Skip to main content
SpringerLink
Log in

Automating Automated Reasoning

The Case of Two Generic Automated Reasoning Tools

  • Chapter
  • First Online:
Description Logic, Theory Combination, and All That

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11560))

  • 658 Accesses

Abstract

The vision of automated support for the investigation of logics, proposed decades ago, has been implemented in many forms, producing numerous tools that analyze various logical properties (e.g., cut-elimination, semantics, and more). However, full ‘automation of automated reasoning’ in the sense of automatic generation of efficient provers has remained a ‘holy grail’ of the field. Creating a generic prover which can efficiently reason in a given logic is challenging, as each logic may be based on a different language, and involve different inference rules, that require different implementation considerations to achieve efficiency, or even tractability. Two recently introduced generic automated provers apply different approaches to tackle this challenge. \({\text {MetTeL}}\), based on the formalism of tableaux, automatically generates a prover for a given tableau calculus, by implementing generic proof-search procedures with optimizations applicable to many tableau calculi. \({\text {Gen2sat}}\), based on the formalism of sequent calculi, shifts the burden of search to the realm of off-the-shelf SAT solvers by applying a uniform reduction of derivability in sequent calculi to SAT. This paper examines these two generic provers, focusing in particular on criteria relevant for comparing their performance and usability. To this end, we evaluate the performance of the tools, and describe the results of a preliminary empirical study where user experiences of expert logicians using the two tools are compared.

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.

    Here sequents are taken to be pairs of sets of formulas, and therefore exchange and contraction are built in.

  2. 2.

    Paralyzer is a tool that transforms Hilbert calculi of a certain general form into equivalent analytic sequent calculi. It was described in [12] and can be found at http://www.logic.at/people/lara/paralyzer.html.

  3. 3.

    Note that a translation of \(\mathcal{T}\) to a sequent calculus is less obvious, as this is a three-sided calculus, where \({\text {Gen2sat}}\) employs ordinary two-sided sequents.

  4. 4.

    https://github.com/yoni206/gen2satvsmettel.

References

  1. Abate, P., Goré, R.: The tableau workbench. Electron. Notes Theor. Comput. Sci. 231, 55–67 (2009)

    Article  MathSciNet  Google Scholar 

  2. Aitken, S., Melham, T.: An analysis of errors in interactive proof attempts. Interact. Comput. 12(6), 565–586 (2000)

    Article  Google Scholar 

  3. Areces, C.E.: Logic engineering: the case of description and hybrid logics. Ph.D. thesis, University of Amsterdam (2000)

    Google Scholar 

  4. Asperti, A., Coen, C.S.: Some considerations on the usability of interactive provers. In: Autexier, S., et al. (eds.) CICM 2010. LNCS (LNAI), vol. 6167, pp. 147–156. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14128-7_13

    Chapter  Google Scholar 

  5. Avron, A.: Simple consequence relations. Inf. Comput. 92(1), 105–139 (1991)

    Article  MathSciNet  Google Scholar 

  6. Avron, A.: Classical Gentzen-type methods in propositional many-valued logics. In: Fitting, M., Orłowska, E. (eds.) Beyond Two: Theory and Applications of Multiple-Valued Logic. Studies in Fuzziness and Soft Computing, vol. 114, pp. 117–155. Physica-Verlag, Heidelberg (2003). https://doi.org/10.1007/978-3-7908-1769-0_5

    Chapter  MATH  Google Scholar 

  7. Baader, F., Sattler, U.: An overview of tableau algorithms for description logics. Stud. Logica 69, 5–40 (2001)

    Article  MathSciNet  Google Scholar 

  8. Baaz, M., Fermüller, C.G., Salzer, G., Zach, R.: MUltlog 1.0: towards an expert system for many-valued logics. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 226–230. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61511-3_84

    Chapter  Google Scholar 

  9. Baumgartner, P., Schmidt, R.A.: Blocking and other enhancements for bottom-up model generation methods. J. Autom. Reason. 1–27 (2019). https://doi.org/10.1007/s10817-019-09515-1

