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Algebras for iteration and infinite computations

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Abstract

We give axioms for an operation that describes iteration in various relational models of computations. The models differ in their treatment of finite, infinite and aborting executions, covering partial, total and general correctness and extensions thereof. Based on the common axioms we derive separation, refinement and program transformation results hitherto known from particular models, henceforth recognised to hold in many different models. We introduce a new model that independently describes the finite, infinite and aborting executions of a computation, and axiomatise an operation that extracts the infinite executions in this model and others. From these unifying axioms we derive explicit representations for recursion and iteration. We show that also the new model is an instance of our general theory of iteration. All results are verified in Isabelle heavily using automated theorem provers.

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References

  1. Aarts C.J., Backhouse R.C., Boiten E.A., Doornbos H., van Gasteren N., van Geldrop R., Hoogendijk P.F., Voermans E., van der Woude J.: Fixed-point calculus. Inf. Process. Lett. 53(3), 131–136 (1995)

    Article  Google Scholar 

  2. Apt K.R., de Boer F.S., Olderog E.-R.: Verification of Sequential and Concurrent Programs. 3rd edn. Springer, London (2009)

    Book  MATH  Google Scholar 

  3. Back R.J.R., von Wright J.: Reasoning algebraically about loops. Acta Informatica 36(4), 295–334 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berghammer R., Zierer H.: Relational algebraic semantics of deterministic and nondeterministic programs. Theor. Comput. Sci. 43, 123–147 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birkhoff G.: Lattice Theory, volume XXV of Colloquium Publications. 3rd edn. American Mathematical Society, Providence, RI (1967)

    Google Scholar 

  6. Blanchette J.C., Böhme S., Paulson L.C.: Extending Sledgehammer with SMT solvers. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) Automated Deduction: CADE-23, volume 6803 of Lecture Notes in Computer Science, pp. 116–130. Springer, Berlin (2011)

    Chapter  Google Scholar 

  7. Blanchette J.C., Nipkow T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving, volume 6172 of Lecture Notes in Computer Science, pp. 131–146. Springer, Berlin (2010)

    Chapter  Google Scholar 

  8. Bloom S.L., Ésik Z.: Iteration Theories: The Equational Logic of Iterative Processes. Springer, Berlin (1993)

    MATH  Google Scholar 

  9. Broy M., Gnatz R., Wirsing M.: Semantics of nondeterministic and noncontinuous constructs. In: Bauer, F.L., Broy, M. (eds.) Program Construction, volume 69 of Lecture Notes in Computer Science, pp. 553–592. Springer, Berlin (1979)

    Google Scholar 

  10. Cohen E.: Separation and reduction. In: Backhouse, R., Oliveira, J.N. (eds.) Mathematics of Program Construction, volume 1837 of Lecture Notes in Computer Science, pp. 45–59. Springer, Berlin (2000)

    Google Scholar 

  11. Conway J.H.: Regular Algebra and Finite Machines. Chapman and Hall, London (1971)

    MATH  Google Scholar 

  12. 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 

  13. De Carufel J.-L., Desharnais J.: Demonic algebra with domain. In: Schmidt, R. (ed) Relations and Kleene Algebra in Computer Science, volume 4136 of Lecture Notes in Computer Science, pp. 120–134. Springer, Berlin (2006)

    Chapter  Google Scholar 

  14. Desharnais J., Möller B., Struth G.: Kleene algebra with domain. ACM Trans. Comput. Log. 7(4), 798–833 (2006)

    Article  MathSciNet  Google Scholar 

  15. Desharnais J., Möller B., Struth G.: Algebraic notions of termination. Log. Methods Comput. Sci. 7(1:1), 1–29 (2011)

    Google Scholar 

  16. Desharnais J., Möller B., Tchier F.: Kleene under a modal demonic star. J. Log. Algebr. Program. 66(2), 127–160 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Desharnais J., Struth G.: Internal axioms for domain semirings. Sci. Comput. Program. 76(3), 181–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Doornbos H.: A relational model of programs without the restriction to Egli-Milner-monotone constructs. In: Olderog, E.-R. (ed.) Programming Concepts, Methods and Calculi, pp. 363–382. North-Holland Publishing Company, Amsterdam (1994)

    Google Scholar 

  19. 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 (2001)

  20. Guttmann W.: General correctness algebra. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds.) Relations and Kleene Algebra in Computer Science, volume 5827 of Lecture Notes in Computer Science, pp. 150–165. Springer, Berlin (2009)

    Chapter  Google Scholar 

  21. Guttmann W.: Partial, total and general correctness. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) Mathematics of Program Construction, volume 6120 of Lecture Notes in Computer Science, pp. 157–177. Springer, Berlin (2010)

    Google Scholar 

  22. Guttmann, W.: Unifying recursion in partial, total and general correctness. In: Qin, S. (eds.) Unifying Theories of Programming, Third International Symposium, UTP 2010, volume 6445 of Lecture Notes in Computer Science, pp. 207–225. Springer, Berlin (2010)

  23. Guttmann W.: Fixpoints for general correctness. J. Log. Algebr. Program. 80(6), 248–265 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guttmann W.: Towards a typed omega algebra. In: de Swart, H. (ed.) Relational and Algebraic Methods in Computer Science, volume 6663 of Lecture Notes in Computer Science, pp. 196–211. Springer, Berlin (2011)

    Chapter  Google Scholar 

  25. Guttmann, W.: Extended designs algebraically. Sci. Comput. Program. (2012) (to appear)

  26. Guttmann W.: Unifying correctness statements. In: Gibbons, J., Nogueira, P. (eds.) Mathematics of Program Construction, volume 7342 of Lecture Notes in Computer Science, pp. 198–219. Springer, Berlin (2012)

    Google Scholar 

  27. Guttmann, W.: Unifying lazy and strict computations. In: Griffin, T.G., Kahl, W. (eds.) Relational and Algebraic Methods in Computer Science, Lecture Notes in Computer Science. Springer, Berlin (2012) (to appear)

  28. Guttmann W., Möller B.: Modal design algebra. In: Dunne, S., Stoddart, W. (eds.) Unifying Theories of Programming, volume 4010 of Lecture Notes in Computer Science, pp. 236–256. Springer, Berlin (2006)

    Google Scholar 

  29. Guttmann W., Möller B.: Normal design algebra. J. Log. Algebr. Program. 79(2), 144–173 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Guttmann W., Struth G., Weber T.: Automating algebraic methods in Isabelle. In: Qin, S., Qiu, Z. (eds.) Formal Methods and Software Engineering, volume 6991 of Lecture Notes in Computer Science, pp. 617–632. Springer, Berlin (2011)

    Google Scholar 

  31. Hayes I.J., Dunne S.E., Meinicke L.: Unifying theories of programming that distinguish nontermination and abort. In: Bolduc, C., Desharnais, J., Ktari, B. (eds.) Mathematics of Program Construction, volume 6120 of Lecture Notes in Computer Science, pp. 178–194. Springer, Berlin (2010)

    Google Scholar 

  32. Hoare, C.A.R., He, J.: Unifying Theories of Programming. Prentice Hall Europe (1998)

  33. Höfner P., Möller B., Solin K.: Omega algebra, demonic refinement algebra and commands. In: Schmidt, R. (ed.) Relations and Kleene Algebra in Computer Science, volume 4136 of Lecture Notes in Computer Science, pp. 222–234. Springer, Berlin (2006)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Kozen D.: Kleene algebra with tests. ACM Trans. Program. Lang. Syst. 19(3), 427–443 (1997)

    Article  Google Scholar 

  37. Kozen D.: On Hoare logic and Kleene algebra with tests. ACM Trans. Comput. Log. 1(1), 60–76 (2000)

    Article  MathSciNet  Google Scholar 

  38. McCune, W.: Mace4 reference manual and guide. Technical Memorandum ANL/MCS-TM-264, Mathematics and Computer Science Division, Argonne National Laboratory, (2003). Mace4 is available at http://www.cs.unm.edu/~mccune/mace4/

  39. Möller B.: The linear algebra of UTP. In: Uustalu, T. (ed.) Mathematics of Program Construction, volume 4014 of Lecture Notes in Computer Science, pp. 338–358. Springer, Berlin (2006)

    Google Scholar 

  40. Möller B.: Kleene getting lazy. Sci. Comput. Program. 65(2), 195–214 (2007)

    Article  MATH  Google Scholar 

  41. Möller B., Struth G.: Algebras of modal operators and partial correctness. Theor. Comput. Sci. 351(2), 221–239 (2006)

    Article  MATH  Google Scholar 

  42. Möller B., Struth G.: WP is WLP. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) Relational Methods in Computer Science 2005, volume 3929 of Lecture Notes in Computer Science, pp. 200–211. Springer, Berlin (2006)

    Google Scholar 

  43. Nelson G.: A generalization of Dijkstra’s calculus. ACM Trans. Program. Lang. Syst. 11(4), 517–561 (1989)

    Article  Google Scholar 

  44. Parnas D.: A generalized control structure and its formal definition. Commun. ACM 26(8), 572–581 (1983)

    Article  MATH  Google Scholar 

  45. Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: Sutcliffe G., Ternovska E., Schulz S. (eds.) Proceedings of the 8th International Workshop on the Implementation of Logics, pp. 3–13 (2010)

  46. von Wright J.: Towards a refinement algebra. Sci. Comput. Program. 51(1–2), 23–45 (2004)

    Article  MATH  Google Scholar 

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  1. Universität Ulm, Ulm, Germany

    Walter Guttmann

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  1. Walter Guttmann
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Guttmann, W. Algebras for iteration and infinite computations. Acta Informatica 49, 343–359 (2012). https://doi.org/10.1007/s00236-012-0162-2

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