Skip to main content
SpringerLink
Log in

Eliminating Models During Model Elimination

  • Conference paper
  • First Online:
Book cover Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

  • 562 Accesses

Abstract

We investigate the integration of SAT technology into clausal connection-tableau systems for classical first-order logic. Clauses present in tableaux during backtracking search are heuristically grounded and added to an incremental SAT solver. If the solver reports an unsatisfiable set of ground clauses at any point, search may be halted and a proof reported. This technique alone is surprisingly effective, but also supports further refinements “for free”. In particular we further investigate depth control of randomised search based on grounded clauses, and a kind of ground lemmata rule derived from the partial SAT model.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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.

    also known as, or closely related to, the connection method [6], model elimination [28], and/or the method of matings [2].

  2. 2.

    Unless there are no such clauses or all clauses stem from the conjecture, in which case positive clauses are used instead.

  3. 3.

    pseudo-random shuffle such that results are reproducible.

  4. 4.

    https://github.com/MichaelRawson/satcop commit 65122a99e08648f5b2e331280d0a0011e73a0836 is discussed here.

  5. 5.

    It is interesting to note that the saturation-based Vampire theorem prover also fails to solve this problem in reasonable time without support from a SAT solver.

  6. 6.

    run with SWI Prolog 7.6.4 [50].

References

  1. Alama, J., Heskes, T., Kühlwein, D., Tsivtsivadze, E., Urban, J.: Premise selection for mathematics by corpus analysis and kernel methods. J. Autom. Reason. 52(2), 191–213 (2014). https://doi.org/10.1007/s10817-013-9286-5

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews, P.B.: Theorem proving via general matings. J. ACM (JACM) 28(2), 193–214 (1981)

    Article  MathSciNet  Google Scholar 

  3. Balyo, T., Froleyks, N., Heule, M.J., Iser, M., Järvisalo, M., Suda, M.: Proceedings of SAT Competition 2020: solver and benchmark descriptions (2020)

    Google Scholar 

  4. Baumgartner, P., Tinelli, C.: The model evolution calculus. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 350–364. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45085-6_32

    Chapter  Google Scholar 

  5. Bayerl, S., Letz, R.: SETHEO: a sequential theorem prover for first-order logic. In: Esprit’87-Achievements and Impacts, part 1, pp. 721–735 (1987)

    Google Scholar 

  6. Bibel, W.: Automated Theorem Proving. Springer, Heidelberg (2013)

    MATH  Google Scholar 

  7. Biere, A.: PicoSAT essentials. J. Satisf. Boolean Model. Comput. 4(2–4), 75–97 (2008)

    MATH  Google Scholar 

  8. Biere, A., Cimatti, A., Clarke, E.M., Strichman, O., Zhu, Y.: Bounded model checking (2003)

    Google Scholar 

  9. Biere, A., Ganesh, V., Grohe, M., Nordström, J., Williams, R.: Theory and practice of SAT solving (Dagstuhl Seminar 15171). In: Dagstuhl Reports. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)

    Google Scholar 

  10. Biere, A., Heule, M., van Maaren, H.: Handbook of satisfiability, vol. 185, IOS press (2009)

    Google Scholar 

  11. Brown, C.E.: Reducing higher-order theorem proving to a sequence of SAT problems. J. Autom. Reason. 51(1), 57–77 (2013). https://doi.org/10.1007/s10817-013-9283-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Claessen, K., Sorensson, N.: New techniques that improve MACE-style model finding. In: Model Computation (2003)

    Google Scholar 

  13. Codish, M., Lagoon, V., Stuckey, P.J.: Logic programming with satisfiability. Theory Pract. Logic Program. 8(1), 121 (2008)

    Article  Google Scholar 

  14. De Moura, L., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Commun. ACM 54(9), 69–77 (2011)

    Article  Google Scholar 

  15. Deshane, T., Hu, W., Jablonski, P., Lin, H., Lynch, C., McGregor, R.E.: Encoding first order proofs in SAT. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 476–491. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73595-3_35

    Chapter  Google Scholar 

  16. Färber, M.: A curiously effective backtracking strategy for connection tableaux. CoRR abs/2106.13722 (2021). https://arxiv.org/abs/2106.13722

  17. Färber, M., Kaliszyk, C., Urban, J.: Machine learning guidance for connection tableaux. J. Autom. Reason. 65(2), 287–320 (2021). https://doi.org/10.1007/s10817-020-09576-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Grabowski, A., Korniłowicz, A., Naumowicz, A.: Four decades of Mizar. J. Autom. Reason. 55(3), 191–198 (2015). https://doi.org/10.1007/s10817-015-9345-1

    Article  MathSciNet  MATH  Google Scholar 

  19. Hoder, K., Voronkov, A.: Sine qua non for large theory reasoning. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 299–314. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22438-6_23

    Chapter  Google Scholar 

  20. Jaffar, J., Lassez, J.L.: Constraint logic programming. In: Proceedings of the 14th ACM SIGACT-SIGPLAN symposium on Principles of programming languages, pp. 111–119 (1987)

    Google Scholar 

  21. Kaliszyk, C.: Efficient low-level connection tableaux. In: De Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 102–111. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24312-2_8

    Chapter  MATH  Google Scholar 

  22. Kaliszyk, C., Urban, J., Michalewski, H., Olšák, M.: Reinforcement learning of theorem proving. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 8836–8847 (2018)

    Google Scholar 

  23. Kaliszyk, C., Urban, J., Vyskočil, J.: Certified connection tableaux proofs for HOL Light and TPTP. In: Proceedings of the 2015 Conference on Certified Programs and Proofs, pp. 59–66 (2015)

    Google Scholar 

  24. Korovin, K.: Instantiation-based automated reasoning: from theory to practice. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 163–166. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_14

    Chapter  Google Scholar 

  25. Korovin, K.: Inst-Gen – a modular approach to instantiation-based automated reasoning. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics. LNCS, vol. 7797, pp. 239–270. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37651-1_10

    Chapter  Google Scholar 

  26. Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 1–35. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_1

    Chapter  Google Scholar 

  27. Letz, R., Stenz, G.: Model elimination and connection tableau procedures. In: Handbook of Automated Reasoning, pp. 2015–2114. Elsevier (2001)

    Google Scholar 

  28. Loveland, D.W.: Mechanical theorem-proving by model elimination. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning, pp. 117–134. Springer, Heidelberg (1968). https://doi.org/10.1007/978-3-642-81955-1_8

    Chapter  Google Scholar 

  29. Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  30. McDonald, A., et al.: Parallel WalkSAT with clause learning. Data analysis project papers (2009)

    Google Scholar 

  31. Otten, J.: leanCoP 2.0 and ileanCoP 1.2: high performance lean theorem proving in classical and intuitionistic logic (system descriptions). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 283–291. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71070-7_23

    Chapter  Google Scholar 

  32. Otten, J.: Restricting backtracking in connection calculi. AI Commun. 23(2–3), 159–182 (2010)

    Article  MathSciNet  Google Scholar 

  33. Otten, J.: MleanCoP: a connection prover for first-order modal logic. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 269–276. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_20

    Chapter  Google Scholar 

  34. Otten, J.: nanoCoP: a non-clausal connection prover. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 302–312. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_21

    Chapter  Google Scholar 

  35. Otten, J.: The pocket reasoner – automatic reasoning on small devices. In: Norwegian Informatics Conference, NIK (2018)

    Google Scholar 

  36. Raths, T., Otten, J.: randoCoP: randomizing the proof search order in the connection calculus. In: First International Workshop on Practical Aspects of Automated Reasoning, pp. 94–103 (2008). http://ceur-ws.org/Vol-373/

