Skip to main content
SpringerLink
Log in

CSI: New Evidence – A Progress Report

  • Conference paper
  • First Online:
Book cover Automated Deduction – CADE 26 (CADE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10395))

Included in the following conference series:

Abstract

CSI is a strong automated confluence prover for rewrite systems which has been in development since 2010. In this paper we report on recent extensions that make CSI more powerful, secure, and useful. These extensions include improved confluence criteria but also support for uniqueness of normal forms. Most of the implemented techniques produce machine-readable proof output that can be independently verified by an external tool, thus increasing the trust in CSI. We also report on CSI\(\mathbf {\hat{~}}\)oho, a tool built on the same framework and similar ideas as CSI that automatically checks confluence of higher-order rewrite systems.

This research is supported by FWF (Austrian Science Fund) project P27528.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Available from http://coco.nue.riec.tohoku.ac.jp/2013, http://coco.nue.riec.tohoku.ac.jp/2014, http://coco.nue.riec.tohoku.ac.jp/2015, http://coco.nue.riec.tohoku.ac.jp/2016, under Entrants.

  2. 2.

    http://cops.uibk.ac.at.

  3. 3.

    http://cl-informatik.uibk.ac.at/software/mkbtt.

  4. 4.

    In fact, CSI uses a modified definition of \(\mathcal {S}^e\) that avoids adding extended rules for rules where the same linear variable appears as an argument of the two top-flattenings of the left-hand and right-hand sides of the rule, using the same AC symbol. In the example, this applies to the first two rules. We still have \({\rightarrow _{\mathcal {S}/\mathsf {AC}}} = {\rightarrow _{\mathcal {S}^e,\mathsf {AC}}} \cdot \sim _\mathsf {AC}\).

  5. 5.

    This can be shown using tree automata techniques [11].

  6. 6.

    A term is a pattern if free variables only have distinct bound variables as arguments.

  7. 7.

    A development step contracts multiple, non-overlapping but possibly nested redexes at once.

  8. 8.

    IsaFoR/CeTA and CPF are available at http://cl-informatik.uibk.ac.at/software/ceta.

  9. 9.

    Full details are available from CSI’s website.

References

  1. Aoto, T., Hirokawa, N., Nagele, J., Nishida, N., Zankl, H.: Confluence competition 2015. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 101–104. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_5

    Chapter  Google Scholar 

  2. Aoto, T., Toyama, Y.: A reduction-preserving completion for proving confluence of non-terminating term rewriting systems. LMCS 8(1: 31), 1–29 (2012). doi:10.2168/LMCS-8(1:31)2012

    MathSciNet  MATH  Google Scholar 

  3. Aoto, T., Toyama, Y.: Ground confluence prover based on rewriting induction. In: Proceedings of 1st FSCD. LIPIcs, vol. 52, pp. 33: 1–33: 12 (2016). doi:10.4230/LIPIcs.FSCD.2016.33

  4. Aoto, T., Toyama, Y., Uchida, K.: Proving confluence of term rewriting systems via persistency and decreasing diagrams. In: Dowek, G. (ed.) RTA 2014. LNCS, vol. 8560, pp. 46–60. Springer, Cham (2014). doi:10.1007/978-3-319-08918-8_4

    Google Scholar 

  5. 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

    Chapter  Google Scholar 

  6. Appel, C., van Oostrom, V., Simonsen, J.G.: Higher-order (non-)modularity. In: Proceedings of 21st RTA. LIPIcs, vol. 6, pp. 17–32 (2010). doi:10.4230/LIPIcs.RTA.2010.17

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

    Book  MATH  Google Scholar 

  8. Felgenhauer, B.: Deciding confluence of ground term rewrite systems in cubic time. In: Proceedings of 23rd RTA. LIPIcs, vol. 15, pp. 165–175 (2012). doi:10.4230/LIPIcs.RTA.2012.165

  9. Felgenhauer, B.: Efficiently deciding uniqueness of normal forms and unique normalization for ground TRSs. In: Proceedings of 5th IWC, pp. 16–20 (2016)

