Advertisement

Term Rewriting in Logics of Partial Functions

  • Matthias Schmalz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6991)

Abstract

We devise a theoretical foundation of directed rewriting, a term rewriting strategy for logics of partial functions, inspired by term rewriting in the Rodin platform. We prove that directed rewriting is sound and show how to supply new rewrite rules in a soundness preserving fashion. In the context of Rodin, we show that directed rewriting makes a significant number of conditional rewrite rules unconditional. Our work not only allows us to point out a number of concrete ways of improving directed rewriting in Rodin, but also has applications in other logics of partial functions. Additionally, we give a semantics for the logic of Event-B.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrial, J.R.: Modeling in Event-B, Cambridge (2010)Google Scholar
  2. 2.
    Abrial, J.R., Butler, M.J., Hallerstede, S., Hoang, T.S., Mehta, F., Voisin, L.: Rodin: an open toolset for modelling and reasoning in Event-B. STTT 12(6), 447–466 (2010)CrossRefGoogle Scholar
  3. 3.
    Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory. Kluwer, Dordrecht (2002)zbMATHGoogle Scholar
  4. 4.
    Barringer, H., Cheng, J.H., Jones, C.B.: A logic covering undefinedness in program proofs. Acta Inf. 21, 251–269 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Butler, M., Maamria, I.: Mathematical extension in Event-B through the Rodin theory component (2010), http://deploy-eprints.ecs.soton.ac.uk/251
  6. 6.
    Darvas, Á., Mehta, F., Rudich, A.: Efficient well-definedness checking. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 100–115. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Dawson, J.E.: Simulating term-rewriting in LPF and in display logic. In: Supplementary Proc. of TPHOLs, pp. 47–62. Australian National University (1998)Google Scholar
  8. 8.
    Gordon, M.J.C., Melham, T.F.: Introduction to HOL, Cambridge (1993)Google Scholar
  9. 9.
    Hindley, J.R., Seldin, J.P.: Lambda-Calculus and Combinators, Cambridge (2008)Google Scholar
  10. 10.
    Jones, C.B., Middelburg, C.A.: A typed logic of partial functions reconstructed classically. Acta Inf. 31(5), 399–430 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Maamria, I., Butler, M.: Rewriting and well-definedness within a proof system. In: PAR. EPTCS, vol. 43, pp. 49–64 (2010)Google Scholar
  12. 12.
    Mehta, F.: A practical approach to partiality – A proof based approach. In: Liu, S., Araki, K. (eds.) ICFEM 2008. LNCS, vol. 5256, pp. 238–257. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Metayer, C., Voisin, L.: The Event-B mathematical language (2009), http://deploy-eprints.ecs.soton.ac.uk/11
  14. 14.
    Nipkow, T.: Term rewriting and beyond - theorem proving in isabelle. Formal Asp. Comput. 1(4), 320–338 (1989)CrossRefzbMATHGoogle Scholar
  15. 15.
    Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  16. 16.
    Owe, O.: Partial logics reconsidered: A conservative approach. Formal Asp. Comput. 5(3), 208–223 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Owre, S., Shankar, N.: The formal semantics of PVS (1999), http://pvs.csl.sri.com/papers/csl-97-2/csl-97-2.ps
  18. 18.
    Paulson, L.C.: Logic and Computation: Interactive Proof with Cambridge LCF, Cambridge (1987)Google Scholar
  19. 19.
    Paulson, L.C.: The foundation of a generic theorem prover. J. Autom. Reasoning 5(3), 363–397 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Schmalz, M.: The logic of Event-B (2011), ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/6xx/698.pdf
  21. 21.
    Schmalz, M.: Term rewriting in logics of partial functions (2011), ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/7xx/732.pdf
  22. 22.
    Shankar, N., Owre, S., Rushby, J.M., Stringer-Calvert, D.W.J.: PVS prover guide (2001), http://pvs.csl.sri.com/doc/pvs-prover-guide.pdf
  23. 23.
    Silva, R., Butler, M.: Supporting reuse of Event-B developments through generic instantiation. In: Breitman, K., Cavalcanti, A. (eds.) ICFEM 2009. LNCS, vol. 5885, pp. 466–484. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. 24.
    Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 307–322. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matthias Schmalz
    • 1
  1. 1.ETH ZurichSwitzerland

Personalised recommendations