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12-The Typed Logic of Partial Functions and the

12-The Typed Logic of Partial Functions and the - The Typed...

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The Typed Logic of Partial Functions and the Vienna Development Method John S. Fitzgerald Centre for Software Reliability, Newcastle University, Newcastle upon Tyne, NE1 7RU, United Kingdom, [email protected] For Harry Fitzgerald, Engineer (1928–2004) Summary. Decisions about the logic underpinning a formal specification language have important consequences for the utility of the formalism. This chapter describes the major features of the typed Logic of Partial Functions (LPF) as it has been im- plemented in support of the Vienna Development Method’s Specification Language, VDM-SL. It compares attempts to realise the logic in different environments: a user- centred proof support tool, a specification interpreter and an automated proof tool. Future directions in integrated proof support for the language are suggested. 1 Introduction The logic that underpins a specification language has great practical signifi- cance, directly affecting the capabilities of the tools that are so necessary to the successful application of the language in professional practice. Decisions regarding logic are therefore influenced by methodological and pragmatic con- cerns, as well as by the desire to provide an intuitive and elegant theory. The Vienna Development Method’s Specification Language (VDM-SL) has a long history of use, both as a vehicle for research and as a tool in the development of computer-based systems. Machine support for the coding and analysis for VDM specifications has been available for VDM-SL for well over a decade, and there is considerable experience at proving properties of specifications and refinements in the formalism. Nevertheless, the logic underpinning VDM, and its susceptibility to machine support, remain the subject of debate and research. The Logic of Partial Functions (LPF) is closely associated with VDM, although its potential for application goes well beyond that particular formal- ism. The aims of this chapter are to give an account of LPF and attempts to realise it in various support environments for VDM, ranging from an inter-
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454 John S. Fitzgerald preter to automated provers, and to identify the next steps in the provision of integrated support for reasoning about VDM specifications. In order to understand the requirements for a logic supporting VDM-SL, it is worth reviewing the distinguishing characteristics of the specification lan- guage and a little of its history (Sect. 2). The key features of Typed LPF, notably the handling of undefined terms, are introduced in Sects. 3 and 4. Us- ing LPF to reason about models expressed in VDM-SL entails the addition of types and other relevant features, discussed in Sect. 5. Contrasting approaches to tool support for LPF-based reasoning in VDM-SL are discussed in Sect. 6, leading to a discussion of future directions and concluding remarks (Sect. 7).
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