    Google Scholar 

  10. Felgenhauer, B., Middeldorp, A., Zankl, H., Oostrom, V.O.: Layer systems for proving confluence. ACM TOCL 16(2: 14), 1–32 (2015). doi:10.1145/2710017

    Article  MathSciNet  MATH  Google Scholar 

  11. 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, Cham (2014). doi:10.1007/978-3-319-04921-2_28

    Chapter  Google Scholar 

  12. Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems. JACM 27(4), 797–821 (1980). doi:10.23638/LMCS-13(2:4)2017

    Article  MathSciNet  MATH  Google Scholar 

  13. Jouannaud, J.P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. Comput. 15(4), 1155–1194 (1986). doi:10.1137/0215084

    Article  MathSciNet  MATH  Google Scholar 

  14. Kahrs, S., Smith, C.: Non-\(\omega \)-overlapping TRSs are UN. In: Proceedings of 1st FSCD. LIPIcs, vol. 52, pp. 22: 1–22: 17 (2016). doi:10.4230/LIPIcs.FSCD.2016.22

  15. 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). doi:10.1007/978-3-642-28717-6_21

    Chapter  Google Scholar 

  16. Klop, J.: Combinatory reduction systems. Ph.D. thesis, Utrecht University (1980)

    Google Scholar 

  17. Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)

    Google Scholar 

  18. Kop, C.: Higher order termination. Ph.D. thesis, Vrije Universiteit, Amsterdam (2012)

    Google Scholar 

  19. Kusakari, K., Isogai, Y., Sakai, M., Blanqui, F.: Static dependency pair method based on strong computability for higher-order rewrite systems. IEICE TIS 92–D(10), 2007–2015 (2009)

    Google Scholar 

  20. Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. TCS 192(1), 3–29 (1998). doi:10.1016/S0304-3975(97)00143-6

    Article  MathSciNet  MATH  Google Scholar 

  21. Miller, D.: A logic programming language with lambda-abstraction, function variables, and simple unification. JLP 1(4), 497–536 (1991). doi:10.1093/logcom/1.4.497

    MathSciNet  MATH  Google Scholar 

  22. Nagele, J., Felgenhauer, B., Middeldorp, A.: Improving automatic confluence analysis of rewrite systems by redundant rules. In: Proceedings of 26th RTA. LIPIcs, vol. 36, pp. 257–268 (2015). doi:10.4230/LIPIcs.RTA.2015.257

  23. Nagele, J., Felgenhauer, B., Zankl, H.: Certifying confluence proofs via relative termination and rule labeling. LMCS (to appear) (2017)

    Google Scholar 

  24. Nagele, J., Middeldorp, A.: Certification of classical confluence results for left-linear term rewrite systems. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 290–306. Springer, Cham (2016). doi:10.1007/978-3-319-43144-4_18

    Chapter  Google Scholar 

  25. Nagele, J., Thiemann, R.: Certification of confluence proofs using CeTA. In: Proceedings of 3rd IWC, pp. 19–23 (2014)

    Google Scholar 

  26. Nelson, G., Oppen, D.: Fast decision procedures based on congruence closure. JACM 27(2), 356–364 (1980). doi:10.1145/322186.322198

    Article  MathSciNet  MATH  Google Scholar 

  27. Nipkow, T.: Higher-order critical pairs. In: Proceedings of 6th LICS, pp. 342–349 (1991). doi:10.1109/LICS.1991.151658

  28. Nipkow, T.: Functional unification of higher-order patterns. In: Proceedings of 8th LICS, pp. 64–74 (1993). doi:10.1109/LICS.1993.287599

  29. van Oostrom, V.: Developing developments. TCS 175(1), 159–181 (1997). doi:10.1016/S0304-3975(96)00173-9

