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Partial, Total and General Correctness

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Mathematics of Program Construction (MPC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6120))

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Abstract

We identify weak semirings, which drop the right annihilation axiom a 0 = 0, as a common foundation for partial, total and general correctness. It is known how to extend weak semirings by operations for finite and infinite iteration and domain. We use the resulting weak omega algebras with domain to define a semantics of while-programs which is valid in all three correctness approaches. The unified, algebraic semantics yields program transformations at once for partial, total and general correctness. We thus give a proof of the normal form theorem for while-programs, which is a new result for general correctness and extends to programs with non-deterministic choice.

By adding specific axioms to the common ones, we obtain partial, total or general correctness as a specialisation. We continue our previous investigation of axioms for general correctness. In particular, we show that a subset of these axioms is sufficient to derive a useful theory, which includes the Egli-Milner order, full recursion, correctness statements and a correctness calculus. We also show that this subset is necessary.

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References

  1. Aarts, C.J.: Galois connections presented calculationally. Master’s thesis, Department of Mathematics and Computing Science, Eindhoven University of Technology (1992)

    Google Scholar 

  2. de Bakker, J.W.: Semantics and termination of nondeterministic recursive programs. In: Michaelson, S., Milner, R. (eds.) Automata, Languages and Programming: Third International Colloquium, pp. 435–477. Edinburgh University Press, Edinburgh (1976)

    Google Scholar 

  3. Berghammer, R., Zierer, H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theoretical Computer Science 43, 123–147 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. Broy, M., Gnatz, R., Wirsing, M.: Semantics of nondeterministic and noncontinuous constructs. In: Bauer, F.L., Broy, M. (eds.) Program Construction. LNCS, vol. 69, pp. 553–592. Springer, Heidelberg (1979)

    Chapter  Google Scholar 

  5. De Carufel, J.-L., Desharnais, J.: Demonic algebra with domain. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 120–134. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Cohen, E.: Separation and reduction. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 45–59. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  7. Dang, H.-H., Höfner, P.: First-order theorem prover evaluation w.r.t. relation- and Kleene algebra. In: Berghammer, R., Möller, B., Struth, G. (eds.) Relations and Kleene Algebra in Computer Science: PhD Programme at RelMiCS10/AKA5, Report 2008-04, pp. 48–52. Institut für Informatik, Universität Augsburg (April 2008)

    Google Scholar 

  8. Desharnais, J., Möller, B., Struth, G.: Algebraic notions of termination. Report 2006-23, Institut für Informatik, Universität Augsburg (October 2006)

    Google Scholar 

  9. Desharnais, J., Möller, B., Struth, G.: Kleene algebra with domain. ACM Transactions on Computational Logic 7(4), 798–833 (2006)

    MathSciNet  Google Scholar 

  10. Desharnais, J., Struth, G.: Domain axioms for a family of near-semirings. In: Meseguer, J., Roşu, G. (eds.) AMAST 2008. LNCS, vol. 5140, pp. 330–345. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  11. Desharnais, J., Struth, G.: Modal semirings revisited. In: Audebaud, P., Paulin-Mohring, C. (eds.) MPC 2008. LNCS, vol. 5133, pp. 360–387. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Dijkstra, E.W.: A Discipline of Programming. Prentice Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  13. Dijkstra, R.M.: Computation calculus bridging a formalization gap. Science of Computer Programming 37(1–3), 3–36 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dunne, S.: Recasting Hoare and He’s Unifying Theory of Programs in the context of general correctness. In: Butterfield, A., Strong, G., Pahl, C. (eds.) 5th Irish Workshop on Formal Methods, Electronic Workshops in Computing. The British Computer Society (July 2001)

    Google Scholar 

  15. Dunne, S., Galloway, A.: Lifting general correctness into partial correctness is ok. In: Davies, J., Gibbons, J. (eds.) IFM 2007. LNCS, vol. 4591, pp. 215–232. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  16. Dunne, S., Hayes, I., Galloway, A.: Reasoning about loops in total and general correctness. In: Butterfield, A. (ed.) Second International Symposium on Unifying Theories of Programming. LNCS, vol. 5713, Springer, Heidelberg (to appear)

    Google Scholar 

  17. Guttmann, W.: General correctness algebra. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS/AKA 2009. LNCS, vol. 5827, pp. 150–165. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Guttmann, W.: Lazy UTP. In: Butterfield, A. (ed.) Second International Symposium on Unifying Theories of Programming. LNCS, vol. 5713. Springer, Heidelberg (to appear)

    Google Scholar 

  19. Guttmann, W., Möller, B.: Modal design algebra. In: Dunne, S., Stoddart, W. (eds.) UTP 2006. LNCS, vol. 4010, pp. 236–256. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Guttmann, W., Möller, B.: Normal design algebra. Journal of Logic and Algebraic Programming 79(2), 144–173 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hoare, C.A.R.: An axiomatic basis for computer programming. Communications of the ACM 12(10), 576–580, 583 (1969)

    Article  MATH  Google Scholar 

  22. Hoare, C.A.R., He, J.: Unifying theories of programming. Prentice-Hall Europe (1998)

    Google Scholar 

  23. Höfner, P., Möller, B.: An algebra of hybrid systems. Journal of Logic and Algebraic Programming 78(2), 74–97 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Höfner, P., Möller, B., Solin, K.: Omega algebra, demonic refinement algebra and commands. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 222–234. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  25. Jacobs, D., Gries, D.: General correctness: A unification of partial and total correctness. Acta Informatica 22(1), 67–83 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kozen, D.: Kleene algebra with tests. ACM Transactions on Programming Languages and Systems 19(3), 427–443 (1997)

    Article  Google Scholar 

  28. Kozen, D.: On Hoare logic and Kleene algebra with tests. ACM Transactions on Computational Logic 1(1), 60–76 (2000)

    Article  MathSciNet  Google Scholar 

  29. Möller, B.: Lazy Kleene algebra. In: Kozen, D. (ed.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)

    Google Scholar 

  30. Möller, B.: The linear algebra of UTP. In: Uustalu, T. (ed.) MPC 2006. LNCS, vol. 4014, pp. 338–358. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  31. Möller, B., Struth, G.: Algebras of modal operators and partial correctness. Theoretical Computer Science 351(2), 221–239 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  32. Möller, B., Struth, G.: WP is WLP. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 200–211. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  33. Moszkowski, B.C.: A complete axiomatization of Interval Temporal Logic with infinite time. In: Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science, pp. 241–252. IEEE, Los Alamitos (2000)

    Google Scholar 

  34. Nelson, G.: A generalization of Dijkstra’s calculus. ACM Transactions on Programming Languages and Systems 11(4), 517–561 (1989)

    Article  Google Scholar 

  35. Solin, K.: A while program normal form theorem in total correctness. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) RelMiCS/AKA 2009. LNCS, vol. 5827, pp. 322–336. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  36. Søndergaard, H., Sestoft, P.: Non-determinism in functional languages. The Computer Journal 35(5), 514–523 (1992)

    Article  MathSciNet  Google Scholar 

  37. von Wright, J.: Towards a refinement algebra. Science of Computer Programming 51(1–2), 23–45 (2004)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut für Programmiermethodik und Compilerbau, Universität Ulm, 89069, Ulm, Germany

    Walter Guttmann

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  1. Walter Guttmann
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  1. Département d’informatique et de génie logiciel, Pavillon Adrien-Pouliot, Université Laval, 1065 Avenue de la Médicine, G1V 0A6, Québec, Canada

    Claude Bolduc , Jules Desharnais  & Béchir Ktari ,  & 

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Guttmann, W. (2010). Partial, Total and General Correctness. In: Bolduc, C., Desharnais, J., Ktari, B. (eds) Mathematics of Program Construction. MPC 2010. Lecture Notes in Computer Science, vol 6120. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13321-3_11

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  • DOI: https://doi.org/10.1007/978-3-642-13321-3_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13320-6

  • Online ISBN: 978-3-642-13321-3

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