on the partial specifications and the predicate that relates them characterise consistency.
Living with Free Type and Class Union
 In The 1995 AsiaPacific Software Engineering Conference (APSEC'95
, 1995
"... There are two constructs in the formal specification language ObjectZ for modelling polymorphic and recursive structures. One construct, the free type, is adopted from the Z specification language. The other, classunion, facilitates polymorphic class declarations. Free type and classunion constru ..."
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Cited by 8 (5 self)
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There are two constructs in the formal specification language ObjectZ for modelling polymorphic and recursive structures. One construct, the free type, is adopted from the Z specification language. The other, classunion, facilitates polymorphic class declarations. Free type and classunion constructs are respectively based upon the functional value point of view and the object reference point of view. Consequently, the roles these two constructs perform in system modelling are different. In this paper, the free type and classunion constructs are compared and discussed through various examples. The aim of this comparison and discussion is to present guidelines on how to appropriately and effectively use these two constructs to specify polymorphic and recursive structures. Keywords: formal specification techniques, objectoriented modelling, polymorphic and recursive structures. 1 Introduction In the formal specification language Z[12], the free type construct is introduced as a conv...
The Z/EVES 2.0 User's Guide
, 1999
"... Contents 1 Introduction 1 2 Using Z/EVES 3 2.1 Running Z/EVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Z/EVES Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.1 The Specification window . . . . . . . . . . . . . . . ..."
Abstract

Cited by 6 (0 self)
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Contents 1 Introduction 1 2 Using Z/EVES 3 2.1 Running Z/EVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Z/EVES Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.1 The Specification window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.2 The Proof window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.3 The Theorems window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.4 The Proof Scripts window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.5 The Edit window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.6 The Clipboard window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Composing and Checking a Specification . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Analysing a Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 T E X Markup . . .
The Z/EVES User's Guide
, 1997
"... this report. Input to Z/EVES is usually shown in typewriter ..."
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Cited by 6 (1 self)
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this report. Input to Z/EVES is usually shown in typewriter
Recursive Definitions in Z
 ZUM’98: The Z Formal Specification Notation, volume 1493 of Lecture Notes in Computer Science
, 1998
"... This paper considers some issues in the theory and practice of defining functions over recursive data types in Z. Principles justifying such definitions are formulated. Z free types are contrasted with the free algebras of universal algebra: the notions turn out to be related but not isomorphic. ..."
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Cited by 5 (0 self)
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This paper considers some issues in the theory and practice of defining functions over recursive data types in Z. Principles justifying such definitions are formulated. Z free types are contrasted with the free algebras of universal algebra: the notions turn out to be related but not isomorphic.
On Mutually Recursive Free Types in Z
 In Proceedings International Conference of Z and B Users, ZB2000, LNCS
, 2000
"... . Mutually recursive free types are one of the innovations in the forthcoming ISO Standard for the Z notation. Their semantics has been specified by extending a formalization of the semantics of traditional Z free types to permit mutual recursion. That development is reflected in the structure o ..."
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Cited by 2 (2 self)
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. Mutually recursive free types are one of the innovations in the forthcoming ISO Standard for the Z notation. Their semantics has been specified by extending a formalization of the semantics of traditional Z free types to permit mutual recursion. That development is reflected in the structure of this paper. An explanation of traditional Z free types is given, along with some examples, and their general form is defined. Their semantics is defined by transformation to other equivalent Z notation. These equivalent constraints provide a basis for inference rules, as illustrated by an example proof. Notation for mutually recursive free types is introduced, and the semantics presented earlier is extended to define their meaning. Example inductive proofs concerning mutually recursive free types are presented. 1 Introduction A specification written in Z [10] names the components of the specified system and expresses constraints between the values of those components. The constrai...
ICL Secure Systems
"... Recent discussions in the Z community have considered the issue of the consistency of the free type construct in Z. A key question is whether free type definitions which met the criterion for consistency given in the Z Reference Manual, [5], are conservative over Zermelo set theory (i.e. ZF without ..."
Abstract
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Recent discussions in the Z community have considered the issue of the consistency of the free type construct in Z. A key question is whether free type definitions which met the criterion for consistency given in the Z Reference Manual, [5], are conservative over Zermelo set theory (i.e. ZF without the axiom of replacement). The main purpose of this paper is to give an introduction to the issues and to show that the answer to this question is “yes” (given the axiom of choice). A byproduct of the arguments we give here is that the criterion given in the Z reference manual may be replaced by an intuitively simpler one without loss of expressive power from the theoretical or practical point of view. 1
Z and HOL
, 1994
"... A simple `shallow' semantic embedding of the Z notation into the HOL logic is described. The Z notation is based on set theory and first order predicate logic. The HOL theorem proving system supports higher order logic. A wellknown case study is used as a running example. The presentation is i ..."
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A simple `shallow' semantic embedding of the Z notation into the HOL logic is described. The Z notation is based on set theory and first order predicate logic. The HOL theorem proving system supports higher order logic. A wellknown case study is used as a running example. The presentation is intended to show people with some knowledge of Z how a tool such as HOL can be used to provide mechanical support for the notation, including mechanization of proofs. No specialized knowledge of HOL is assumed.