  37. Reger, G., Bjorner, N., Suda, M., Voronkov, A.: AVATAR modulo theories. In: Benzmüller, C., Sutcliffe, G., Rojas, R. (eds.) GCAI 2016. 2nd Global Conference on Artificial Intelligence. EPiC Series in Computing, vol. 41, pp. 39–52. EasyChair (2016). https://doi.org/10.29007/k6tp. https://easychair.org/publications/paper/7

  38. Reger, G., Suda, M.: The uses of SAT solvers in Vampire. In: Kovács, L., Voronkov, A. (eds.) Proceedings of the 1st and 2nd Vampire Workshops, Vampire@VSL 2014, Vienna, Austria, July 23, 2014 / Vampire@CADE 2015, Berlin, Germany, 2 August 2015. EPiC Series in Computing, vol. 38, pp. 63–69. EasyChair (2015). https://easychair.org/publications/paper/ZG9

  39. Reger, G., Suda, M.: Global subsumption revisited (briefly). In: Kovacs, L., Voronkov, A. (eds.) Vampire 2016. Proceedings of the 3rd Vampire Workshop. EPiC Series in Computing, vol. 44, pp. 61–73. EasyChair (2017). https://doi.org/10.29007/qcd7. https://easychair.org/publications/paper/QDj

  40. Reger, G., Suda, M., Voronkov, A.: Finding finite models in multi-sorted first-order logic. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 323–341. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_20

    Chapter  Google Scholar 

  41. Riazanov, A., Voronkov, A.: Limited resource strategy in resolution theorem proving. J. Symb. Comput. 36(1–2), 101–115 (2003)

    Article  MathSciNet  Google Scholar 

  42. Robbins, E., King, A., Howe, J.M.: Backjumping is exception handling. Theory Pract. Logic Program. 21, 1–20 (2020)

    MathSciNet  Google Scholar 

  43. Schulz, S.: A comparison of different techniques for grounding near-propositional CNF formulae. In: FLAIRS Conference, pp. 72–76 (2002)

    Google Scholar 

  44. Schulz, S.: Light-weight integration of SAT solving into first-order reasoners – first experiments. Vampire, pp. 9–19 (2017)

    Google Scholar 

  45. Schulz, S., Cruanes, S., Vukmirović, P.: Faster, higher, stronger: E 2.3. In: Fontaine, P. (ed.) CADE 2019. LNCS (LNAI), vol. 11716, pp. 495–507. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29436-6_29

    Chapter  Google Scholar 

  46. Smullyan, R.M.: First-order logic. Courier Corporation (1995)

    Google Scholar 

  47. Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reason. 43(4), 337 (2009). https://doi.org/10.1007/s10817-017-9407-7

    Article  MATH  Google Scholar 

  48. Urban, J.: MPTP 0.2: design, implementation, and initial experiments. J. Autom. Reason. 37(1–2), 21–43 (2006). https://doi.org/10.1007/s10817-006-9032-3

    Article  MATH  Google Scholar 

  49. Voronkov, A.: AVATAR: the architecture for first-order theorem provers. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 696–710. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_46

    Chapter  Google Scholar 

  50. Wielemaker, J.: SWI-Prolog version 7 extensions. In: Workshop on Implementation of Constraint and Logic Programming Systems and Logic-based Methods in Programming Environments, vol. 109. Citeseer (2014)

    Google Scholar 

  51. Zombori, Z., Urban, J., Brown, C.E.: Prolog technology reinforcement learning prover. In: Peltier, N., Sofronie-Stokkermans, V. (eds.) IJCAR 2020. LNCS (LNAI), vol. 12167, pp. 489–507. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-51054-1_33

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Manchester, Manchester, UK

    Michael Rawson & Giles Reger

Authors
  1. Michael Rawson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giles Reger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Rawson .

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Rawson, M., Reger, G. (2021). Eliminating Models During Model Elimination. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-86059-2_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86058-5

  • Online ISBN: 978-3-030-86059-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation