Advertisement

Approximation Logics for Subclasses of Probabilistic Conditional Independence and Hierarchical Dependence on Incomplete Data

  • Sebastian LinkEmail author
Chapter

Abstract

Probabilistic conditional independence constitutes a principled approach to handle knowledge and uncertainty in artificial intelligence, and is fundamental in probability theory and multivariate statistics. Similarly, first-order hierarchical dependence provides an expressive framework to capture the semantics of an application domain within a database system, and is essential for the design of databases. For complete data it is well known that the implication problem associated with probabilistic conditional independence is not axiomatizable by a finite set of Horn rules (Studený, Conditional independence relations have no finite complete characterization. In: Kubik, S., Visek, J. (eds.) Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 377–396. Kluwer, Dordrecht, 1992), and the implication problem for first-order hierarchical dependence is undecidable (Herrmann, Inf. Comput. 122(2):221–235, 1995). Moreover, both implication problems do not coincide (Studený, Conditional independence relations have no finite complete characterization. In: Kubik, S., Visek, J. (eds.) Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 377–396. Kluwer, Dordrecht, 1992) and neither of them is equivalent to the implication problem of some fragment of Boolean propositional logic (Sagiv et al., J. ACM 28(3):435–453, 1981). In this article, generalized saturated conditional independence and full first-order hierarchical dependence over incomplete data are investigated as expressive subclasses of probabilistic conditional independence and first-order hierarchical dependence, respectively. The associated implication problems are axiomatized by a finite set of Horn rules, and both shown to coincide with that of a propositional fragment under interpretations in the well-known approximation logic \(\mathcal{S}\)-3. Here, the propositional variables in the set \(\mathcal{S}\) are interpreted classically, and correspond to random variables as well as attributes on which incomplete data is not permitted to occur.

Notes

Acknowledgements

I would like to thank the anonymous reviewer for feedback that largely improved the presentation of the paper. This research is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.

