Correctness of Program Transformations as a Termination Problem

  • Conrad Rau
  • David Sabel
  • Manfred Schmidt-Schauß
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)


The diagram-based method to prove correctness of program transformations includes the computation of (critical) overlappings between the analyzed program transformation and the (standard) reduction rules which result in so-called forking diagrams. Such diagrams can be seen as rewrite rules on reduction sequences which abstract away the expressions and allow additional expressive power, like transitive closures of reductions. In this paper we clarify the meaning of forking diagrams using interpretations as infinite term rewriting systems. We then show that the termination problem of forking diagrams as rewrite rules can be encoded into the termination problem for conditional integer term rewriting systems, which can be solved by automated termination provers. Since the forking diagrams can be computed automatically, the results of this paper are a big step towards a fully automatic prover for the correctness of program transformations.


Transitive Closure Program Transformation Termination Problem Reduction Sequence Standard Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Barendregt, H.P.: The Lambda Calculus. Its Syntax and Semantics. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  2. 2.
    Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Automated certified proofs with CiME3. In: Schmidt-Schauß, M. (ed.) RTA 22. LIPIcs, vol. 10, pp. 21–30. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  3. 3.
    Ford, J., Mason, I.A.: Formal foundations of operational semantics. Higher Order Symbol. Comput. 16(3), 161–202 (2003)zbMATHCrossRefGoogle Scholar
  4. 4.
    Fuhs, C., Giesl, J., Plücker, M., Schneider-Kamp, P., Falke, S.: Proving Termination of Integer Term Rewriting. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 32–47. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic Termination Proofs in the Dependency Pair Framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean Termination Tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Kutzner, A., Schmidt-Schauß, M.: A nondeterministic call-by-need lambda calculus. In: Felleisen, M., Hudak, P., Queinnec, C. (eds.) 3rd ICFP, pp. 324–335. ACM (1998)Google Scholar
  8. 8.
    Machkasova, E., Turbak, F.A.: A Calculus for Link-Time Compilation. In: Smolka, G. (ed.) ESOP 2000. LNCS, vol. 1782, pp. 260–274. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  9. 9.
    Mann, M.: Congruence of bisimulation in a non-deterministic call-by-need lambda calculus. Electron. Notes Theor. Comput. Sci. 128(1), 81–101 (2005)CrossRefGoogle Scholar
  10. 10.
    Morris, J.H.: Lambda-Calculus Models of Programming Languages. PhD thesis. MIT (1968)Google Scholar
  11. 11.
    Niehren, J., Sabel, D., Schmidt-Schauß, M., Schwinghammer, J.: Observational semantics for a concurrent lambda calculus with reference cells and futures. Electron. Notes Theor. Comput. Sci. 173, 313–337 (2007)CrossRefGoogle Scholar
  12. 12.
    Niehren, J., Schwinghammer, J., Smolka, G.: A concurrent lambda calculus with futures. Theoret. Comput. Sci. 364(3), 338–356 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Peyton Jones, S.: Haskell 98 language and libraries: the Revised Report. Cambridge University Press (2003)Google Scholar
  14. 14.
    Peyton Jones, S., Gordon, A., Finne, S.: Concurrent Haskell. In: Steele, G. (ed.) 23th POPL, pp. 295–308. ACM (1996)Google Scholar
  15. 15.
    Plotkin, G.D.: Call-by-name, call-by-value, and the lambda-calculus. Theoret. Comput. Sci. 1, 125–159 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Rau, C., Schmidt-Schauß, M.: Towards correctness of program transformations through unification and critical pair computation. In: Fernandez, M. (ed.) 24th UNIF. EPTCS, vol. 42, pp. 39–54 (2010)Google Scholar
  17. 17.
    Rau, C., Schmidt-Schauß, M.: A unification algorithm to compute overlaps in a call-by-need lambda-calculus with variable-binding chains. In: Baader, F., Morawska, B., Otop, J. (eds.) 25th UNIF, pp. 35–41 (2011)Google Scholar
  18. 18.
    Sabel, D., Schmidt-Schauß, M.: A call-by-need lambda-calculus with locally bottom-avoiding choice: Context lemma and correctness of transformations. Math. Structures Comput. Sci. 18(03), 501–553 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sabel, D., Schmidt-Schauß, M.: A contextual semantics for Concurrent Haskell with futures. In: Hanus, M. (ed.) 13th PPDP, pp. 101–112. ACM (2011)Google Scholar
  20. 20.
    Schmidt-Schauß, M., Sabel, D.: On generic context lemmas for higher-order calculi with sharing. Theoret. Comput. Sci. 411(11-13), 1521–1541 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Schmidt-Schauß, M., Schütz, M., Sabel, D.: Safety of Nöcker’s strictness analysis. J. Funct. Programming 18(04), 503–551 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Wells, J.B., Plump, D., Kamareddine, F.: Diagrams for Meaning Preservation. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 88–106. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Conrad Rau
    • 1
  • David Sabel
    • 1
  • Manfred Schmidt-Schauß
    • 1
  1. 1.Goethe-UniversityFrankfurt am MainGermany

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