Skip to main content
SpringerLink
Log in

Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Capturing propositional logic, constraint satisfaction problems and systems of polynomial equations, we introduce the notion of systems with finite instantiation by partial assignments, fipa-systems for short, which are independent of special representations of problem instances, but which are based on an axiomatic approach with instantiation (or substitution) by partial assignments as the fundamental notion. Fipa-systems seem to constitute the most general framework allowing for a theory of resolution with nontrivial upper and lower bounds. For every fipa-system we generalise relativised hierarchies originating from generalised Horn formulas [14,26,33,43], and obtain hierarchies of problem instances with recognition and satisfiability decision in polynomial time and linear space, quasi-automatising relativised and generalised tree resolution and utilising a general “quasi-tight” lower bound for tree resolution. And generalising width-restricted resolution from [7,14,25,33], for every fipa-system a (stronger) family of hierarchies of unsatisfiable instances with polynomial time recognition is introduced, weakly automatising relativised and generalised full resolution and utilising a general lower bound for full resolution generalising [7,17,25,33].

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. M. Alekhnovich, S. Buss, S. Moran and T. Pitassi, Minimum propositional proof length is NP-hard to linearly approximate, Journal of Symbolic Logic 66 (2001) 171–191.

    Google Scholar 

  2. M. Alekhnovich and A.A. Razborov, Resolution is not automatizable unless W[P] is tractable, in: Proceedings of the 42nd IEEE FOCS (2001) pp. 210–219.

  3. M. Alekhnovich and A.A. Razborov, Satisfiability, branch-width and Tseitin tautologies, in: Proceedings of the 43rd IEEE FOCS (2002) pp. 593–603.

  4. S. Arnborg, Efficient algorithms for combinatorial problems on graphs with bounded decomposability-a survey, BIT 25 (1985) 2–23.

    Google Scholar 

  5. A.B. Baker, Intelligent backtracking on constraint satisfaction problems: experimental and theoretical results, Ph.D. Thesis (University of Oregon, 1995). The ps-file can be downloaded using the URL http://www.cirl.uoregon.edu/baker/thesis.ps.gz.

  6. P. Beame and T. Pitassi, Simplified and improved resolution lower bounds, in: Proc. of 37th Symposium on Foundations of Computer Science (FOCS' 96) (1996) pp. 274–282.

  7. E. Ben-Sasson and A.Wigderson, Short proofs are narrow-resolution made simple, Technical Report TR99-022, ECCC Electronic Colloquium on Computational Complexity (1999).

  8. U. Bertele and F. Brioschi, Nonserial Dynamic Programming (Academic Press, New York, 1972).

    Google Scholar 

  9. H.L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth, Theoretical Computer Science 209 (1998) 1–45.

    Google Scholar 

  10. M.L. Bonet, C. Domingo, R. Gavaldá, A. Maciel and T. Pitassi, Non-automatizability of bounded depth Frege proofs, in: Proc. of 14th Annual IEEE Conference on Computational Complexity (1999).

  11. M.L. Bonet and N. Galesi, A study of proof search algorithms for resolution and polynomial calculus, in: Proc. of 40th Annual Symposium on Foundations of Computer Science (FOCS) (1999) pp. 422–431.

  12. M.L. Bonet, T. Pitassi and R. Raz, No feasible interpolation for TC 0-Frege proofs, in: Proc. of 38th Symposium on Foundations of Computer Science (FOCS' 97) (1997) pp. 254–263.

  13. N. Bourbaki, Algebra I, Chapters 1-3, Elements of Mathematics (Springer, 1989).

  14. H.K. Büning, On generalized Horn formulas and k-resolution, Theoretical Computer Science 116 (1993) 405–413.

    Google Scholar 

  15. M. Buro and H.K. Büning, On resolution with short clauses, Annals of Mathematics and Artificial Intelligence 18(2-4) (1996) 243–260.

    Google Scholar 

  16. C.L. Chang, The unit proof and the input proof in theorem proving, Journal of the Association for Computing Machinery 17(4) (1970) 698–707.

    Google Scholar 

  17. M. Clegg, J. Edmonds and R. Impagliazzo, Using the Groebner basis algorithm to find proofs of unsatisfiability, in: Proceedings of the 28th ACM Symposium on Theory of Computation (1996) pp. 174–183.

  18. B. Courcelle, J.A. Makowsky and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique width (1999). Can be obtained from http://www.cs.technion.ac.il/~janos/.

  19. M. Davis and H. Putnam, A computing procedure for quantification theory, Journal of the ACM 7 (1960) 201–215.

    Google Scholar 

  20. R. Dechter and J. Pearl, Network-based heuristics for constraint-satisfaction problems, Artificial Intelligence 34 (1988) 1–38.

    Google Scholar 

  21. R. Dechter and I. Rish, Directional resolution: the Davis-Putnam procedure, revisited, in: Proceedings of KR-94 (1994). An extended version can be obtained from http://www.ics. uci.edu/~irinar.

  22. R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173 (Springer, New York, 2nd edn., 2000).

    Google Scholar 

  23. J.L. Esteban and J. Torán, Space bounds for resolution, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, eds. C. Meinel and S. Tison, Lecture Notes in Computer Science, Vol. 1563 (Springer, 1999).

  24. T. Feder and M.Y. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory, SIAM J. Comput. 28(1) (1998) 57–104.

    Google Scholar 

  25. Z. Galil, On the validity and complexity of bounded resolution, in: Proceedings of Seventh Annual ACM Symposium on Theory of Computing (1975) pp. 72–82.

  26. G. Gallo and M.G. Scutellà, Polynomially solvable satisfiability problems, Information Processing Letters 29 (1988) 221–227.

