Advertisement

Automated Proofs of Unique Normal Forms w.r.t. Conversion for Term Rewriting Systems

  • Takahito AotoEmail author
  • Yoshihito Toyama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11715)

Abstract

The notion of normal forms is ubiquitous in various equivalent transformations. Confluence (CR), one of the central properties of term rewriting systems (TRSs), concerns uniqueness of normal forms. Yet another such property, which is weaker than confluence, is the property of unique normal forms w.r.t. conversion (UNC). Recently, automated confluence proof of TRSs has caught attentions; some powerful confluence tools integrating multiple methods for (dis)proving the CR property of TRSs have been developed. In contrast, there have been little efforts on (dis)proving the UNC property automatically yet. In this paper, we report on a UNC prover combining several methods for (dis)proving the UNC property. We present an equivalent transformation of TRSs preserving UNC, as well as some new criteria for (dis)proving UNC.

Notes

Acknowledgements

Thanks are due to the anonymous reviewers of the previous versions of the paper. This work is partially supported by JSPS KAKENHI No. 18K11158.

References

  1. 1.
    Aoto, T., Toyama, Y.: Top-down labelling and modularity of term rewriting systems. Research Report IS-RR-96-0023F, School of Information Science, JAIST (1996)Google Scholar
  2. 2.
    Aoto, T., Toyama, Y.: On composable properties of term rewriting systems. In: Hanus, M., Heering, J., Meinke, K. (eds.) ALP/HOA -1997. LNCS, vol. 1298, pp. 114–128. Springer, Heidelberg (1997).  https://doi.org/10.1007/BFb0027006CrossRefGoogle Scholar
  3. 3.
    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).  https://doi.org/10.1007/978-3-642-02348-4_7CrossRefzbMATHGoogle Scholar
  4. 4.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  5. 5.
    Bergstra, J.A., Klop, J.W.: Conditional rewrite rules: confluence and termination. J. Comput. Syst. Sci. 32, 323–362 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dauchet, M., Heuillard, T., Lescanne, P., Tison, S.: Decidability of the confluence of finite ground term rewrite systems and of other related term rewrite systems. Inf. Comput. 88, 187–201 (1990)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gramlich, B.: Modularity in term rewriting revisited. Theor. Comput. Sci. 464, 3–19 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jaffar, J.: Efficient unification over infinite terms. New Gener. Comput. 2, 207–219 (1984)CrossRefGoogle Scholar
  10. 10.
    Kahrs, S., Smith, C.: Non-\(\omega \)-overlapping TRSs are UN. In: Proceedings of 1st FSCD. LIPIcs, vol. 52, pp. 22:1–22:17. Schloss Dagstuhl (2016)Google Scholar
  11. 11.
    Klein, D., Hirokawa, N.: Confluence of non-left-linear TRSs via relative termination. In: Bjørner, N., Voronkov, A. (eds.) LPAR 2012. LNCS, vol. 7180, pp. 258–273. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-28717-6_21CrossRefGoogle Scholar
  12. 12.
    Klop, J.: Combinatory Reduction Systems, Mathematical Centre Tracts, vol. 127. CWI, Amsterdam (1980)Google Scholar
  13. 13.
    Klop, J.W., de Vrijer, R.: Extended term rewriting systems. In: Kaplan, S., Okada, M. (eds.) CTRS 1990. LNCS, vol. 516, pp. 26–50. Springer, Heidelberg (1991).  https://doi.org/10.1007/3-540-54317-1_79CrossRefGoogle Scholar
  14. 14.
    Nagele, J., Felgenhauer, B., Middeldorp, A.: CSI: new evidence – a progress report. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 385–397. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63046-5_24CrossRefGoogle Scholar
  15. 15.
    O’Donnell, M.J. (ed.): Computing in Systems Described by Equations. LNCS, vol. 58. Springer, Heidelberg (1977).  https://doi.org/10.1007/3-540-08531-9CrossRefzbMATHGoogle Scholar
  16. 16.
    van Oostrom, V.: Developing developments. Theor. Comput. Sci. 175(1), 159–181 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Radcliffe, N.R., Moreas, L.F.T., Verma, R.M.: Uniqueness of normal forms for shallow term rewrite systems. ACM Trans. Comput. Logic 18(2), 17:1–17:20 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rapp, F., Middeldorp, A.: Automating the first-order theory of rewriting for left-linear right-ground rewrite systems. In: Proceedings of 1st FSCD. LIPIcs, vol. 52, pp. 36:1–36:17. Schloss Dagstuhl (2016)Google Scholar
  19. 19.
    Toyama, Y.: Commutativity of term rewriting systems. In: Fuchi, K., Kott, L. (eds.) Programming of Future Generation Computers II, North-Holland, pp. 393–407 (1988)Google Scholar
  20. 20.
    Toyama, Y.: Confluent term rewriting systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, p. 1. Springer, Heidelberg (2005).  https://doi.org/10.1007/978-3-540-32033-3_1. Slides at http://www.nue.ie.niigata-u.ac.jp/toyama/user/toyama/slides/toyama-RTA05.pdfCrossRefzbMATHGoogle Scholar
  21. 21.
    Toyama, Y., Oyamaguchi, M.: Conditional linearization of non-duplicating term rewriting systems. IEICE Trans. Inf. Syst. E84-D(4), 439–447 (2001)Google Scholar
  22. 22.
    de Vrijer, R.: Conditional linearization. Indagationes Math. 10(1), 145–159 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Natural SciencesNiigata UniversityNiigataJapan
  2. 2.RIECTohoku UniversitySendaiJapan

Personalised recommendations