Abstract
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.
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References
R. D. Arthan. On free type definitions in Z. In J. E. Nicholls, editor, Z User Workshop, York, December 1991. Springer.
R. D. Arthan. Recursive definitions in Z. In J. P. Bowen, A. Fett, and M. G. Hinchey, editors, ZUM’98: The Z Formal Specification Notation, LNCS 1493, Berlin, September 1998. Springer.
Robert S. Boyer and J. Strother Moore. A computational logic. Academic Press, 1979.
H. B. Enderton. Elements of Set Theory. Academic Press, 1977.
D. Kapur and M. Subramaniam. Automating induction over mutually recursive functions. In M. Wirsing and M. Nivat, editors, Proceedings of the 5th International Conference on Algebraic Methodology and Software Technology (AMAST’96), LNCS 1101, Munich, 1996. Springer.
P. Liu and R.-J. Chang. A new structural induction scheme for proving properties of mutually recursive concepts. In Proceedings of the 6th National Conference on Artificial Intelligence, volume 1, pages 144–148. AAAI, 1987.
Thomas F. Melham. Automating recursive type definitions in higher order logic. In G. Birtwistle and P. A. Subrahmanyam, editors, Current Trends in Hardware Verification and Automated Theorem Proving, pages 341–386. Springer, 1989.
A. Smith. On recursive free types in Z. In J. E. Nicholls, editor, Z User Workshop, York, December 1991. Springer.
J. M. Spivey. The Z Notation: A Reference Manual. Prentice Hall, first edition, 1989.
J. M. Spivey. The Z Notation: A Reference Manual. Prentice Hall, second edition, 1992.
I. Toyn. CADiZ web pages. http://www.cs.york.ac.uk/~ian/cadiz/.
I. Toyn. Innovations in the notation of standard Z. In J. P. Bowen, A. Fett, and M. G. Hinchey, editors, ZUM’98: The Z Formal Specification Notation, LNCS 1493, Berlin, September 1998. Springer.
I. Toyn. A tactic language for reasoning about Z specifications. In Third Northern Formal Methods Workshop, Ilkley, September 1998.
I. Toyn, editor. Z Notation ISO, 1999. Final Committee Draft, available at http://www.cs.york.ac.uk/ian/zstan/.
S. H. Valentine. Inconsistency and undefinedness in Z-a practical guide. In J. P. Bowen, A. Fett, and M. G. Hinchey, editors, ZUM’98: The Z Formal Specification Notation, LNCS 1493, pages 233–249, Berlin, September 1998. Springer.
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Toyn, I., Valentine, S.H., Duffy, D.A. (2000). On Mutually Recursive Free Types in Z. In: ZB 2000: Formal Specification and Development in Z and B. ZB 2000. Lecture Notes in Computer Science, vol 1878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44525-0_5
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DOI: https://doi.org/10.1007/3-540-44525-0_5
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