    Google Scholar 

  27. A. Goerdt, Regular resolution versus unrestricted resolution, SIAM Journal on Computing 22(4) (1993) 661–683.

    Google Scholar 

  28. J.F. Groote and J.P. Warners, The propositional formula checker HeerHugo, Journal of Automated Reasoning 24 (2000) 101–125.

    Google Scholar 

  29. J. Harrison, Stålmarck's algorithm as a HOL derived rule, in: Theorem Proving in Higher Order Logics: 9th International Conference, TPHOLs'96 (1996) pp. 221–234.

  30. K. Iwama, Complexity of finding short resolution proofs, in: Mathematical Foundations of Computer Science 1997 (MFCS'97), Proc. of 22nd International Symposium (Springer, 1997) pp. 309–318.

  31. B. Krishnamurthy and R.N.Moll, Examples of hard tautologies in the propositional calculus, in: Proc. of 13th ACM Symposium on Theory of Computing (STOC'81) (1981) pp. 28–37.

  32. O. Kullmann, Heuristics for SAT algorithms: Searching for some foundations, Manuscript (1999) 23 pages, previous version in SAT'98.

  33. O. Kullmann, Investigating a general hierarchy of polynomially decidable classes of CNF's based on short tree-like resolution proofs, Technical Report TR99-041, Electronic Colloquium on Computational Complexity (ECCC) (1999).

  34. O. Kullmann, An improved version of width restricted resolution, in: Electronical Proceedings of Sixth International Symposium on Artificial Intelligence and Mathematics (2000) 11 pages. See “Accepted Contributions” at http://rutcor.rutgers.edu/~amai/.

  35. O. Kullmann, Investigations on autark assignments, Discrete Applied Mathematics 107 (2000) 99–137.

    Google Scholar 

  36. O. Kullmann, On the use of autarkies for satisfiability decision, in: LICS 2001 Workshop on Theory and Applications of Satisfiability Testing (SAT 2001), eds. H. Kautz and B. Selman, Electronic Notes in Discrete Mathematics (ENDM), Vol. 9 (Elsevier Science, 2001).

  37. O. Kullmann, Some general considerations on heuristics for search problems (in preparation).

  38. O. Kullmann, Generalised satisfiability decision: foundations and autarkies, Journal version of [36] concerned with general systems: to be submitted.

  39. C.M. Li and Anbulagan, Look-ahead versus look-back for satisfiability problems, in: Proceedings of CP-97, Lecture Notes in Computer Science, Vol. 1330 (Springer, 1997) pp. 342–356.

  40. P. Liberatore, On the complexity of choosing the branching literal in DPLL, Artificial Intelligence 116 (2000) 315–326.

    Google Scholar 

  41. L. Lovász, M. Naor, I. Newman and A. Wigderson, Search problems in the decision tree model, in: Proc. of 32nd Symposium on Foundations of Computer Science (FOCS' 91) (1991) pp. 576–585.

  42. D.G. Mitchell, Hard problems for CSP algorithms, in: Proceedings of 15th National Conference on Artificial Intelligence (AAAI-98) (1998).

  43. D. Pretolani, Hierarchies of polynomially solvable satisfiability problems, Annals of Mathematics and Artificial Intelligence 17(3-4) (1996) 339–357.

    Google Scholar 

  44. R.A. Reckhow, On the lengths of proofs in the propositional calculus, Ph.D. Thesis, Department of Computer Science, University of Toronto, Toronto (1975).

    Google Scholar 

  45. D.J. Rose, Triangulated graphs and the elimination process, Journal of Mathematical Analysis and Applications 32 (1970) 597–609.

    Google Scholar 

  46. M. Sheeran and G. Stålmarck, A tutorial on Stålmarck's proof procedure for propositional logic, in: Proc. of FMCAD'98, Lecture Notes in Computer Science, Vol. 1522 (1998) pp. 82–99.

  47. J.P.M. Silva and K.A. Sakallah, GRASP-A new search algorithm for satisfiability, Technical Report CSE-TR-292-96, Department of Electrical Engineering and Computer Science, University of Michigan (1996).

  48. G. Stålmarck and M. Säflund, Modeling and verifying systems and software in propositional logic, in: Proc. of Safety of Computer Control Systems (SAFECOMP'90) ed. B. Daniels (1990) pp. 31–36.

  49. J. Torán, Lower bounds for space in resolution, in: Computer Science Logic, Proc. of 13th International Workshop, CSL'99, Lecture Notes in Computer Science, Vol. 1683 (Springer, 1999) pp. 362–373.

  50. G. Tseitin, On the complexity of derivation in propositional calculus, in: Seminars in Mathematics, Vol. 8 (V.A. Steklov Mathematical Institute, Leningrad). English translation: Studies in Mathematics and Mathematical Logic, Part II, ed. A.O. Slisenko (1970) pp. 115–125.

    Google Scholar 

  51. A. Urquhart, The complexity of propositional proofs, The Bulletin of Symbolic Logic 1(4) (1995) 425–467.

    Google Scholar 

  52. S. Yamasaki and S. Doshita, The satisfiability problem for a class consisting of Horn sentences and some non-Horn sentences in propositional logic, Information and Control 59 (1983) 1–12.

    Google Scholar 

  53. H. Zhang, SATO: an efficient propositional prover, in: Proceedings of International Conference on Automated Deduction (CADE-97) (1997).

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, University of Wales Swansea, Swansea, SA2 8PP, UK

    Oliver Kullmann

Authors
  1. Oliver Kullmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kullmann, O. Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems. Annals of Mathematics and Artificial Intelligence 40, 303–352 (2004). https://doi.org/10.1023/B:AMAI.0000012871.08577.0b

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:AMAI.0000012871.08577.0b

Search

Navigation