Skip to main content
SpringerLink
Log in

A Proof-Theoretic Perspective on SMT-Solving for Intuitionistic Propositional Logic

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11714))

Abstract

Claessen and Rósen have recently presented an automated theorem prover, intuit, for intuitionistic propositional logic which utilises a SAT-solver. We present a sequent calculus perspective of the theory underpinning intuit by showing that it implements a generalisation of the implication-left rule from the sequent calculus LJT, also known as G4ip and popularised by Roy Dyckhoff.

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

Similar content being viewed by others

Notes

  1. 1.

    We follow Troelstra and Schwichtenberg [14] where the calculus is called G4ip; the original (\(L\!\rightarrow \rightarrow \)) rule by Dyckhoff [2] has \(\varGamma ,\beta \rightarrow \gamma \Rightarrow \alpha \rightarrow \beta \) as the left premise.

  2. 2.

    This is the number of connectives, each conjunction counting for two [2].

  3. 3.

    These effectful additions account for the distinction between \(R^0\), \(R^1\) and \(R^2\) above.

References

  1. Claessen, K., Rosén, D.: SAT modulo intuitionistic implications. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds.) LPAR 2015. LNCS, vol. 9450, pp. 622–637. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48899-7_43

    Chapter  Google Scholar 

  2. Dyckhoff, R.: Contraction-free sequent calculi for intuitionistic logic. J. Symb. Log. 57(3), 795–807 (1992)

    Article  MathSciNet  Google Scholar 

  3. Dyckhoff, R., Pinto, L.: Implementation of a loop-free method for construction of counter-models for intuitionistic propositional logic. University of St Andrews Report CS/96/8 (1996)

    Google Scholar 

  4. Farooque, M., Graham-Lengrand, S., Mahboubi, A., A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus. In: Momigliano, A., Pientka, B., Pollack, R. (eds.) Proceedings of the 2013 International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice, LFMTP 2013. ACM Press, September 2013

    Google Scholar 

  5. Ferrari, M., Fiorentini, C., Fiorino, G.: Contraction-free linear depth sequent calculi for intuitionistic propositional logic with the subformula property and minimal depth counter-models. J. Autom. Reason. 51(2), 129–149 (2013)

    Article  MathSciNet  Google Scholar 

  6. Gentzen, G.: Untersuchungen über das logische schließen. i. Mathematische Zeitschrift 39, 176–210 (1934)

    Google Scholar 

  7. Gentzen, G.: Investigations into logical deduction i. In: Szabo, M. (ed.) The Collected Papers of Gerhard Gentzen. North-Holland, Amsterdam (1969)

    Google Scholar 

  8. Graham-Lengrand, S.: Psyche: a proof-search engine based on sequent calculus with an LCF-style architecture. In: Galmiche, D., Larchey-Wendling, D. (eds.) TABLEAUX 2013. LNCS (LNAI), vol. 8123, pp. 149–156. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40537-2_14

    Chapter  MATH  Google Scholar 

  9. Graham-Lengrand, S.: Polarities & Focussing: a journey from Realisability to Automated Reasoning. Habilitation thesis, Université Paris-Sud (2014)

    Google Scholar 

  10. Li, J., Zhu, S., Pu, G., Zhang, L., Vardi, M.Y.: Sat-based explicit LTL reasoning and its application to satisfiability checking. Form. Methods Syst. Des. (2019). https://doi.org/10.1007/s10703-018-00326-5

  11. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)

    Article  MathSciNet  Google Scholar 

  12. Raths, T., Otten, J., Kreitz, C.: The ILTP problem library for intuitionistic logic. J. Autom. Reason. 38(1), 261–271 (2007)

    Article  MathSciNet  Google Scholar 

  13. Statman, R.: Intuitionistic propositional logic is polynomial-space complete. Theor. Comput. Sci. 9, 67–72 (1979)

    Article  MathSciNet  Google Scholar 

  14. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, volume 43 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  15. Tseitin, G.S.: On the Complexity of Derivation in Propositional Calculus, pp. 466–483. Springer, Heidelberg (1983). https://doi.org/10.1007/978-3-642-81955-1_28

    Book  Google Scholar 

  16. Wolper, P.: Temporal logic can be more expressive. Inf. Control 56(1/2), 72–99 (1983)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgment

This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 689176.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Milan, Milan, Italy

    Camillo Fiorentini

  2. Research School of Computer Science, Australian National University, Canberra, Australia

    Rajeev Goré

  3. SRI International, Menlo Park, USA

    Stéphane Graham-Lengrand

Authors
  1. Camillo Fiorentini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rajeev Goré
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stéphane Graham-Lengrand
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Camillo Fiorentini .

Editor information

Editors and Affiliations

  1. IBISC, Univ. Evry, Université Paris-Saclay, Evry, France

    Serenella Cerrito

  2. Middlesex University London, London, UK

    Andrei Popescu

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Fiorentini, C., Goré, R., Graham-Lengrand, S. (2019). A Proof-Theoretic Perspective on SMT-Solving for Intuitionistic Propositional Logic. In: Cerrito, S., Popescu, A. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2019. Lecture Notes in Computer Science(), vol 11714. Springer, Cham. https://doi.org/10.1007/978-3-030-29026-9_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-29026-9_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-29025-2

  • Online ISBN: 978-3-030-29026-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation