Skip to main content
SpringerLink
Log in

Optimal Length Resolution Refutations of Difference Constraint Systems

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

This paper is concerned with determining the optimal length resolution refutation (OLRR) of a system of difference constraints over an integral domain. The problem of finding short explanations for unsatisfiable difference constraint systems (DCS) finds applications in a number of design domains including program verification, proof theory, real-time scheduling and operations research. It is well-known that resolution refutation is a sound and complete procedure to establish the unsatisfiability of boolean formulas in clausal form. The literature has also established that a variant of the resolution procedure called Fourier-Motzkin elimination is a sound and complete procedure for establishing the unsatisfiability of linear constraint systems (LCS). Our work in this paper first establishes that every DCS has a short (polynomial in the size of the input) resolution refutation and then shows that there exists a polynomial time algorithm to compute the optimal size refutation. One of the consequences of our work is that the Minimum Unsatisfiable Subset (MUS) of a DCS can be computed in polynomial time; computing the MUS of an unsatisfiable constraint set is an extremely important aspect of certifying algorithms.

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

Similar content being viewed by others

References

  1. Adams, J., Balas, E., Zawack, D.: The shifting bottleneck procedure for job shop scheduling. Manage. Sci. 34, 391–401 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aggoun, A., Beldiceanu, N.: Extending chip to solve complex scheduling and placement problems. J. Math. Comput. Model. 17(7), 57–73 (1993)

    Article  Google Scholar 

  3. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)

    Google Scholar 

  4. Alekhnovich, M., Buss, S., Moran, S., Pitassi, T.: Minimum propositional proof length is NP-hard to linearly approximate. In: Mathematical Foundations of Computer Science (MFCS). Lecture Notes in Computer Science. Springer, New York (1998)

    Google Scholar 

  5. Asarin, E., Bozga, M., Kerbrat, A., Maler, O., Pnueli, A., Rasse, A.: Data structures for verification of timed automata. In: Proceedings of the Hybrid and Real-Time Systems (1997)

  6. Baptiste, P., Pape, C.L., Nuijten, W.: Constraint-based optimization and approximation for job-shop scheduling. In: AAAI-SIGMAN Workshop on Intelligent Manufacturing Systems (1995)

  7. Beame, P., Pitassi, T.: Simplified and improved resolution lower bounds. In: 37th Annual Symposium on Foundations of Computer Science, pp. 274–282. IEEE, Burlington (1996)

    Google Scholar 

  8. Beame, P., Pitassi, T.: Propositional proof complexity: past, present, future. Bull. EATCS 65, 66–89 (1998)

    MATH  MathSciNet  Google Scholar 

  9. Bockmayr, A., Eisenbrand, F.: Combining logic and optimization in cutting plane theory. In: FroCos, pp. 1–17 (2000)

  10. Brzowski, J.A., Seger, C.J.: Asynchronous Circuits. Springer, New York (1994)

    Google Scholar 

  11. Büning, H.K.: On subclasses of minimal unsatisfiable formulas. Discrete Appl. Math. 107(1–3), 83–98 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Buss, P.: Resolution and the weak pigeonhole principle. In: CSL: 11th Workshop on Computer Science Logic, LNCS. Springer, New York (1997)

    Google Scholar 

  13. Buss, S.R. (ed.): Handbook of Proof Theory. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  14. Chandru, V., Rao, M.: Linear programming. In: Algorithms and Theory of Computation Handbook, CRC Press, 1999. CRC, Boca Raton (1999)

    Google Scholar 

  15. Cook, S.A., Reckhow, R.A.: Time bounded random access machines. J. Comput. Syst. Sci. 7(4), 354–375 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  16. Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms, 2nd edn. MIT and McGraw-Hill, Boston (1992)

    Google Scholar 

  17. Cotton, S., Asarin, E., Maler, O., Niebert, P.: Some progress in satisfiability checking for difference logic. In: FORMATS/FTRTFT, pp. 263–276 (2004)

  18. Dantzig, G.B., Eaves, B.C.: Fourier-Motzkin elimination and its dual. J. Comb. Theory (A) 14, 288–297 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. Demtrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)

    Article  MathSciNet  Google Scholar 

  20. Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  21. Duterre, B., de Moura, L.: The yices smt solver. Tech. rep., SRI International (2006)

  22. Ford, J., Shankar, N.: Formal verification of a combination decision procedure. In: CADE, pp. 347–362 (2002)

  23. Fourier, J.B.J.: Reported In: Analyse de travaux de l’Academie Royale des Sciences, pendant l’annee 1824, Partie Mathematique, Histoire de ’Academie Royale de Sciences de l’Institue de France 7 (1827) xlvii-lv. (Partial English translation in: D.A. Kohler, Translation of a Report by Fourier on his work on Linear Inequalities. Opsearch 10 (1973) 38–42.). Academic, London (1824)

  24. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  25. Haken, A.: The intractability of resolution. Theor. Comp. Sci. 39(2–3), 297–308 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  26. Han, C.C., Lin, K.J.: Job scheduling with temporal distance constraints. Tech. Rep. UIUCDCS-R-89-1560, University of Illinois at Urbana-Champaign, Department of Computer Science (1989)

  27. Hochbaum, D.S., Naor, J.S.: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput. 23(6), 1179–1192 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Huynh, T., Lassez, C., Lassez, J.L.: Fourier algorithm revisited. In: Kirchner, H., Wechler, W. (eds.) Proceedings Second International Conference on Algebraic and Logic Programming. Lecture Notes in Computer Science, vol. 463, pp. 117–131. Springer, Nancy (1990)

    Google Scholar 

  29. Iwama, K., Miyano, E.: Intractability of read-once resolution. In: Proceedings of the 10th Annual Conference on Structure in Complexity Theory (SCTC ’95), pp. 29–36. IEEE Computer Society, Los Alamitos (1995)

    Chapter  Google Scholar 

  30. Johnson, D.S., Preparata, F.P.: The densest hemisphere problem. Theoretical Comput. Sci. 6(1), 93–107 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  31. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Google Scholar 

  32. Kratsch, D., McConnell, R.M., Mehlhorn, K., Spinrad, J.: Certifying algorithms for recognizing interval graphs and permutation graphs. In: SODA, pp. 158–167 (2003)

  33. Lagarias, J.C.: The computational complexity of simultaneous Diophantine approximation problems. SIAM J. Comput. 14(1), 196–209 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  34. Lahiri, S.K., Musuvathi, M.: An efficient decision procedure for UTVPI constraints. In: FroCos, pp. 168–183 (2005)

  35. Lassez, J.L., Maher, M.: On fourier’s algorithm for linear constraints. J. Autom. Reason 9, 373–379 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  36. Levi, S.T., Tripathi, S.K., Carson, S.D., Agrawala, A.K.: The Maruti hard real-time operating system. ACM Special Interest Group on Operating Systems. Oper. Syst. Rev. 23(3), 90–106 (1989)

    Article  Google Scholar 

  37. Lynce, I., Silva, J.P.M.: On computing minimum unsatisfiable cores. In: SAT (2004)

  38. de Moura, L.M., Owre, S., Ruess, H., Rushby, J.M., Shankar, N.: The ICS decision procedures for embedded deduction. In: IJCAR, pp. 218–222 (2004)

  39. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1999)

    MATH  Google Scholar 

  40. Pinedo, M.: Scheduling: Theory, Algorithms, and Systems. Prentice-Hall, Englewood Cliffs (1995)

    MATH  Google Scholar 

  41. 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 

  42. Robinson, A., Voronkov, A. (eds.): Handbook of Automated Reasoning. Elsevier, Amsterdam (2001)

    MATH  Google Scholar 

  43. Robinson, J.A.: A machine-oriented logic based on the resolution principle. J ACM 12(1), 23–41 (1965)

    Article  MATH  Google Scholar 

  44. Rushby, J.M.: Tutorial: automated formal methods with pvs, sal, and yices. In: SEFM, p. 262 (2006)

  45. Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1987)

    Google Scholar 

  46. Seshia, S.A., Bryant, R.E.: Deciding quantifier-free presburger formulas using parameterized solution bounds. In: LICS, pp. 100–109 (2004)

  47. Seshia, S.A., Lahiri, S.K., Bryant, R.E.: A hybrid sat-based decision procedure for separation logic with uninterpreted functions. In: DAC, pp. 425–430 (2003)

  48. Seshia, S.A., Subramani, K., Bryant, R.E.: On solving boolean combinations of UTVPI constraints. Journal on Satisfiability, Boolean Modeling and Computation 3(1,2), 67–90 (2007)

    MATH  MathSciNet  Google Scholar 

  49. Subramani, K.: On deciding the non-emptiness of 2SAT polytopes with respect to first order queries. Math. Log. Q. 50(3):281–292 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  50. Subramani, K.: An analysis of totally clairvoyant scheduling. J. Sched. 8(2), 113–133 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  51. Subramani, K.: A comprehensive framework for specifying clairvoyance, constraints and periodicty in real-time scheduling. Comput. J. 48(3), 259–272 (2005)

    Article  Google Scholar 

  52. Tseitin, G.: On the complexity of derivation in propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic, pp. 115–125 (1970)

  53. Urquhart, A.: The complexity of propositional proofs. Bull. Symb. Log. 1(4), 425–467 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  54. Wasserman, H., Blum, M.: Software reliability via run-time result-checking. J ACM 44(6), 826–849 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  55. Williams, H.: Fourier-motzkin elimination extension to integer programming. J. Comb. Theory 21, 118–123 (1976)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LDCSEE, West Virginia University, Morgantown, WV, USA

    K. Subramani

Authors
  1. K. Subramani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Subramani.

Additional information

This research was supported in part by a research grant from the Air-Force Office of Scientific Research under contract FA9550-06-1-0050 and in part by the National Science Foundation through Award CCF-0827397. A portion of this research was conducted at the Stanford Research Institute, where the author was a Visiting Fellow.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Subramani, K. Optimal Length Resolution Refutations of Difference Constraint Systems. J Autom Reasoning 43, 121–137 (2009). https://doi.org/10.1007/s10817-009-9139-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-009-9139-4

Keywords

Search

Navigation