Skip to main content
SpringerLink
Log in

Generalized Craig Interpolation for Stochastic Satisfiability Modulo Theory Problems

  • Conference paper
Reachability Problems (RP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8762))

Included in the following conference series:

Abstract

Craig interpolation is widely used in solving reachability and model-checking problems by SAT or SMT techniques, as it permits the computation of invariants as well as discovery of meaningful predicates in CEGAR loops based on predicate abstraction. Extending such algorithms from the qualitative to the quantitative setting of probabilistic models seems desirable. In 2012, Teige et al. [1] succeeded to define an adequate notion of generalized, stochastic interpolants and to expose an algorithm for efficiently computing them for stochastic Boolean satisfiability problems, i.e., SSAT. In this work we present a notion of Generalized Craig Interpolant for the stochastic SAT modulo theories framework, i.e., SSMT, and introduce a mechanism to compute such stochastic interpolants for non-polynomial SSMT problems based on a sound and, w.r.t. the arithmetic reasoner, relatively complete resolution calculus. The algorithm computes interpolants in SAT, SMT, SSAT, and SSMT problems. As this extends the scope of SSMT-based model-checking of probabilistic hybrid automata from the bounded to the unbounded case, we demonstrate our interpolation principle on an unbounded probabilistic reachability problem in a probabilistic hybrid automaton.

Research supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center SFB/TR 14 AVACS ( http://www.avacs.org ).

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Teige, T., Fränzle, M.: Generalized Craig interpolation. Logical Methods in Computer Science 8(2) (2012)

    Google Scholar 

  2. Fränzle, M., Hermanns, H., Teige, T.: Stochastic satisfiability modulo theory: A novel technique for the analysis of probabilistic hybrid systems. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 172–186. Springer, Heidelberg (2008)

    Google Scholar 

  3. Papadimitriou, C.H.: Games against nature. J. Comput. Syst. Sci. 31(2), 288–301 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  4. Teige, T.: Stochastic Satisfiability Modulo Theories: A Symbolic Technique for the Analysis of Probabilistic Hybrid Systems. PhD thesis, Dpt. of Computing Science, Carl von Ossietzky Universität, Oldenburg, Germany (August 2012)

    Google Scholar 

  5. Majercik, S.M., Littman, M.L.: Maxplan: A new approach to probabilistic planning. In: Simmons, R.G., Veloso, M.M., Smith, S.F. (eds.) AIPS, pp. 86–93. AAAI (1998)

    Google Scholar 

  6. Majercik, S.M., Littman, M.L.: Contingent planning under uncertainty via stochastic satisfiability. Artif. Intell. 147(1-2), 119–162 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bacchus, F., Dalmao, S., Pitassi, T.: DPLL with caching: A new algorithm for #sat and Bayesian inference. Electronic Colloquium on Computational Complexity (ECCC) 10(003) (2003)

    Google Scholar 

  8. Freudenthal, E., Karamcheti, V.: QTM: Trust management with quantified stochastic attributes. Technical Report NYU Computer Science Technical Report TR2003-848, Courant Institute of Mathematical Sciences, New York University (2003)

    Google Scholar 

  9. Teige, T., Eggers, A., Fränzle, M.: Constraint-based analysis of concurrent probabilistic hybrid systems: An application to networked automation systems. Nonlinear Analysis: Hybrid Systems 5(2), 343–366 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Mahdi, A., Fränzle, M.: Resolution for stochastic SAT modulo theories. Technical report, Dpt. of Computing Science, Carl von Ossietzky Universität Oldenburg, Germany (December 2013)

    Google Scholar 

  11. Benhamou, F., Granvilliers, L.: Combining local consistency, symbolic rewriting and interval methods. In: Pfalzgraf, J., Calmet, J., Campbell, J. (eds.) AISMC 1996. LNCS, vol. 1138, pp. 144–159. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  12. Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symb. Log. 62(3), 981–998 (1997)

    Article  MATH  Google Scholar 

  13. Kupferschmid, S., Becker, B.: Craig interpolation in the presence of non-linear constraints. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 240–255. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  14. McMillan, K.L.: Interpolation and SAT-based model checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  15. Christ, J., Hoenicke, J., Nutz, A.: Proof tree preserving interpolation. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 124–138. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  16. Brillout, A., Kroening, D., Wahl, T.: Craig interpolation for quantifier-free Presburger arithmetic. CoRR abs/0811.3521 (2008)

    Google Scholar 

  17. Griggio, A., Le, T.T.H., Sebastiani, R.: Efficient interpolant generation in satisfiability modulo linear integer arithmetic. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 143–157. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  18. Goel, A., Krstić, S., Tinelli, C.: Ground interpolation for combined theories. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 183–198. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  19. Cimatti, A., Griggio, A., Sebastiani, R.: Efficient generation of Craig interpolants in satisfiability modulo theories. ACM Trans. Comput. Log. 12(1), 7 (2010)

    Article  MathSciNet  Google Scholar 

  20. Lynch, C., Tang, Y.: Interpolants for linear arithmetic in SMT. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 156–170. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  21. Vizel, Y., Ryvchin, V., Nadel, A.: Efficient generation of small interpolants in CNF. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 330–346. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  22. Mahdi, A., Fränzle, M.: Generalized Craig interpolation for SSMT. Technical report, Dpt. of Comuting Sceince, Carl von Ossietzky Universität, Oldenburg, Germany (2014)

    Google Scholar 

  23. Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J., Wrightson, G. (eds.) Automation of Reasoning 2: Classical Papers on Computational Logic 1967-1970, pp. 466–483. Springer, Heidelberg (1983)

    Chapter  Google Scholar 

  24. Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. Journal on Satisfiability, Boolean Modeling and Computation – Special Issue on SAT/CP Integration 1, 209–236 (2007)

    Google Scholar 

  25. Teige, T., Fränzle, M.: Resolution for stochastic boolean satisfiability. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 625–639. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  26. Benhamou, F., McAllester, D.A., Van Hentenryck, P.: CLP(Intervals) revisited. In: ILPS, pp. 124–138. MIT Press (1994)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Esparza, J., Kiefer, S., Schwoon, S.: Abstraction refinement with craig interpolation and symbolic pushdown systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 489–503. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  29. Cooper, D.C.: Theorem proving in arithmetic without multiplication. Machine Intelligence 7, 91–99 (1972)

    MATH  Google Scholar 

  30. Ferrante, J., Rackoff, C.: A decision procedure for the first order theory of real addition with order. SIAM J. Comput. 4(1), 69–76 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tarski, A.: A decision method for elementary algebra and geometry. RAND Corporation, Santa Monica, Calif. (1948)

    Google Scholar 

  32. Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1/2), 29–35 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Eggers, A., Fränzle, M., Herde, C.: SAT modulo ODE: A direct SAT approach to hybrid systems. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 171–185. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  34. Zhang, L., She, Z., Ratschan, S., Hermanns, H., Hahn, E.M.: Safety verification for probabilistic hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 196–211. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  35. Fränzle, M., Hahn, E.M., Hermanns, H., Wolovick, N., Zhang, L.: Measurability and safety verification for stochastic hybrid systems. In: HSCC, pp. 43–52. ACM (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Carl von Ossietzky Universität, Ammerländer Heerstraße 114-118, 26111, Oldenburg, Germany

    Ahmed Mahdi & Martin Fränzle

Authors
  1. Ahmed Mahdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Fränzle
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Joël Ouaknine

  2. Department of Computer Science, University of Liverpool, Liverpool, UK

    Igor Potapov

  3. Department of Computer Science, University of Oxford, UK

    James Worrell

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Mahdi, A., Fränzle, M. (2014). Generalized Craig Interpolation for Stochastic Satisfiability Modulo Theory Problems. In: Ouaknine, J., Potapov, I., Worrell, J. (eds) Reachability Problems. RP 2014. Lecture Notes in Computer Science, vol 8762. Springer, Cham. https://doi.org/10.1007/978-3-319-11439-2_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-11439-2_16

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-11438-5

  • Online ISBN: 978-3-319-11439-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation