Skip to main content
SpringerLink
Log in

Axiomatisability and hardness for universal Horn classes of hypergraphs

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

We consider hypergraphs as symmetric relational structures. In this setting, we characterise finite axiomatisability for finitely generated universal Horn classes of loop-free hypergraphs. An Ehrenfeucht–Fraïssé game argument is employed to show that the results continue to hold when restricted to first order definability amongst finite structures. We are also able to show that every interval in the homomorphism order on hypergraphs contains a continuum of universal Horn classes and conclude the article by characterising the intractability of deciding membership in universal Horn classes generated by finite loop-free hypergraphs.

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. Allender, E., Bauland, M., Immerman, N., Schnoor, H., Vollmer, H.: The complexity of satisfiability problems: refining Schaefer’s Theorem. J. Comput. Syst. Sci. 75, 245–254 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alechina, N., Gurevich, Y.: Syntax vs. semantics on finite structures. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds.) Structures in Logic and Computer Science. A Selection of Essays in Honor of Ehrenfeucht, A., LNCS, vol. 1261, pp. 14–33. Springer, Berlin (1997)

    Chapter  Google Scholar 

  3. Barto, L.: The constraint satisfaction problem and universal algebra. Bull. Symb. Logic 21, 319–337 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms and the constraint satisfaction problem. Log. Methods Comput. Sci. 8(1), 7:1–7:26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang–Jensen and Hell). SIAM J. Comput. 38, 1782–1802 (2008/2009)

  6. Barto, L., Krokhin, A., Willard, R.: Polymorphisms and how to use them. In: The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-ups, vol. 7, pp. 1–44 (2017)

  7. Bonato, A.: Continuum many universal Horn classes of graphs of bounded chromatic number. Algebra Universalis 40, 105–108 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brewster, R.C., Lee, J-B., Moore, B., Noel, J.A., Siggers, M.: On the complexity of two decision problems related to graph homomorphism reconfiguration. Manuscript. arXiv:1712.00200 (2017)

  9. Bulatov, A.A., Jeavons, P.G., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34, 720–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer, New York (1981)

    Book  MATH  Google Scholar 

  11. Caicedo, X.: Finitely axiomatizable quasivarieties of graphs. Algebra Universalis 34, 314–321 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, H., Larose, B.: Asking the metaquestions in constraint tractability. ACM Trans. Comput. Theory 9, 11:1–11:27 (2017)

    MathSciNet  Google Scholar 

  13. Clark, D.M., Davey, B.A., Haviar, M., Pitkethly, J.G., Talukder, M.R.: Standard topological quasi-varieties. Houston J. Math. 4, 859–887 (2003)

    MathSciNet  MATH  Google Scholar 

  14. Clark, D.M., Davey, B.A., Jackson, M.G., Pitkethly, J.G.: The axiomatizability of topological prevarieties. Adv. Math. 218, 1604–1653 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Erdős, P.: Graph theory and probability. Can. J. Math. 11, 34–48 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  16. Erdős, P., Hajnal, A.: On chromatic number of graphs and set-systems. Acta Mathematica Academiae Scientiarum Hungaricae Tomus 17, 61–99 (1966)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Gorbunov, V.A.: Algebraic Theory of Quasivarieties. Consultants Bureau, New York (1998)

    MATH  Google Scholar 

  19. Ham, L., Jackson, M.: All or nothing: toward a promise problem dichotomy for constraint problems. In: Beck, J.C. (ed.) CP 2017, LNCS, vol. 10416, pp. 139–156 (2017)

  20. Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory Ser. B 48, 92–110 (1990)

    Article  MATH  Google Scholar 

  21. Immerman, N.: Descriptive Complexity. Springer, New York (1999)

    Book  MATH  Google Scholar 

  22. Jackson, M.: Flexible constraint satisfiability and a problem in semigroup theory. Manuscript. arXiv:1512.03127 (2015)

  23. Jackson, M., Kowalski, T., Niven, T.: Digraph related constructions and the complexity of digraph homomorphism problems. Int. J. Algebra Comput. 26, 1395–1433 (2016)

    Article  MATH  Google Scholar 

  24. Jackson, M., Trotta, B.: Constraint satisfaction, irredundant axiomatisability and continuous colouring. Studia Logica 101, 65–94 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Jackson, M., McKenzie, R.: Interpreting graph colorability in finite semigroups. Int. J. Algebra Comput. 16, 119–140 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jackson, M., Volkov, M.V.: Relatively inherently nonfinitely Q-based finite semigroups. Trans. Am. Math. Soc. 361, 2181–2206 (2009)

    Article  MATH  Google Scholar 

  27. Larose, B.: Algebraic methods and the complexity of digraph CSPs: a survey. In: The Constraint Satisfaction Problem: Complexity and Approximability. Dagstuhl Follow-ups, vol. 7, pp. 267–285 (2017)

  28. Libkin, L.: Elements of Finite Model Theory. Texts in Theoretical Computer Science. Springer, Berlin (2004)

    Book  Google Scholar 

  29. Maltsev, A.I.: On the immersion of the associative systems in groups. I. Mat. Sb. 6, 331–336 (1939). (Russian)

    Google Scholar 

  30. Maltsev, A.I.: On the immersion of the associative systems in groups. II. Mat. Sb. 8, 251–264 (1940). (Russian)

    Google Scholar 

  31. Maltsev, A.I.: Algebraic Systems. Springer, Berlin (1973)

    Google Scholar 

  32. Margolis, S.W., Sapir, M.V.: Quasi-identities of finite semigroups and symbolic dynamics. Israel J. Math. 92, 317–331 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. McNulty, G.F.: Fragments of first order logic. I. Universal Horn logic. J. Symb. Logic 42, 221–237 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nešetřil, J.: The homomorphism structure of classes of graphs. Comb. Probab. Comput. 8, 177–184 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  35. Nešetřil, J., Pultr, A.: On classes of relations and graphs determined by subobjects and factorobjects. Discrete Math. 22, 287–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rosen, E.: Some aspects of model theory and finite structures. Bull. Symb. Logic 8, 380–403 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sapir, M.V.: On the quasivarieties generated by finite semigroups. Semigroup Forum 20, 73–88 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  38. Szekely, Z.: Computational complexity of the finite algebra membership problem for varieties. Int. J. Algebra Comput. 12, 811–823 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Trotta, B.: Residual properties of simple graphs. Bull. Aust. Math. Soc. 82, 488–504 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, La Trobe University, Melbourne, VIC, 3086, Australia

    Lucy Ham & Marcel Jackson

Authors
  1. Lucy Ham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcel Jackson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcel Jackson.

Additional information

Presented by R. Willard.

M. Jackson was supported by ARC Discovery Project DP1094578 and Future Fellowship FT120100666.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ham, L., Jackson, M. Axiomatisability and hardness for universal Horn classes of hypergraphs. Algebra Univers. 79, 30 (2018). https://doi.org/10.1007/s00012-018-0515-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-018-0515-y

Mathematics Subject Classification

Keywords

Search

Navigation