# Operation Refinement and Monotonicity in the Schema Calculus

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## Abstract

The schema calculus of Z provides a means for expressing structured, modular specifications. Extending this modularity to program development requires the monotonicity of these operators with respect to refinement. This paper provides a thorough mathematical analysis of monotonicity with respect to four schema operations for three notions of operation refinement. The mathematical connection between the equational schema logic and monotonicity is discussed and evaluated.

## Keywords

Schema Type Operation Schema Elimination Rule Introduction Rule Equational Logic
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## References

- [1]C. Bolton, J. Davies, and J. C. P. Woodcock. On the refinement and simulation of data types and processes. In K. Araki, A. Galloway, and K. Taguchi, editors,
*Integrated Formal Methods (IFM’99)*. Springer, 1999.Google Scholar - [2]A. Cavalcanti.
*A Refinement Calculus for Z*. PhD thesis, University of Oxford, 1997.Google Scholar - [3]A. Cavalcanti and J. C. P. Woodcock. ZRC — a refinement calculus for Z.
*Formal Aspects of Computing*, 10(3):267–289, 1998.zbMATHCrossRefGoogle Scholar - [4]J. Derrick and E. Boiten.
*Refinement in Z and Object-Z: Foundations and Advanced Applications*. Formal Approaches to Computing and Information Technology — FACIT. Springer, May 2001.Google Scholar - [5]M. Deutsch, M. C. Henson, and S. Reeves. An analysis of total correctness refinement models for partial relation semantics I.
*University of Essex*,*technical report CSM-362*, 2001. To appear in the Logic Journal of the IGPL.Google Scholar - [6]M. Deutsch, M. C. Henson, and S. Reeves. Results on formal stepwise design in Z. In
*9th Asia Pacific Software Engineering Conference (APSEC 2002)*, pages 33–42. IEEE Computer Society Press, December 2002.Google Scholar - [7]M. Deutsch, M. C. Henson, and S. Reeves. Operation refinement and monotonicity in the schema calculus.
*University of Essex*,*technical report CSM-381*, February 2003.Google Scholar - [8]A. Diller.
*Z: An Introduction to Formal Methods*. J. Wiley and Sons, 2nd edition, 1994.Google Scholar - [9]L. Groves.
*Evolutionary Software Development in the Refinement Calculus*. PhD thesis, Victoria University, 2000.Google Scholar - [10]L. Groves. Refinement and the Z schema calculus. In
*REFINE 2002: Refinement Workshop*. BCS FACS, July 2002.Google Scholar - [11]J. Grundy.
*A Method of Program Refinement*. PhD thesis, University of Cambridge, 1993.Google Scholar - [12]I. Hayes.
*Specification Case Studies*. Prentice Hall, 2nd edition, 1993.Google Scholar - [13]M. C. Henson and S. Reeves. Investigating Z.
*Logic and Computation*, 10(1):43–73, 2000.zbMATHCrossRefMathSciNetGoogle Scholar - [14]M. C. Henson and S. Reeves. Program development and specification refinement in the schema calculus. In J. P. Bowen, S. Dunne, A. Galloway, and S. King, editors,
*ZB 2000: Formal Speci.cation and Development in Z and B*, volume 1878 of*Lecture Notes in Computer Science*, pages 344–362. Springer, 2000.CrossRefGoogle Scholar - [15]M. C. Henson and S. Reeves. A logic for schema-based program development.
*University of Essex*,*technical report CSM-361*, 2001. To appear in the Journal of Formal Aspects of Computing.Google Scholar - [16]J. Jacky. Formal specification of control software for a radiation therapy machine.
*Radiation Oncology Department, University of Washington*,*technical report 94-07-01*, 1994.Google Scholar - [17]S. King. Z and the Refinement Calculus. In D. Bjørner, C. A. R. Hoare, and H. Langmaack, editors,
*VDM’ 90 VDM and Z — Formal Methods in Software Development*, volume 428 of*Lecture Notes in Computer Science*, pages 164–188. Springer-Verlag, April 1990.Google Scholar - [18]B. P. Mahony. The least conjunctive refinement and promotion in the refinement calculus.
*Formal Aspects of Computing*, 11:75–105, 1999.zbMATHCrossRefGoogle Scholar - [19]B. Potter, J. Sinclair, and D. Till.
*An Introduction to Formal Specification and Z*. Prentice Hall, 2nd edition, 1996.Google Scholar - [20]J. M. Spivey.
*The Z Notation: A Reference Manual*. Prentice Hall, 2nd edition, 1992.Google Scholar - [21]B. Strulo. How firing conditions help inheritance. In J. P. Bowen and M. G. Hinchey, editors,
*ZUM’ 95: The Z Formal Specification Notation*, volume 967 of*Lecture Notes in Computer Science*, pages 264–275. Springer Verlag, 1995.Google Scholar - [22]M. Utting. Private communication. Department of Computer Science, University of Waikato, Hamilton, New Zealand, June 2002.Google Scholar
- [23]N. Ward. Adding specification constructors to the refinement calculus. In J. C. P. Woodcock and P. G. Larsen, editors,
*Formal Methods Europe (FME’ 93)*, volume 670 of*Lecture Notes in Computer Science*, pages 652–670. Springer-Verlag, 1993.CrossRefGoogle Scholar - [24]J. C. P. Woodcock. Calculating properties of Z specifications.
*ACM SIGSOFT Software Engineering Notes*, 14(5):43–54, 1989.CrossRefGoogle Scholar - [25]J. C. P. Woodcock. Implementing promoted operations in Z. In C. B. Jones, R. C. Shaw, and T. Denvir, editors,
*5th Refinement Workshop*, Workshops in Computing, pages 367–378. Springer-Verlag, 1992.Google Scholar - [26]J. C. P. Woodcock and J. Davies.
*Using Z: Specification, Refinement and Proof*. Prentice Hall, 1996.Google Scholar - [27]J. B. Wordsworth.
*Software Development with Z — A Practical Approach to Formal Methods in Software Engineering*. Internalional Computer Science Series. Addison-Wesley, 1992.Google Scholar

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