Generalised Substitution Language and Differentials
Embedded continuous control systems can be thought of as implementing complex(piecewise and pipelined) differential functions. Each ‘piece’ of the function may be preconditioned with a ‘domain of applicability’, which prescribes the circumstances the piece was designed to handle. The preconditions often involve rate of change, i.e. differentials, as well as range constraints. In this paper we present an adaptation of the substitution calculus which can be used to reason about such systems. Our approach is based on generalising the traditional view that a component is a fragment of a sequential programme. We consider a component to be an autonomous transformation which is ‘clocked’ to perform its computation at regular intervals, over and over again. In the case of such a component we can generalise the notion of weakest precondition to traces (sequences of values) of inputs and outputs. In our approach we characterise such traces by ‘step’ predicates over adjacent elements in the trace. We also generalise our calculus to cover nth order differentials. Since analysis can be performed at a comparable complexity to regular wp, our techniques are a powerful tool in the validation of continuous control systems.
KeywordsAcceptance Criterion Generalise Substitution Proof Obligation Order Constraint Output Trace
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- 2.J-R Abrial. The B Book-Assigning Programs to Meanings. Cambridge University Press, 1996.Google Scholar
- 4.M. Broy. The Specification of System Components by State Transition Diagrams. Technical Report TUM-I9729, Technische Univeritat Munchen, 1997.Google Scholar
- 6.M. Broy. A Logical Basis for Component-Based Systems Engineering. International Summer School, Marktoberdorf, July-August 1999.Google Scholar
- 7.M. Broy. From States to History. In International Summer School, Marktoberdorf, 2000.Google Scholar
- 8.Z. Chaochen, A.P. Ravn, and M.R. Hansen. An Extended Duration Calculus for Hybrid Real Time Systems. In Hybrid Systems, Lecture Notes in Computer Science, pages 36–59, 1993.Google Scholar
- 9.E.W. Dijkstra. Guarded Commands, Nondetermincy and Formal Derivation of Programs. Communications of the ACM, 18:453–457, August 1975.Google Scholar
- 10.A. Galloway. Communicating Generalised Substitution Language. In Proceedings of the International Conference on Perspectives of System Informatics, PSI’01, 2001.Google Scholar
- 11.A. Galloway and J. Blow. Multi Layered Domain Specific Formal Languages. In Proceedings of the Workshop on Formal Specification of Computer Based Systems, FSCBS’01, April 2001.Google Scholar
- 12.A. J. Galloway, T. J. Cockram, and J. A. McDermid. Experiences with the Application of Discrete Formal Methods to the Development of Engine Control Software. In Proceedings of DCCS (Distributed Computer Control Systems) 98. IFAC — International Federation of Automatic Control, 1998.Google Scholar
- 13.J. Blow, A. Galloway, J.A. McDermid, M. Dowding and T. Cockram. The Industrial Use of a Formal Method in a Gas Turbine Engine Electronic Control System. In Proceedings of the Workshop on Formal Specifications of Computer Based Systems, FSCBS’00, April 2000.Google Scholar
- 14.John McDermid and Andy Galloway et al. Towards Industrially Applicable Formal Methods: Three Small Steps, and One Giant Leap. In The International Conference on Formal Engineering Methods (ICFEM) 1998. IEEE Press, 1998.Google Scholar
- 15.UK Ministry of Defence. Defence Standard 00-55 — The Procurement of Safety Critical Software in Defence Equipment. 1997.Google Scholar