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Tuple Interpretations for Termination of Term Rewriting

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Abstract

Interpretation methods constitute a foundation of the termination analysis of term rewriting. From time to time, remarkable instances of interpretation methods appeared, such as polynomial interpretations, matrix interpretations, arctic interpretations, and their variants. In this paper, we introduce a general framework, the tuple interpretation method, that subsumes these variants as well as many previously unknown interpretation methods as instances. Employing the notion of derivers, we prove the soundness of the proposed method in an elegant way. We implement the proposed method in the termination prover NaTT and verify its significance through experiments.

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Notes

  1. For clarity, we omit the rules that define \(\mathtt{mul}\); the analyses we make on \({\mathcal {R}}_\mathtt{fact}\) later on can be easily extended for coping with those rules.

  2. In the literature, sorted signatures are given more assumptions such as monotonicity or regularity. For the purpose of this paper, these assumptions are not necessary.

References

  1. Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236(1–2), 133–178 (2000). https://doi.org/10.1016/S0304-3975(99)00207-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Book  MATH  Google Scholar 

  3. Barrett, C.W., Tinelli, C.: Satisfiability modulo theories. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking, pp. 305–343. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-10575-8_11

  4. Ben-Amram, A.M., Codish, M.: A SAT-based approach to size change termination with global ranking functions. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 218–232. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_16

  5. 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). https://doi.org/10.1017/S0960129511000120

    Article  MathSciNet  MATH  Google Scholar 

  6. Contejean, É., 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 fuer Informatik, Dagstuhl (2011). https://doi.org/10.4230/LIPIcs.RTA.2011.21

  7. Courtieu, P., Gbedo, G., Pons, O.: Improved matrix interpretation. In: van Leeuwen, J., Muscholl, A., Peleg, D., Pokorný, J., Rumpe, B. (eds.) SOFSEM 2010. LNCS, vol. 5901, pp. 283–295. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11266-9_24

  8. de Moura, L.M., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24

  9. Dershowitz, N.: 33 examples of termination. In: Comon, H., Jounnaud, J.P. (eds.) Term Rewriting. pp. 16–26. Springer, Cham (1995). https://doi.org/10.1007/3-540-59340-3_2

  10. Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40(2–3), 195–220 (2008). https://doi.org/10.1007/s10817-007-9087-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: Maximal termination. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 110–125. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70590-1_8

  12. Giesl, J., Thiemann, R., Schneider-Kamp, P.: The dependency pair framework: Combining techniques for automated termination proofs. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS, vol. 3452, pp. 301–331. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-32275-7_21

  13. Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. J. Autom. Reason. 37(3), 155–203 (2006). https://doi.org/10.1007/s10817-006-9057-7

    Article  MathSciNet  MATH  Google Scholar 

  14. Giesl, J., Aschermann, C., Brockschmidt, M., Emmes, F., Frohn, F., Fuhs, C., Hensel, J., Otto, C., Plücker, M., Schneider-Kamp, P., Ströder, T., Swiderski, S., Thiemann, R.: Analyzing program termination and complexity automatically with AProVE. J. Autom. Reason. 58(1), 3–31 (2017). https://doi.org/10.1007/s10817-016-9388-y

    Article  MathSciNet  MATH  Google Scholar 

  15. Giesl, J., Rubio, A., Sternagel, C., Waldmann, J., Yamada, A.: The termination and complexity competition. In: Beyer, D., Huisman, M., Kordon, F., Steffen, B. (eds.) TACAS 2019 (3). LNCS, vol. 11429, pp. 156–166. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17502-3_10

  16. Gutiérrez, R., Lucas, S.: mu-term: Verify termination properties automatically (system description). In: Peltier, N., Sofronie-Stokkermans, V. (eds.) IJCAR 2020 (2). LNCS, vol. 12167, pp. 436–447. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51054-1_28

  17. Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-25979-4_18

  18. Hirokawa, N., Middeldorp, A.: Polynomial interpretations with negative coefficients. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNAI, vol. 3249, pp. 185–198. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30210-0_16

  19. Hofbauer, D., Waldmann, J.: Termination of string rewriting with matrix interpretations. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 328–342. Springer, Heidelberg (2006). https://doi.org/10.1007/11805618_25

  20. Jouannaud, J., Rubio, A.: The higher-order recursive path ordering. In: LICS 1999, pp. 402–411. IEEE Computer Society (1999). https://doi.org/10.1109/LICS.1999.782635

  21. Kop, C., Vale, D.: Tuple interpretations for higher-order complexity. In: Kobayashi, N. (ed.) FSCD 2021. LIPIcs, vol. 195, pp. 31:1–31:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern (2021). https://doi.org/10.4230/LIPIcs.FSCD.2021.31

  22. Kop, C., van Raamsdonk, F.: Dynamic dependency pairs for algebraic functional systems. Log. Methods Comput. Sci. (2012). https://doi.org/10.2168/LMCS-8(2:10)2012

  23. Koprowski, A., Waldmann, J.: Max/plus tree automata for termination of term rewriting. Acta Cybern. 19(2), 357–392 (2009)

    MathSciNet  MATH  Google Scholar 

  24. 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). https://doi.org/10.1007/978-3-642-02348-4_21

  25. Kusakari, K., Nakamura, M., Toyama, Y.: Argument filtering transformation. In: Nadathur, G. (ed.) PPDP 1999. LNCS, vol. 1702, pp. 47–61. Springer, Heidelberg (1999). https://doi.org/10.1007/10704567_3

  26. Lankford, D.: Canonical algebraic simplification in computational logic. Tech. Rep. ATP-25, University of Texas (1975)

  27. Lucas, S., Gutiérrez, R.: Automatic synthesis of logical models for order-sorted first-order theories. J. Autom. Reason. 60(4), 465–501 (2018). https://doi.org/10.1007/s10817-017-9419-3

    Article  MathSciNet  MATH  Google Scholar 

  28. Manna, Z., Ness, S.: On the termination of Markov algorithms. In: The 3rd Hawaii International Conference on System Science, pp. 789–792 (1970)

  29. Sternagel, C., Thiemann, R.: The certification problem format. In: Benzmüller, C., Paleo, B.W. (eds.) UITP 2014. EPTCS, vol. 167, pp. 61–72 (2014). https://doi.org/10.4204/EPTCS.167.8

  30. Sternagel, C., Thiemann, R.: Formalizing monotone algebras for certification of termination and complexity proofs. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 441–455. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-319-08918-8_30

  31. Stump, A., Sutcliffe, G., Tinelli, C.: StarExec: A cross-community infrastructure for logic solving. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR. LNCS, vol. 8562, pp. 367–373. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_28

  32. 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). https://doi.org/10.1007/978-3-642-03359-9_31

  33. Thiemann, R., Schöpf, J., Sternagel, C., Yamada, A.: Certifying the Weighted Path Order (Invited Talk). In: Ariola, Z.M. (ed.) FSCD 2020. LIPIcs, vol. 167, pp. 4:1–4:20. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl (2020). https://doi.org/10.4230/LIPIcs.FSCD.2020.4

  34. TPDB: The termination problem data base. http://termination-portal.org/wiki/TPDB

  35. Watson, T., Goguen, J., Thatcher, J., Wagner, E.: An initial algebra approach to the specification, correctness, and implementation of abstract data types. In: Current Trends in Programming Methodology. Prentice Hall, Englewood Cliffs (1976)

  36. Yamada, A.: Multi-dimensional interpretations for termination of term rewriting. In: Platzer, A., Sutcliffe, G. (eds.) CADE-28. LNCS, vol. 12699, pp. 273–290. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79876-5_16

  37. Yamada, A., Kusakari, K., Sakabe, T.: Nagoya Termination Tool. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-319-08918-8_32

  38. Yamada, A., Kusakari, K., Sakabe, T.: A unified order for termination proving. Sci. Comput. Program. 111, 110–134 (2015). https://doi.org/10.1016/j.scico.2014.07.009

    Article  Google Scholar 

  39. Zantema, H.: Termination of term rewriting: interpretation and type elimination. J. Symb. Comput. 17(1), 23–50 (1994). https://doi.org/10.1006/jsco.1994.1003

    Article  MathSciNet  MATH  Google Scholar 

  40. Zantema, H.: The termination hierarchy for term rewriting. Appl. Algebr. Eng. Commun. Comput. 12(1/2), 3–19 (2001). https://doi.org/10.1007/s002000100061

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author would like to thank Aart Middeldorp for discussions on the power of existing methods, and Aaron Stump and his team for the StarExec environment, without which the experiments reported in the paper would not be feasible. The author also thanks the anonymous reviewers of previous versions of the paper for their detailed comments that improved the presentation a lot. This work was partly supported by the Austrian Science Fund (FWF) projects Y757 and P27502, and the Japan Science and Technology Agency (JST) project ERATO MMSD.

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  1. Cyber Physical Security Research Center (CPSEC), National Institute of Advanced Industrial Science and Technology (AIST), 2-3-26, Aomi, Koto-ku, Tokyo, 135-0064, Japan

    Akihisa Yamada

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Yamada, A. Tuple Interpretations for Termination of Term Rewriting. J Autom Reasoning 66, 667–688 (2022). https://doi.org/10.1007/s10817-022-09640-4

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