    Article  MathSciNet  MATH  Google Scholar 

  30. van Oostrom, V., Raamsdonk, F.: Weak orthogonality implies confluence: the higher-order case. In: Nerode, A., Matiyasevich, Y.V. (eds.) LFCS 1994. LNCS, vol. 813, pp. 379–392. Springer, Heidelberg (1994). doi:10.1007/3-540-58140-5_35

    Chapter  Google Scholar 

  31. Oyamaguchi, M., Hirokawa, N.: Confluence and critical-pair-closing systems. In: Proceedings of 3rd IWC, pp. 29–33 (2014)

    Google Scholar 

  32. Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. JACM 28(2), 233–264 (1981). doi:10.1145/322248.322251

    Article  MathSciNet  MATH  Google Scholar 

  33. van Raamsdonk, F.: On termination of higher-order rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 261–275. Springer, Heidelberg (2001). doi:10.1007/3-540-45127-7_20

    Chapter  Google Scholar 

  34. Regnier, L.: Une équivalence sur les lambda-termes. TCS 126(2), 281–292 (1994). doi:10.1016/0304-3975(94)90012-4

    Article  MATH  Google Scholar 

  35. Rosen, B.: Tree-manipulating systems and Church-Rosser theorems. JACM 20(1), 160–187 (1973). doi:10.1145/321738.321750

    Article  MathSciNet  MATH  Google Scholar 

  36. Rubio, A.: A fully syntactic AC-RPO. I&C 178(2), 515–533 (2002). doi:10.1006/inco.2002.3158

    MathSciNet  MATH  Google Scholar 

  37. Sakai, M., Oyamaguchi, M., Ogawa, M.: Non-E-overlapping, weakly shallow, and non-collapsing TRSs are confluent. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 111–126. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_7

    Chapter  Google Scholar 

  38. Shintani, K., Hirokawa, N.: CoLL: A confluence tool for left-linear term rewrite systems. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 127–136. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_8

    Chapter  Google Scholar 

  39. Sternagel, C., Thiemann, R.: Formalizing Knuth-Bendix orders and Knuth-Bendix completion. In: Proceedings of 24th RTA. LIPIcs, vol. 21, pp. 287–302 (2013).doi:10.4230/LIPIcs.RTA.2013.287

  40. Sternagel, C., Thiemann, R.: The certification problem format. In: Proceedings of 11th UITP. EPTCS, vol. 167, pp. 61–72 (2014). doi:10.4204/EPTCS.167.8

  41. 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

    Chapter  Google Scholar 

  42. Toyama, Y.: Commutativity of term rewriting systems. In: Fuchi, K., Kott, L. (eds.) Programming of Future Generation Computers II, pp. 393–407. North-Holland Publishing, North Holland (1988)

    Google Scholar 

  43. Toyama, Y., Oyamaguchi, M.: Church-Rosser property and unique normal form property of non-duplicating term rewriting systems. In: Proceedings of the 4th CTRS withDershowitz N., Lindenstrauss N. (eds.) CTRS 1994. LNCS, vol. 968 (1995). doi:10.1007/3-540-60381-6_19

  44. 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

    Chapter  Google Scholar 

  45. Zankl, H., Felgenhauer, B., Middeldorp, A.: Labelings for decreasing diagrams. JAR 54(2), 101–133 (2015). doi:10.1007/s10817-014-9316-y

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank Sarah Winkler for contributing code and expertise related to AC termination and AC critical pairs.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Innsbruck, Innsbruck, Austria

    Julian Nagele, Bertram Felgenhauer & Aart Middeldorp

Authors
  1. Julian Nagele
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bertram Felgenhauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aart Middeldorp
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julian Nagele .

Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, Washington, USA

    Leonardo de Moura

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Nagele, J., Felgenhauer, B., Middeldorp, A. (2017). CSI: New Evidence – A Progress Report. In: de Moura, L. (eds) Automated Deduction – CADE 26. CADE 2017. Lecture Notes in Computer Science(), vol 10395. Springer, Cham. https://doi.org/10.1007/978-3-319-63046-5_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-63046-5_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-63045-8

  • Online ISBN: 978-3-319-63046-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation