Skip to main content
SpringerLink
Log in

Optimization techniques for craig interpolant compaction in unbounded model checking

  • Published:
Formal Methods in System Design Aims and scope Submit manuscript

Abstract

This paper addresses the problem of reducing the size of Craig interpolants generated within inner steps of SAT-based Unbounded Model Checking. Craig interpolants are obtained from refutation proofs of unsatisfiable SAT runs, in terms of and/or circuits of linear size, w.r.t. the proof. Existing techniques address proof reduction, whereas interpolant circuit compaction is typically considered as an implementation problem, tackled with standard logic synthesis techniques. We propose a three step interpolant computation process, specifically oriented to scalability, in which we: (1) exploit an existing technique to detect and remove redundancies in refutation proofs, (2) apply customized light weight combinational logic reductions (constant propagation, ODC-based simplifications, and BDD-based sweeping) directly on the proof graph data structure, (3) introduce an ad-hoc combinational reduction procedure for large interpolant circuits, based on the incrementality and divide-and-conquer paradigms. The main contributions of our approach are represented by the overall approach, the proposed combinational reduction technique, and a detailed experimental evaluation of the interpolant generation process, on a set of benchmarks from the Hardware Model Checking Competitions 2013 and 2014.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. we use the transposed graph \({\varPi }^T\) here, as reverse edges are needed, formally, to identify sub-trees

  2. Another motivation for our choice is the fact that AIGER is the netlist interchange format chosen for Hardware Model Checking Competitions [5]

  3. Forward and backward directions refer to the interpolant circuit. Forward thus follows edges in \(\varPi \), whereas backward follows edges in \(\varPi ^T\).

  4. Balancing is a linear-time DAG-rewriting, based on identifying unbalanced sub-trees of AND (resp OR) gates.

  5. The overall purpose of POpt6 is to enforce local ODC reductions, by trying different orderings within AND (resp. OR) groups of nodes. So it is basically a variant of POpt5 with higher overhead, and more compaction expected.

  6. PdTrav, though disqualified for an unsound result, globally ranked second.

References

  1. Craig W (1957) Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J Symb Log 22(3):269–285

    Article  MATH  MathSciNet  Google Scholar 

  2. Pacific Journal of Mathematics RC (1959) An interpolation theorem in the predicate calculus. Pac J Math 9:155–164

    Article  Google Scholar 

  3. McMillan KL (2003) Interpolation and SAT-based model checking. In: Proceedings of the computer aided verification, vol 2725 of LNCS. Springer, Boulder, CO, USA, pp 1–13

  4. Bradley AR (2011) Sat-based model checking without unrolling. VMCAI. Springer, Austin, TX, pp 70–87

    Google Scholar 

  5. Biere A, Jussila T. The model checking competition web page, http://fmv.jku.at/hwmcc

  6. McMillan KL, Jhala R (2005) Interpolation and SAT-based model checking. In: T. Ball, R.B. Jones (eds) Proceeding of the computer aided verification, vol 3725 of LNCS. Springer, Edinburgh, pp 39–51

  7. Marques-Silva J (2005) Improvements to the implementation of Interpolant-Based Model Checking. In: D. Borrione, W. Paul (eds) Proceedings of the correct hardware design and verification methods, vol 3725 of LNCS. Springer, Edinburgh, pp 367–370

  8. Cabodi G, Murciano M, Nocco S, Quer S (2008) Boosting interpolation with dynamic localized abstraction and redundancy removal. ACM Trans Des Autom Electron Syst 13(1):309–340

    Article  Google Scholar 

  9. Cabodi G, Camurati P, Murciano M (2008) Automated abstraction by incremental refinement in interpolant-based model checking. In: Proceedings of the internaional conference on computer-aided design. ACM Press, San Jose, CA, pp 129–136

  10. Li B, Somenzi F (2006) Efficient abstraction refinement in interpolation-based unbounded model checking. In: Tools and algorithms for the construction and analysis of systems. vol 3920, pp 227–241

