Skip to main content
SpringerLink
Log in

Incremental Search for Conflict and Unit Instances of Quantified Formulas with E-Matching

  • Conference paper
  • First Online:
Verification, Model Checking, and Abstract Interpretation (VMCAI 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12597))

  • 779 Accesses

Abstract

We present a new method to find conflicting instances of quantified formulas in the context of SMT solving. Our method splits the search for such instances in two parts. In the first part, E-matching is used to find candidate instances of the quantified formulas. In principle, any existing incremental E-matching technique can be used. The incrementality avoids duplicating work for each small change of the E-graph. Together with the candidate instance, E-matching also provides an existing node in the E-graph corresponding to each term in this instance. In the second part, these nodes are used to evaluate the candidate instance, i.e., without creating new terms. The evaluation can be done in constant time per instance. Our method detects conflicting instances and unit-propagating instances (clauses that propagate new literals). This makes our method suitable for a tight integration with the DPLL(\(\mathcal {T}\)) framework, very much in the style of an additional theory solver.

Partially supported by the German Research Council (DFG) under HO 5606/1-2.

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 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
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.

    From now on \(\sim \) denotes the congruence closure of the equality edges in the E-graph and not the transitive closure as in the previous section. Note that this is also defined for \(p\sigma \), if it does not exist in the E-graph.

  2. 2.

    https://ultimate.informatik.uni-freiburg.de/smtinterpol/

  3. 3.

    https://smt-comp.github.io/2020/

  4. 4.

    https://www.starexec.org

References

  1. Barbosa, H., Fontaine, P., Reynolds, A.: Congruence closure with free variables. In: Legay, A., Margaria, T. (eds.) TACAS 2017, Part II. LNCS, vol. 10206, pp. 214–230. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54580-5_13

    Chapter  Google Scholar 

  2. Bradley, A.R., Manna, Z., Sipma, H.B.: What’s decidable about arrays? In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 427–442. Springer, Heidelberg (2005). https://doi.org/10.1007/11609773_28

  3. de Moura, L., Bjørner, N.: Efficient E-matching for SMT solvers. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 183–198. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73595-3_13

    Chapter  Google Scholar 

  4. de Moura, L., Bjørner, N.: Engineering DPLL(T) + Saturation. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 475–490. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71070-7_40

    Chapter  Google Scholar 

  5. Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52(3), 365–473 (2005)

    Article  MathSciNet  Google Scholar 

  6. Dutertre, B., de Moura, L.: The Yices SMT solver. Technical report, SRI International (2006). https://yices.csl.sri.com/papers/tool-paper.pdf

  7. Dutertre, B., de Moura, L.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006). https://doi.org/10.1007/11817963_11

    Chapter  Google Scholar 

  8. Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiability modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02658-4_25

    Chapter  Google Scholar 

  9. Hoenicke, J., Schindler, T.: Artifacts for incremental search for conflict and unit instances of quantified formulas with E-matching. Technical report, Zenodo (2021). https://doi.org/10.5281/zenodo.4277777

  10. Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27(2), 356–364 (1980)

    Article  MathSciNet  Google Scholar 

  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. Nonnengart, A., Weidenbach, C.: Computing small clause normal forms. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), pp. 335–367. Elsevier and MIT Press, New York (2001)

    Chapter  Google Scholar 

  13. Reynolds, A., Barbosa, H., Fontaine, P.: Revisiting enumerative instantiation. In: Beyer, D., Huisman, M. (eds.) TACAS 2018, Part II. LNCS, vol. 10806, pp. 112–131. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89963-3_7

    Chapter  Google Scholar 

  14. Reynolds, A., King, T., Kuncak, V.: Solving quantified linear arithmetic by counterexample-guided instantiation. Form. Methods Syst. Des. 51(3), 500–532 (2017). https://doi.org/10.1007/s10703-017-0290-y

    Article  MATH  Google Scholar 

  15. Reynolds, A., Tinelli, C., de Moura, L.M.: Finding conflicting instances of quantified formulas in SMT. In: Formal Methods in Computer-Aided Design, FMCAD 2014, Lausanne, Switzerland, 21–24 October 2014, pp. 195–202. IEEE (2014)

    Google Scholar 

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

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Freiburg, Freiburg im Breisgau, Germany

    Jochen Hoenicke & Tanja Schindler

Authors
  1. Jochen Hoenicke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tanja Schindler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Jochen Hoenicke or Tanja Schindler .

Editor information

Editors and Affiliations

  1. University of Copenhagen, Copenhagen, Denmark

    Fritz Henglein

  2. Tel Aviv University, Tel Aviv, Israel

    Sharon Shoham

  3. Technion, Haifa, Israel

    Yakir Vizel

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

Hoenicke, J., Schindler, T. (2021). Incremental Search for Conflict and Unit Instances of Quantified Formulas with E-Matching. In: Henglein, F., Shoham, S., Vizel, Y. (eds) Verification, Model Checking, and Abstract Interpretation. VMCAI 2021. Lecture Notes in Computer Science(), vol 12597. Springer, Cham. https://doi.org/10.1007/978-3-030-67067-2_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-67067-2_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-67066-5

  • Online ISBN: 978-3-030-67067-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation