The Taming of the (X)OR

  • Peter Baumgartner
  • Fabio Massacci
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1861)


Many key verification problems such as boundedmodel-checking, circuit verification and logical cryptanalysis are formalized with combined clausal and affine logic (i.e. clauses with xor as the connective) and cannot be efficiently (if at all) solved by using CNF-only provers.

We present a decision procedure to efficiently decide such problems. The Gauss-DPLL procedure is a tight integration in a unifying framework of a Gauss-Elimination procedure (for affine logic) and a Davis-Putnam-Logeman-Loveland procedure (for usual clause logic).

The key idea, which distinguishes our approach from others, is the full interaction bewteen the two parts which makes it possible to maximize (deterministic) simplification rules by passing around newly created unit or binary clauses in either of these parts.We show the correcteness and the termination of Gauss-DPLL under very liberal assumptions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BCC+99._A. Biere, A. Cimatti, E. Clarke, M. Fujita, and Y. Zhu. Symbolic model checking using SAT procedures instead of BDDs. In Proc. of ACM/IEEE DAC-99, pages 317–320. ACM Press, 1999.Google Scholar
  2. BDW95.
    B. Becker, R. Drechsler, and R. Werchner. On the relation between BDDs and FDDs. Inf. and Comp., 123(2):185–197, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  3. BRB90.
    K. Brace, R. Rudell, and R. Bryant. Efficient implementation of a BDD package. In Proc. of ACM/IEEE DAC-90, pages 40–45. IEEE Press, 1990.Google Scholar
  4. BS97.
    R. Bayardo and R. Schrag. Using CSP look-back techniques to solve real-world SAT instances. In Proc. of AAAI-97, pages 203–208. AAAI Press/TheMIT Press, 1997.Google Scholar
  5. CA96.
    J. Crawford and L. Auton. Experimental results on the crossover point in random 3SAT. AIJ, 81(1–2):31–57, 1996.MathSciNetGoogle Scholar
  6. CL73.
    C. Chang and R. Lee. Symbolic Logic and Mechanical Theorem Proving. Academic Press, 1973.Google Scholar
  7. Cla90.
    L. Claesen, ed. Formal VLSI Correctness Verification: VLSI Design Methods, volume II. Elsevier, 1990.Google Scholar
  8. DBR97.
    R. Drechsler, B. Becker, and S. Ruppertz. Manipulation algorithms for K*BMDs. In Proc. of TACAS-97, LNCS 1217, pages 4–18. Springer-Verlag, 1997.Google Scholar
  9. DLL62.
    M. Davis, G. Logeman, and D. Loveland. A machine program for theorem proving. CACM, 5(7), 1962.Google Scholar
  10. GMS98.
    E. Giunchiglia, A. Massarotto, and R. Sebastiani. Act and the rest will follow: Expliting nondeterminismin planning as satisfiability. In Proc. of AAAI-98, pages 948–952. The MIT Press, 1998.Google Scholar
  11. GW00.
    J. Groote and J. Warners. The propositional formula checker HeerHugo. JAR, 2000. To appear.Google Scholar
  12. HS98.
    U. Hustadt and R. Schmidt. Simplification and backjumping in modal tableau. In Proc. of TABLEAUX-98, LNAI 1397, pages 187–201. Springer-Verlag, 1998.Google Scholar
  13. JT96.
    D. Johnson and M. Trick, eds. Cliques, Coloring, satisfiability: the second DIMACS implementation challenge, volume 26 of AMS Series in Discr. Math. and TCS. AMS, 1996.Google Scholar
  14. KS96.
    H. Kautz and B. Selman. Pushing the envelope: Planning, propositional logic and stocastic search. In Proc. of AAAI-96, pages 1194–1201. The MIT Press, 1996.Google Scholar
  15. Li99.
    Chu-Min Li. A constraint-based approach to narrow search trees for satisfiability. IPL, 71(2):75–80, 1999.zbMATHCrossRefGoogle Scholar
  16. Li00.
    Chun-Min Li. Integrating equivalency reasoning into Davis-Putnam procedure. To appear in Proc. of AAAI-00.Google Scholar
  17. Mas98.
    Fabio Massacci. Simplification: A general constraint propagation technique for propositional and modal tableaux. In Proc. of TABLEAUX-98, LNAI 1397, pages 217–231. Springer-Verlag, 1998.Google Scholar
  18. Mas99.
    Fabio Massacci. Using Walk-SAT and Rel-sat for cryptographic key search. In Proc. of IJCAI-99, pages 290–295.Morgan Kaufmann, 1999.Google Scholar
  19. MM00.
    Fabio Massacci and Laura Marraro. Logical cryptanalysis as a SAT-problem: Encoding and analysis of the u.s. Data Encryption Standard. JAR, 2000. To appear.Google Scholar
  20. Sch78.
    T. Schaefer. The complexity of satisfiability problems. In Proc. of STOC-78, pages 216–226. ACM Press, 1978.Google Scholar
  21. SKM97.
    Bart Selman, Henry Kautz, and David McAllester. Ten challenges in propositional resoning and search. In Proc. of IJCAI-97, pages 50–54. Morgan Kaufmann, 1997.Google Scholar
  22. Wil90.
    J. Wilson. Compact normal forms in propositional logics and integer programming formulations. Comp. and Op. Res., 17(3):309–314, 1990.zbMATHCrossRefGoogle Scholar
  23. WvM99.
    J. Warners and H. van Maaren. A two phase algorithm for solving a class of hard satisfiability problems. Op. Res. Lett., 23(3–5):81–88, 1999.Google Scholar
  24. WvM00.
    J. Warners and H. van Maaren. Recognition of tractable satisfiability problems through balanced polynomial representations. Discr. Appl. Math., 2000.Google Scholar
  25. Zha97.
    H. Zhang. SATO: An Efficient Propositional Theorem Prover. In Proc. of CADE 97, LNAI 1249, pages 272–275, 1997. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Peter Baumgartner
    • 1
  • Fabio Massacci
    • 2
  1. 1.Institut für InformatikUniversität Koblenz-LandauKoblenzGermany
  2. 2.Dip. di Ingegneria dell’InformazioneUniversitá di SienaSienaItaly

Personalised recommendations