  11. D’Silva V, Purandare M, Kroening D (2008) Approximation refinement for interpolation-based model checking. In: F. Logozzo, D. Peled, L.D. Zuck (eds) Verification, model checking and abstract interpretation. Lecture notes in computer science, vol 4905, Springer, pp 68–82

  12. Davis M, Putnam H (1960) A computing procedure for quantification theory. J ACM 7:201–215

    Article  MATH  MathSciNet  Google Scholar 

  13. Davis M, Logemann G, Loveland D (1962) A machine procedure for theorem-proving. J ACM 5:394–397

    Article  MATH  MathSciNet  Google Scholar 

  14. Marques-Silva JP, Sakalla KA (1996) GRASP— A new search algorithm for satisfiability. In: Proceedings of the international conference on tool with artificial intelligence

  15. Zhang L, Malik S (2003) Validating sat solvers using an independent resolution-based checker: practical implementations and other applications. In: Proceedings of the conference on design, automation and test in Europe, vol 1. DATE ’03, IEEE Computer Society 10880, Washington, DC

  16. Bar-Ilan O, Fuhrmann O, Hoory S, Shacham O, Strichman O (2009) Linear-time reductions of resolution proofs, vol 5394. Springer, Berlin, pp 114–128

    Google Scholar 

  17. Rollini SF, Bruttomesso R, Sharygina N (2010) An efficient and flexible approach to resolution proof reduction. In: Haifa verification conference (HVC), Springer, Haifa

  18. Fontaine P, Merz S, Paleo BW (2011) Compression of propositional resolution proofs via partial regularization. In: Proceedings of the automated deduction - CADE-23 - 23rd international conference on automated deduction, Wroclaw, Poland, July 31– August 5, 2011, pp 237–251

  19. Gupta A. (2012) Improved single pass algorithms for resolution proof reduction. In: ATVA, pp 107–121

  20. Cotton S. (2010) Two techniques for minimizing resolution proofs. In: Proceedings of the theory and applications of satisfiability testing - SAT 2010, 13th international conference, SAT 2010, Edinburgh, UK, July 11–14, 2010, pp 306–312

  21. Saluja N, Khatri SP (2004) A robust algorithm for approximate compatible observability don’t care computation. In: Proceedings design automation conference, pp 422–427

  22. Savoj H, Brayton RK (1990) The use of observability and external don’t cares for the simplification of multi-level networks. In: Proceedings of the design automation conference, pp 297–301

  23. Savoj H, Brayton RK, Touati HJ (1991) Extracting local don’t cares for network optimization. In: ICCAD, pp 514–517

  24. Mishchenko A, Brayton RK (2007) Sat-based complete don’t-care computation for network optimization. CoRR Arxiv:0710.4695

  25. McMillan KL (2002) Applying sat methods in unbounded symbolic model checking. In: E Brinksma, KG Larsen (eds) CAV, Lecture notes in computer science, vol 2404. Springer, pp 250–264

  26. McMillan KL, Labs CB (2005) Don’t-care computation using k-clause approximation. In: International Workshop on Logic & Synthesis

  27. Brayton RK, Mishchenko A (2010) Abc: An academic industrial-strength verification tool. In: T Touili, B Cook, P Jackson (eds) CAV, Lecture notes in computer science, vol 6174, Springer, pp 24–40

  28. Mishchenk A (2005) ABC: A system for sequential synthesis and verification, http://www.eecs.berkeley.edu/~alanmi/abc/

  29. Cabodi G, Loiacono C, Vendraminetto D (2013) Optimization techniques for craig interpolant compaction in unbounded model checking. In: Proceedings of the design automation & test in Europe conference, IEEE Computer Society, Grenoble, France, pp 1417–1422

  30. Bruttomesso R, Rollini SF, Sharygina N, Tsitovich A (2010) Flexible interpolation with local proof transformations. In: International conference of computer aided design (ICCAD), IEEE Computer Society, San Jose, USA