  10. Beckert, B., Grebing, S., Böhl, F.: A usability evaluation of interactive theorem provers using focus groups. In: Canal, C., Idani, A. (eds.) SEFM 2014. LNCS, vol. 8938, pp. 3–19. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15201-1_1

    Chapter  Google Scholar 

  11. Caridroit, T., Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: A SAT-based approach for solving the modal logic S5-satisfiability problem. In: Singh, S.P., Markovitch, S. (eds.) Thirty-First AAAI Conference on Artificial Intelligence, pp. 3864–3870. AAAI Press (2017)

    Google Scholar 

  12. Ciabattoni, A., Lahav, O., Spendier, L., Zamansky, A.: Automated support for the investigation of paraconsistent and other logics. In: Artemov, S., Nerode, A. (eds.) LFCS 2013. LNCS, vol. 7734, pp. 119–133. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-35722-0_9

    Chapter  MATH  Google Scholar 

  13. Ciabattoni, A., Lahav, O., Spendier, L., Zamansky, A.: Taming paraconsistent (and other) logics: an algorithmic approach. ACM Trans. Comput. Logic 16(1), 5:1–5:23 (2014)

    Article  MathSciNet  Google Scholar 

  14. Ciabattoni, A., Spendier, L.: Tools for the investigation of substructural and paraconsistent logics. In: Fermé, E., Leite, J. (eds.) JELIA 2014. LNCS (LNAI), vol. 8761, pp. 18–32. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11558-0_2

    Chapter  MATH  Google Scholar 

  15. Comon-Lundh, H., Shmatikov, V.: Intruder deductions, constraint solving and insecurity decision in presence of exclusive or. In: Logic in Computer Science (LICS 2003), pp. 271–280. IEEE Computer Society (2003)

    Google Scholar 

  16. Conradie, W., Goranko, V., Vakarelov, D.: Algorithmic correspondence and completeness in modal logic I: the core algorithm SQEMA. Log. Methods Comput. Sci. 2, 1–5 (2006)

    Article  MathSciNet  Google Scholar 

  17. Cotrini, C., Gurevich, Y.: Basic primal infon logic. J. Logic Comput. 26(1), 117 (2016)

    Article  MathSciNet  Google Scholar 

  18. Degtyarev, A., Voronkov, A.: The inverse method. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 179–272. MIT Press (2001)

    Google Scholar 

  19. Dixon, C., Konev, B., Schmidt, R.A., Tishkovsky, D.: Labelled tableaux for temporal logic with cardinality constraints. In: Proceedings of the 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2012), pp. 111–118. IEEE Computer Society (2012)

    Google Scholar 

  20. Doherty, P., Łukaszewicz, W., Szałas, A.: Computing circumscription revisited: a reduction algorithm. J. Autom. Reason. 18(3), 297–336 (1997)

    Article  MathSciNet  Google Scholar 

  21. Fitting, M.: Tableau methods of proof for modal logics. Notre Dame J. Formal Logic 13(2), 237–247 (1972)

    Article  MathSciNet  Google Scholar 

  22. Gaschnig., J.: Performance measurement and analysis of certain search algorithms. Ph.D. thesis, Carnegie-Mellon University (1979)

    Google Scholar 

  23. Gasquet, O., Herzig, A., Longin, D., Sahade, M.: LoTREC: logical tableaux research engineering companion. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 318–322. Springer, Heidelberg (2005). https://doi.org/10.1007/11554554_25

    Chapter  MATH  Google Scholar 

  24. Gentzen, G.: Untersuchungen über das logische Schliessen. Mathematische Zeitschrift 39, 176–210 (1934)

    Article  Google Scholar 

  25. Ginsberg, M.L., McAllester, D.A.: GSAT and dynamic backtracking. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) Principles of Knowledge Representation and Reasoning (KR 1994), pp. 226–237. Morgan Kaufmann (1994)

    Google Scholar 

  26. Giunchiglia, E., Tacchella, A., Giunchiglia, F.: SAT-based decision procedures for classical modal logics. J. Autom. Reason. 28(2), 143–171 (2002)

    Article  MathSciNet  Google Scholar 

  27. Gorín, D., Pattinson, D., Schröder, L., Widmann, F., Wißmann, T.: Cool – a generic reasoner for coalgebraic hybrid logics (system description). In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 396–402. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_31

    Chapter  Google Scholar 

  28. Gould, J.D., Lewis, C.: Designing for usability: key principles and what designers think. Commun. ACM 28(3), 300–311 (1985)

    Article  Google Scholar 

  29. Hähnle, R.: Tableaux and related methods. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 100–178. Elsevier and MIT Press (2001)

    Google Scholar 

  30. Heuerding, A., Jäger, G., Schwendimann, S., Seyfried, M.: The logics workbench LWB: a snapshot. Euromath Bull. 2(1), 177–186 (1996)

    Google Scholar 

  31. Humberstone, I.L.: The modal logic of ‘all and only’. Notre Dame J. Formal Logic 28(2), 177–188 (1987)

    Article  MathSciNet  Google Scholar 

  32. Hustadt, U., Schmidt, R.A.: Simplification and backjumping in modal tableau. In: de Swart, H. (ed.) TABLEAUX 1998. LNCS (LNAI), vol. 1397, pp. 187–201. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-69778-0_22

    Chapter  Google Scholar 

  33. Kaminski, M., Tebbi, T.: InKreSAT: modal reasoning via incremental reduction to SAT. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 436–442. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_31

    Chapter  Google Scholar 

  34. Kawai, H.: Sequential calculus for a first order infinitary temporal logic. Math. Logic Q. 33(5), 423–432 (1987)

    Article  MathSciNet  Google Scholar 

  35. Khodadadi, M., Schmidt, R.A., Tishkovsky, D.: MetTeL. http://www.mettel-prover.org

  36. Khodadadi, M., Schmidt, R.A., Tishkovsky, D.: A refined tableau calculus with controlled blocking for the description logic \(\cal{SHOI}\). In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013. LNCS (LNAI), vol. 8123, pp. 188–202. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40537-2_17

    Chapter  MATH  Google Scholar 

  37. Khodadadi, M., Schmidt, R.A., Tishkovsky, D., Zawidzki, M.: Terminating tableau calculi for modal logic K with global counting operators (2012). http://www.mettel-prover.org/papers/KEn12.pdf

  38. Kleene, S.C.: Introduction to Metamathematics. Van Nostrand, New York (1950)

    MATH  Google Scholar 

  39. Koopmann, P., Schmidt, R.A.: LETHE: saturation-based reasoning for non-standard reasoning tasks. In: Dumontier, M., et al. (eds.) OWL Reasoner Evaluation (ORE-2015), CEUR Workshop Proceedings, vol. 1387, pp. 23–30 (2015)

    Google Scholar 

  40. Lagniez, J.M., Le Berre, D., de Lima, T., Montmirail, V.: On checking Kripke models for modal logic K. In: Fontaine, P., Schulz, S., Urban, J. (eds.) Practical Aspects of Automated Reasoning (PAAR 2016), CEUR Workshop Proceedings, vol. 1635, pp. 69–81 (2016)

    Google Scholar 

  41. Lahav, O., Zohar, Y.: SAT-based decision procedure for analytic pure sequent calculi. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 76–90. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_6

    Chapter  Google Scholar 

  42. Le Berre, D., Parrain, A.: The SAT4J library, release 2.2. J. Satisf. Boolean Model. Comput. 7, 59–64 (2010)

    Google Scholar 

  43. Lukasiewicz, J., Tarski, A.: Investigations into the sentential calculus. Borkowski 12, 131–152 (1956)

    Google Scholar 

  44. Miller, D., Pimentel, E.: A formal framework for specifying sequent calculus proof systems. Theor. Comput. Sci. 474, 98–116 (2013)

    Article  MathSciNet  Google Scholar 

  45. Minica, S., Khodadadi, M., Schmidt, R.A., Tishkovsky, D.: Synthesising and implementing tableau calculi for interrogative epistemic logics. In: Fontaine, P., Schmidt, R.A., Schulz, S. (eds.) Practical Aspects of Automated Reasoning (PAAR-2012). EPiC Series in Computing, vol. 21, pp. 109–123. EasyChair (2012)

    Google Scholar 

  46. Nielsen, J.: Usability inspection methods. In: Plaisant, C. (ed.) Conference on Human Factors in Computing Systems (CHI 1994), pp. 413–414. ACM (1994)

    Google Scholar 

  47. Nigam, V., Pimentel, E., Reis, G.: An extended framework for specifying and reasoning about proof systems. J. Logic Comput. 26, 539–576 (2014)

    Article  MathSciNet  Google Scholar 

  48. Nigam, V., Reis, G., Lima, L.: Quati: an automated tool for proving permutation lemmas. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 255–261. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_18

    Chapter  MATH  Google Scholar 

  49. Ohlbach, H.J.: Computer support for the development and investigation of logics. Logic J. IGPL 4(1), 109–127 (1996)

    Article  MathSciNet  Google Scholar 

  50. Ohlbach, H.J.: SCAN—elimination of predicate quantifiers. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 161–165. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61511-3_77

    Chapter  Google Scholar 

  51. Prosser, P.: Hybrid algorithms for the constraint satisfaction problem. Comput. Intell. 9, 268–299 (1993)

    Article  Google Scholar 

  52. Rothenberg, R.: A class of theorems in Łukasiewicz logic for benchmarking automated theorem provers. In: TABLEAUX, Automated Reasoning with Analytic Tableaux and Related Methods, Position Papers, vol. 7, pp. 101–111 (2007)

    Google Scholar 

  53. Schmidt, R.A., Stell, J.G., Rydeheard, D.: Axiomatic and tableau-based reasoning for Kt(H, R). In: Goré, R., Kooi, B., Kurucz, A. (eds.) Advances in Modal Logic, vol. 10, pp. 478–497. College Publications (2014)

    Google Scholar 

  54. Schmidt, R.A., Tishkovsky, D.: Automated synthesis of tableau calculi. Log. Methods Comput. Sci. 7(2), 1–32 (2011)

    Article  MathSciNet  Google Scholar 

  55. Schmidt, R.A., Tishkovsky, D.: Using tableau to decide description logics with full role negation and identity. ACM Trans. Comput. Logic 15(1), 7:1–7:31 (2014)

    Article  MathSciNet  Google Scholar 

  56. Schmidt, R.A., Waldmann, U.: Modal tableau systems with blocking and congruence closure. In: De Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 38–53. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24312-2_4

    Chapter  Google Scholar 

  57. Sette, A.M.: On the propositional calculus P1. Math. Japonicae 18(13), 173–180 (1973)

    MathSciNet  MATH  Google Scholar 

  58. Smullyan, R.M.: First Order Logic. Springer, Berlin (1971)

    MATH  Google Scholar 

  59. Stell, J.G., Schmidt, R.A., Rydeheard, D.E.: A bi-intuitionistic modal logic: foundations and automation. J. Log. Algebraic Methods Program. 85(4), 500–519 (2016)

    Article  MathSciNet  Google Scholar 

  60. Tishkovsky, D., Schmidt, R.A.: Rule refinement for semantic tableau calculi. In: Schmidt, R.A., Nalon, C. (eds.) TABLEAUX 2017. LNCS (LNAI), vol. 10501, pp. 228–244. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66902-1_14

    Chapter  Google Scholar 

  61. Tishkovsky, D., Schmidt, R.A., Khodadadi, M.: \(MetTeL\): a tableau prover with logic-independent inference engine. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 242–247. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22119-4_19

    Chapter  MATH  Google Scholar 

  62. Tishkovsky, D., Schmidt, R.A., Khodadadi, M.: MetTeL2: towards a tableau prover generation platform. In: Fontaine, P., Schmidt, R.A., Schulz, S. (eds.) Practical Aspects of Automated Reasoning (PAAR-2012). EPiC Series in Computing, vol. 21, pp. 149–162. EasyChair (2012)

    Google Scholar 

  63. Tishkovsky, D., Schmidt, R.A., Khodadadi, M.: The tableau prover generator MetTeL2. In: del Cerro, L.F., Herzig, A., Mengin, J. (eds.) JELIA 2012. LNCS (LNAI), vol. 7519, pp. 492–495. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33353-8_41

    Chapter  Google Scholar 

  64. Wansing, H.: Sequent systems for modal logics. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 8, pp. 61–145. Springer, Dordrecht (2002)

    Chapter  Google Scholar 

  65. Zawidzki, M.: Deductive systems and decidability problem for hybrid logics. Ph.D. thesis, Faculty of Philosophy and History, University of Lodz (2013)

    Google Scholar 

  66. Zhao, Y., Schmidt, R.A.: Forgetting concept and role symbols in \(\cal{ALCOIH}\mu ^+(\nabla ,\sqcap )\)-ontologies. In: Kambhampati, S. (ed.) International Joint Conference on Artificial Intelligence (IJCAI 2016), pp. 1345–1352. AAAI Press/IJCAI (2016)

    Google Scholar 

  67. Zohar, Y.: Gen2sat. http://www.cs.tau.ac.il/research/yoni.zohar/gen2sat.html

  68. Zohar, Y., Zamansky, A.: Gen2sat: an automated tool for deciding derivability in analytic pure sequent calculi. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 487–495. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_33

    Chapter  MATH  Google Scholar 

Download references

Acknowledgments

We thank Francesco Genco, Yotam Feldman, Roman Kuznets, Giselle Reis, João Marcos, and Bruno Woltzenlogel Paleo for providing valuable feedback on both tools. e also thank Mohammad Khodadadi for useful discussions and setting up the \({\text {MetTeL}}\) website. he research of the first and fourth authors was supported by The Israel Science Foundation (grant no. 817-15). The research of the second and third authors was supported by UK EPSRC research grant EP/H043748/1.

Last but not least, we extend our best wishes to Franz Baader on the occasion of his 60th birthday. It is an immense privilege to have been asked to contribute to this volume.

Author information

Authors and Affiliations

  1. Computer Science Department, Stanford University, Stanford, USA

    Yoni Zohar

  2. School of Computer Science, University of Manchester, Manchester, UK

    Dmitry Tishkovsky & Renate A. Schmidt

  3. Information Systems Department, University of Haifa, Haifa, Israel

    Anna Zamansky

Authors
  1. Yoni Zohar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Tishkovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Renate A. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Anna Zamansky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Bremen, Bremen, Germany

    Carsten Lutz

  2. University of Manchester, Manchester, UK

    Uli Sattler

  3. University of Iowa, Iowa City, IA, USA

    Cesare Tinelli

  4. TU Dresden, Dresden, Germany

    Anni-Yasmin Turhan

  5. University of Liverpool, Liverpool, UK

    Frank Wolter

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Zohar, Y., Tishkovsky, D., Schmidt, R.A., Zamansky, A. (2019). Automating Automated Reasoning. In: Lutz, C., Sattler, U., Tinelli, C., Turhan, AY., Wolter, F. (eds) Description Logic, Theory Combination, and All That. Lecture Notes in Computer Science(), vol 11560. Springer, Cham. https://doi.org/10.1007/978-3-030-22102-7_29

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-22102-7_29

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-22101-0

  • Online ISBN: 978-3-030-22102-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation