Certification of Classical Confluence Results for Left-Linear Term Rewrite Systems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)

Abstract

This paper presents the first formalization of three classic confluence criteria for first-order term rewrite systems by Huet and Toyama. We have formalized proofs, showing that (1) linear strongly closed systems, (2) left-linear parallel closed systems, and (3) left-linear almost parallel closed systems are confluent. The third result is extended to commutation. The proofs were carried out in the proof assistant Isabelle/HOL as part of the library IsaFoR and integrated into the certifier CeTA, significantly increasing the number of certifiable proofs produced by automatic confluence tools.

References

  1. 1.
    Aoto, T.: Disproving confluence of term rewriting systems by interpretation and ordering. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS, vol. 8152, pp. 311–326. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40885-4_22 CrossRefGoogle Scholar
  2. 2.
    Aoto, T., Yoshida, J., Toyama, Y.: Proving confluence of term rewriting systems automatically. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 93–102. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02348-4_7 CrossRefGoogle Scholar
  3. 3.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  4. 4.
    Blanqui, F., Koprowski, A.: CoLoR, a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates. Math. Struct. Comput. Sci. 21(4), 827–859 (2011). doi:10.1017/S0960129511000120 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Automated certified proofs with CiME3. In: Schmidt-Schauß, M. (ed.) RTA 2011. LIPIcs, vol. 10. pp. 21–30. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2011). doi:10.4230/LIPIcs.RTA.2011.21
  6. 6.
    Felgenhauer, B., Thiemann, R.: Reachability analysis with state-compatible automata. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 347–359. Springer, Heidelberg (2014). doi:10.1007/978-3-319-04921-2_28 CrossRefGoogle Scholar
  7. 7.
    Geser, A., Middeldorp, A., Ohlebusch, E., Zantema, H.: Relative undecidability in the term rewriting, part 2: The confluence hierarchy. Inf. Comput. 178(1), 132–148 (2002). doi:10.1006/inco.2002.3150 MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hirokawa, N., Klein, D.: Saigawa: a confluence tool. In: Hirokawa, N., Middeldorp, A. (eds.) IWC 2012, p. 49 (2012). http://cl-informatik.uibk.ac.at/events/iwc-2012/
  9. 9.
    Hirokawa, N., Middeldorp, A.: Commutation via relative termination. In: Hirokawa, N., van Oostrom, V. (eds.) IWC 2013, pp. 29–33 (2013). http://www.jaist.ac.jp/~hirokawa/iwc2013/
  10. 10.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980). doi:10.1145/322217.322230 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Klein, D., Hirokawa, N.: Confluence of non-left-linear TRSs via relative termination. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 258–273. Springer, Heidelberg (2012). doi:10.1007/978-3-642-28717-6_21 CrossRefGoogle Scholar
  12. 12.
    Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press (1970)Google Scholar
  13. 13.
    Nagele, J., Felgenhauer, B., Middeldorp, A.: Improving automatic confluence analysis of rewrite systems by redundant rules. In: Fernández, M. (ed.) RTA 2015. LIPIcs, vol. 36, pp. 257–268. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2015). doi:10.4230/LIPIcs.RTA.2015.257
  14. 14.
    Nagele, J., Thiemann, R.: Certification of confluence proofs using CeTA. In: Aoto, T., Kesner, D. (eds.) IWC 2014, pp. 19–23 (2014). http://www.nue.riec.tohoku.ac.jp/iwc2014/
  15. 15.
    Nagele, J., Zankl, H.: Certified rule labeling. In: Fernández, M. (ed.) RTA 2015. LIPIcs, vol. 36, pp. 269–284. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2015). doi:10.4230/LIPIcs.RTA.2015.269
  16. 16.
    Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHGoogle Scholar
  17. 17.
    van Oostrom, V.: Developing developments. Theoret. Comput. Sci. 175(1), 159–181 (1997). doi:10.1016/S0304-3975(96)00173-9 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shintani, K., Hirokawa, N.: CoLL: A confluence tool for left-linear term rewrite systems. In: Felty, A., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 127–136. Springer, Heidelberg (2015). doi:10.1007/978-3-319-21401-6_8 Google Scholar
  19. 19.
    Sternagel, C., Thiemann, R.: The certification problem format. In: Benzmüller, C., Woltzenlogel Paleo, B. (eds.) UITP 2014. EPTCS, vol. 167, pp. 61–72. Open Publishing Association (2014). doi:10.4204/EPTCS.167.8
  20. 20.
    Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical ComputerScience, vol. 55. Cambridge University Press, Cambridge (2003)Google Scholar
  21. 21.
    Thiemann, R., Sternagel, C.: Certification of termination proofs using CeTA. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 452–468. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03359-9_31 CrossRefGoogle Scholar
  22. 22.
    Toyama, Y.: Commutativity of term rewriting systems. In: Fuchi, K., Kott, L. (eds.) Programming of Future Generation Computers II, pp. 393–407. North-Holland (1988)Google Scholar
  23. 23.
    Zankl, H., Felgenhauer, B., Middeldorp, A.: CSI – a confluence tool. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 499–505. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22438-6_38 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of InnsbruckInnsbruckAustria

Personalised recommendations