  31. Sharygina N (2013) PeRIPLO. Formal verification at University of Lugano, http://verify.inf.unisi.ch/periplo.html

  32. Bloem R, Malik S, Schlaipfer M, Weissenbacher G (2014) Reduction of resolution refutations and interpolants via subsumption. In: Proceedings of the hardware and software: verification and testing - 10th international Haifa verification conference, HVC 2014, Haifa, Israel, November 18–20, 2014, pp 188–203

  33. Chockler H, Ivril A, Matsliah A (2012) Computing interpolants without proofs. In: Proceedings of the 7th international Haifa verification conference on hardware and software: verification and testing. HVC ’12

  34. Vizel Y, Ryvchin V, Nadel A (2013) Efficient generation of small interpolants in cnf. In: CAV, pp 330–346

  35. Vizel Y, Gurfinkel A (2014) Interpolating property directed reachability. CAV, pp 260–276

  36. Moskewicz M, Madigan C, Zhao Y, Zhang L, Malik S (2001) Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th design automation conference, IEEE Computer Society, Las Vegas, NV

  37. Eén N, Sörensson N (2009) The minisat SAT solver, http://minisat.se

  38. Biere A, Cimatti A, Clarke EM, Fujita M, Zhu Y (1999) Symbolic model checking using SAT procedures instead of BDDs. In: Proceedings of the 36th design automation conference, IEEE Computer Society, New Orleans, LA, pp 317–320

  39. Eén N (May 2007) Cut sweeping. Technical report, Cadence Research Labs, Berkeley, USA

  40. Kuehlmann A, Krohm F (1997) Equivalence checking using cuts and heaps. In: Proceedings of the 34th design automation conference, IEEE Computer Society, Anaheim, CA, pp 263–268

  41. Kuehlmann A (2004) Dynamic transition relation simplification for bounded property checking. In: Proceedings of the international conference on computer-aided design, IEEE Computer Society, San Jose, CA, pp 50–57

  42. Bjesse P, Boralv A (2004) DAG-aware circuit compression for formal verification. In: Proceedings of the international conference on computer-aided design, IEEE Computer Society, San Jose, CA

  43. Brayton RK, Chatterjee S, Mishchenko A (2006) DAG-aware aig rewriting: a fresh look at combinational logic synthesis. In: Proceedings design automation conference, pp 532–536

  44. Mishchenko A, Chatterjee S, Brayton R (2006) Dag-aware aig rewriting: a fresh look at combinational logic synthesis. In: In DAC 06— Proceedings of the 43rd annual conference on design automation, ACM Press, pp 532–536

  45. Cabodi G, Nocco S, Quer S (2011) Benchmarking a model checker for algorithmic improvements and tuning for performance. Form Methods Syst Des 39(2):205–227

    Article  MATH  Google Scholar 

  46. Formal-Methods-Group (2014) PdTrav Tool, http://fmgroup.polito.it/index.php/download/viewcategory/3-pdtrav-package

  47. Biere A, Heljanko K (2014) The model checking competition web page, http://fmv.jku.at/hwmcc14

Download references

Acknowledgments

This work was supported in part by SRC contract 2009-TJ-1968

Author information

Authors and Affiliations

  1. Politecnico di Torino, Dipartimento di Automatica e Informatica, Turin, Italy

    Gianpiero Cabodi, Carmelo Loiacono & Danilo Vendraminetto

Authors
  1. Gianpiero Cabodi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carmelo Loiacono
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Danilo Vendraminetto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianpiero Cabodi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cabodi, G., Loiacono, C. & Vendraminetto, D. Optimization techniques for craig interpolant compaction in unbounded model checking. Form Methods Syst Des 46, 135–162 (2015). https://doi.org/10.1007/s10703-015-0229-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10703-015-0229-0

Keywords

Search

Navigation