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A Lambda-Free Higher-Order Recursive Path Order

  • Jasmin Christian Blanchette
  • Uwe Waldmann
  • Daniel Wand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)

Abstract

We generalize the recursive path order (RPO) to higher-order terms without \(\lambda \)-abstraction. This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work. It has many useful properties, including well-foundedness, transitivity, stability under substitution, and the subterm property. It appears promising as the basis of a higher-order superposition calculus.

Notes

Acknowledgment

We are grateful to Stephan Merz, Tobias Nipkow, and Christoph Weidenbach for making this research possible; to Heiko Becker and Dmitriy Traytel for proving some theorems on the lexicographic and multiset extensions in Isabelle/HOL; to Carsten Fuhs for his invaluable feedback and his experiments; to Alexander Steen for pointing us to related work; and to Simon Cruanes, Mark Summerfield, Dmitriy Traytel, and the anonymous reviewers for suggesting textual improvements.

Blanchette has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 713999, Matryoshka). Wand is supported by the Deutsche Forschungsgemeinschaft (DFG) grant Hardening the Hammer (NI 491/14-1).

References

  1. 1.
    Aoto, T., Yamada, T.: Termination of simply typed term rewriting by translation and labelling. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 380–394. Springer, Heidelberg (2003). doi: 10.1007/3-540-44881-0_27 CrossRefGoogle Scholar
  2. 2.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Benzmüller, C., Miller, D.: Automation of higher-order logic. In: Siekmann, J.H. (ed.) Computational Logic, Handbook of the History of Logic, vol. 9, pp. 215–254. Elsevier (2014)Google Scholar
  5. 5.
    Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formalized Reasoning 9(1), 101–148 (2016)MathSciNetGoogle Scholar
  6. 6.
    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.) ITP 2010. LNCS, vol. 6172, pp. 131–146. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14052-5_11 CrossRefGoogle Scholar
  7. 7.
    Blanchette, J.C., Waldmann, U., Wand, D.: Formalization of recursive path orders for lambda-free higher-order terms. Archive of Formal Proofs, formal proof development (2016). https://isa-afp.org/entries/Lambda_Free_RPOs.shtml
  8. 8.
    Blanchette, J.C., Waldmann, U., Wand, D.: A lambda-free higher-order recursive path order. Technical report (2016). http://people.mpi-inf.mpg.de/jblanche/lambda_free_rpo_rep.pdf
  9. 9.
    Blanqui, F.: Termination and confluence of higher-order rewrite systems. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 47–61. Springer, Heidelberg (2000). doi: 10.1007/10721975_4 CrossRefGoogle Scholar
  10. 10.
    Blanqui, F., Jouannaud, J., Rubio, A.: The computability path ordering. Log. Meth. Comput. Sci. 11(4) (2015)Google Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bofill, M., Borralleras, C., Rodríguez-Carbonell, E., Rubio, A.: The recursive path and polynomial ordering for first-order and higher-order terms. J. Log. Comput. 23(1), 263–305 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bofill, M., Rubio, A.: Paramodulation with non-monotonic orderings and simplification. J. Autom. Reasoning 50(1), 51–98 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Codish, M., Giesl, J., Schneider-Kamp, P., Thiemann, R.: SAT solving for termination proofs with recursive path orders and dependency pairs. J. Autom. Reasoning 49(1), 53–93 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Contejean, É., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Automated certified proofs with CiME3. In: Schmidt-Schauß, M. (ed.) Rewriting Techniques and Applications (RTA 2011). Leibniz International Proceedings in Informatics (LIPIcs), vol. 10, pp. 21–30. Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2011)Google Scholar
  16. 16.
    Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Commun. ACM 22(8), 465–476 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(T): Fast decision procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 175–188. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-27813-9_14 CrossRefGoogle Scholar
  18. 18.
    Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15(2), 81–91 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hirokawa, N., Middeldorp, A., Zankl, H.: Uncurrying for termination and complexity. J. Autom. Reasoning 50(3), 279–315 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Huet, G., Oppen, D.C.: Equations and rewrite rules: A survey. In: Book, R.V. (ed.) Formal Language Theory: Perspectives and Open Problems, pp. 349–405. Academic Press (1980)Google Scholar
  21. 21.
    Hughes, R.J.M.: Super-combinators: A new implementation method for applicative languages. In: ACM Symposium on LISP and Functional Programming (LFP 1982), pp. 1–10. ACM Press (1982)Google Scholar
  22. 22.
    Jouannaud, J.-P., Rubio, A.: A recursive path ordering for higher-order terms in \(\eta \)-long \(\beta \)-normal form. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 108–122. Springer, Heidelberg (1996). doi: 10.1007/3-540-61464-8_46 CrossRefGoogle Scholar
  23. 23.
    Jouannaud, J., Rubio, A.: Polymorphic higher-order recursive path orderings. J. ACM 54(1), 2:1–2:48 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kennaway, R., Klop, J.W., Sleep, M.R., de Vries, F.: Comparing curried and uncurried rewriting. J. Symb. Comput. 21(1), 15–39 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kop, C.: Higher Order Termination. Ph.D. thesis, Vrije Universiteit Amsterdam (2012)Google Scholar
  26. 26.
    Kop, C., Raamsdonk, F.: A higher-order iterative path ordering. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 697–711. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-89439-1_48 CrossRefGoogle Scholar
  27. 27.
    Lifantsev, M., Bachmair, L.: An LPO-based termination ordering for higher-order terms without \(\lambda \)-abstraction. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 277–293. Springer, Heidelberg (1998). doi: 10.1007/BFb0055142 CrossRefGoogle Scholar
  28. 28.
    Lysne, O., Piris, J.: A termination ordering for higher order rewrite systems. In: Hsiang, J. (ed.) RTA 1995. LNCS, vol. 914, pp. 26–40. Springer, Heidelberg (1995). doi: 10.1007/3-540-59200-8_45 CrossRefGoogle Scholar
  29. 29.
    Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 371–443. Elsevier and MIT Press (2001)Google Scholar
  30. 30.
    Nipkow, T.: Higher-order critical pairs. In: Logic in Computer Science (LICS 1991), pp. 342–349. IEEE Computer Society (1991)Google Scholar
  31. 31.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  32. 32.
    Sternagel, C., Thiemann, R.: Generalized and formalized uncurrying. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 243–258. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-24364-6_17 CrossRefGoogle Scholar
  33. 33.
    Thiemann, R., Allais, G., Nagele, J.: On the formalization of termination techniques based on multiset orderings. In: Tiwari, A. (ed.) Rewriting Techniques and Applications (RTA 2012). LIPIcs, vol. 15, pp. 339–354. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2012)Google Scholar
  34. 34.
    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
  35. 35.
    Toyama, Y.: Termination of S-expression rewriting systems: lexicographic path ordering for higher-order terms. In: Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 40–54. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-25979-4_3 CrossRefGoogle Scholar
  36. 36.
    Turner, D.A.: A new implementation technique for applicative languages. Softw.: Pract. Experience 9(1), 31–49 (1979)zbMATHGoogle Scholar
  37. 37.
    Yamada, T.: Confluence and termination of simply typed term rewriting systems. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 338–352. Springer, Heidelberg (2001). doi: 10.1007/3-540-45127-7_25 CrossRefGoogle Scholar
  38. 38.
    Zantema, H.: Termination. In: Bezem, M., Klop, J.W., de Vrijer, R. (eds.) Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55, pp. 181–259. Cambridge University Press (2003)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Jasmin Christian Blanchette
    • 1
    • 2
    • 3
  • Uwe Waldmann
    • 3
  • Daniel Wand
    • 3
    • 4
  1. 1.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  2. 2.Inria Nancy – Grand EstVillers-lès-NancyFrance
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany
  4. 4.Institut für InformatikTechnische Universität MünchenMunichGermany

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