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Investigating Z
, 2000
"... In this paper we introduce and investigate an improved kernel logic ZC for the specification language Z. Unlike the standard accounts, this logic is consistent and is easily shown to be sound. We show how a complete schema calculus can be derived within this logic and in doing so we reveal a high de ..."
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Cited by 11 (4 self)
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In this paper we introduce and investigate an improved kernel logic ZC for the specification language Z. Unlike the standard accounts, this logic is consistent and is easily shown to be sound. We show how a complete schema calculus can be derived within this logic and in doing so we reveal a high degree of logical organisation within the language. Finally, our approach eschews all nonstandard concepts introduced in the standard approach, notably object level notions of substitution and entities which share properties both of constants and variables. We show, in addition, that these unusual notions are derivable in ZC and are, therefore, unnecessary innovations. Keywords: Specification language Z; Logic and semantics of specification languages. 1 Introduction In this paper we introduce and investigate an improved kernel logic ZC for the specification language Z, a logic in which, in particular, we can derive a schema calculus: a logic for the entire range of schema expressions permit...
Revising Z: Part I  logic and semantics
 Formal Apects of Computing
, 1999
"... . This is the first of two related papers. We introduce a simple specification logic ZC comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within ZC . As a result we ..."
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Cited by 4 (1 self)
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. This is the first of two related papers. We introduce a simple specification logic ZC comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within ZC . As a result we obtain a sound logic for Z, including a basic schema calculus. Keywords: Specification language Z; Logics and semantics of specification languages 1. Introduction 1.1. Background The specification language Z has been in existence, and has been very widely used, for more than a decade. There is, however, no definitive logical account of Z, although progress has been made (e.g. [WB92], [Bri95] 1 , [Nic95], [HM97], [Toy97], [BM96], [WD96], [Mar98]). These attempts to provide Z with a logic are not accompanied by metamathematical results such as a soundness proof; indeed, the logic in [Nic95] (and [BM96]) is inconsistent (see [Hen98] and note that [Mar98] repairs the error). Our aim in thi...
Revising Z: Part II  logical development
, 1999
"... . This is the second of two related papers. In "Revising Z: Part I  logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification langua ..."
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Cited by 3 (1 self)
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. This is the second of two related papers. In "Revising Z: Part I  logic and semantics" (this journal) we introduced a simple specification logic ZC comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within ZC . As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility. Keywords: Specification language Z; Logics and semantics of specification languages 1. Introduction In the earlier companion paper [HR99] we introduced a specification system ZC , a typed set theory incorporating the notion of a schema type and we established a number of metamathematical...
Z Logic And Its Consequences
 CAI: Computing and Informatics
, 2003
"... This paper provides an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have cowritten since 1997. We provide an informal as well as formal introd ..."
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Cited by 1 (0 self)
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This paper provides an introduction to the specification language Z from a logical perspective. The possibility of presenting Z in this way is a consequence of a number of joint publications on Z logic that Henson and Reeves have cowritten since 1997. We provide an informal as well as formal introduction to Z logic and show how it may be used, and extended, to investigate issues such as equational logic, the logic of preconditions, the issue of monotonicity and both operation and data refinement. For Peter Wexler  in memoriam 1
The standard logic of Z is inconsistent
 Formal Aspects of Computing Journal
, 1998
"... . We demonstrate the logic contained in the draft Z standard is inconsistent. 1. Introduction The specification language Z has been very widely used and commended, but it has not yet received the kind of mathematical attention one would hope for a formal method. Despite more than a decade of effor ..."
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. We demonstrate the logic contained in the draft Z standard is inconsistent. 1. Introduction The specification language Z has been very widely used and commended, but it has not yet received the kind of mathematical attention one would hope for a formal method. Despite more than a decade of effort, the standard source for the formalisation of Z [Nic95] still contains many unclarities and incompletenesses. Worse, as this short note demonstrates, the proposed logic it contains is inconsistent. 2. The inconsistency proof Firstly, the rule (BindSel) enables us to obtain the usual closure axiom for bindings. Suppose the alphabet set of b is f\Delta \Delta \Delta l i \Delta \Delta \Deltag. Then we have: \Delta \Delta \Delta b:l i = b:l i \Delta \Delta \Delta \Delta \Delta \Delta b:l i = b:l i \Delta \Delta \Delta b = hj \Delta \Delta \Delta l i V b:l i \Delta \Delta \Delta ji Secondly, we may use this closure axiom to obtain more natural equality congruence rules for binding projecti...