References

  1. 1.
    Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, Boston, MA (1995)zbMATHGoogle Scholar
  2. 2.
    Abramsky, S., Kontinen, J., Väänänen, J.A., Vollmer, H.: Dependence logic: theory and applications (Dagstuhl seminar 13071). Dagstuhl Reports 3 (2), 45–54 (2013)Google Scholar
  3. 3.
    Atzeni, P., Morfuni, N.: Functional dependencies and constraints on null values in database relations. Inf. Control. 70 (1), 1–31 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beeri, C.: On the membership problem for functional and multivalued dependencies in relational databases. ACM Trans. Database Syst. 5 (3), 241–259 (1980)CrossRefzbMATHGoogle Scholar
  5. 5.
    Beeri, C., Fagin, R., Howard, J.H.: A complete axiomatization for functional and multivalued dependencies in database relations. In: Proceedings of the SIGMOD International Conference on Management of Data, pp. 47–61. ACM, Toronto (1977)Google Scholar
  6. 6.
    Biskup, J., Link, S.: Appropriate inferences of data dependencies in relational databases. Ann. Math. Artif. Intell. 63 (3–4), 213–255 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Biskup, J., Hartmann, S., Link, S.: Probabilistic conditional independence under schema certainty and uncertainty. In: Proceedings of the 6th International Conference on Scalable Uncertainty Management (SUM). Lecture Notes in Computer Science, vol. 7520, pp. 365–378. Springer, Berlin (2012)Google Scholar
  8. 8.
    Codd, E.F.: Extending the database relational model to capture more meaning. ACM Trans. Database Syst. 4 (4), 397–434 (1979)CrossRefGoogle Scholar
  9. 9.
    Codd, E.F.: Missing information (applicable and inapplicable) in relational databases. SIGMOD Rec. 15 (4), 53–78 (1986)CrossRefGoogle Scholar
  10. 10.
    Date, C., Darwen, H.: A guide to the SQL standard. Addison-Wesley, Reading, MA (1997)Google Scholar
  11. 11.
    Dawid, A.P.: Conditional independence in statistical theory. J. R. Stat. Soc. Ser. B Methodol. 41 (1), 1–31 (1979)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Delobel, C.: Normalization and hierarchical dependencies in the relational data model. ACM Trans. Database Syst. 3 (3), 201–222 (1978)CrossRefGoogle Scholar
  13. 13.
    Durand, A., Kontinen, J.: Hierarchies in dependence logic. ACM Trans. Comput. Log. 13 (4), 31 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fagin, R.: Functional dependencies in a relational data base and propositional logic. IBM J. Res. Dev. 21 (6), 543–544 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fagin, R.: Multivalued dependencies and a new normal form for relational databases. ACM Trans. Database Syst. 2 (3), 262–278 (1977)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ferrarotti, F., Hartmann, S., Link, S.: Reasoning about functional and full hierarchical dependencies over partial relations. Inf. Sci. 235, 150–173 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fienberg, S.: The analysis of incomplete multi-way contingency tables. Biometrics 28 (1), 177–202 (1972)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Galil, Z.: An almost linear-time algorithm for computing a dependency basis in a relational database. J. ACM 29 (1), 96–102 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Galliani, P.: Inclusion and exclusion dependencies in team semantics - on some logics of imperfect information. Ann. Pure Appl. Logic 163 (1), 68–84 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Geiger, D., Pearl, J.: Logical and algorithmic properties of independence and their application to Bayesian networks. Ann. Math. Artif. Intell. 2, 165–178 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Geiger, D., Pearl, J.: Logical and algorithmic properties of conditional independence and graphical models. Ann. Stat. 21 (4), 2001–2021 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grädel, E., Väänänen, J.A.: Dependence and independence. Stud. Logica 101 (2), 399–410 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Graetzer, G.: General Lattice Theory. Birkhäuser, Boston, MA (1998)Google Scholar
  24. 24.
    Halpern, J.: Reasoning About Uncertainty. MIT, Cambridge, MA (2005)zbMATHGoogle Scholar
  25. 25.
    Hartmann, S., Link, S.: On a problem of Fagin concerning multivalued dependencies in relational databases. Theor. Comput. Sci. 353 (1–3), 53–62 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hartmann, S., Link, S.: Characterising nested database dependencies by fragments of propositional logic. Ann. Pure Appl. Log. 152 (1–3), 84–106 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hartmann, S., Link, S.: When data dependencies over SQL tables meet the Logics of Paradox and S-3. In: Proceedings of the 29th ACM SIGMOD-SIGART-SIGACT Symposium on Principles of Database Systems (PoDS), pp. 317–326. ACM, Indianapolis, IN (2010)Google Scholar
  28. 28.
    Hartmann, S., Link, S.: The implication problem of data dependencies over SQL table definitions: axiomatic, algorithmic and logical characterizations. ACM Trans. Database Syst. 37 (2), pp. 13:1–13:40 (2012)Google Scholar
  29. 29.
    Herrmann, C.: On the undecidability of implications between embedded multivalued database dependencies. Inf. Comput. 122 (2), 221–235 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Herrmann, C.: Corrigendum to “on the undecidability of implications between embedded multivalued database dependencies”. Inf. Comput. 204 (12), 1847–1851 (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Hartmann, S., Kirchberg, M., Link, S.: Design by example for SQL table definitions with functional dependencies. VLDB J. 21 (1), 121–144 (2012)CrossRefGoogle Scholar
  32. 32.
    Köhler, H., Link, S.: Logics for approximating implication problems of saturated conditional independence. In: Fermé, E., Leite, J. (eds.) 14th European Conference on Logics in Artificial Intelligence JELIA 2014, Funchal, Madeira, September 24–26, 2014. Proceedings. Lecture Notes in Computer Science, vol. 8761, pp. 224–238. Springer, Berlin (2014)Google Scholar
  33. 33.
    Köhler, H., Link, S.: Saturated conditional independence with fixed and undetermined sets of incomplete random variables. In: Zhang, M., Tian, J. (eds.) 30th International Conference on Uncertainty in Artificial Intelligence, UAI 2014, Quebec City, Quebec, July 23–27, 2014. Proceedings, pp. 224–238. AUAI Press, Edinburgh (2014)Google Scholar
  34. 34.
    Kontinen, J., Link, S., Väänänen, J.A.: Independence in database relations. In: Proceedings of the 20th International Workshop on Logic, Language, Information, and Computation (WoLLIC). Lecture Notes in Computer Science, vol. 8071, pp. 179–193. Springer, Berlin (2013)Google Scholar
  35. 35.
    Le, V.B.T., Link, S., Ferrarotti, F.: Effective recognition and visualization of semantic requirements by perfect SQL samples. In: Proceedings of the 32th International Conference on Conceptual Modeling (ER). Lecture Notes in Computer Science, vol. 8217, pp. 227–240. Springer, Berlin (2013)Google Scholar
  36. 36.
    Levesque, H.: A knowledge-level account of abduction. In: Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI), pp. 1061–1067. Morgan Kaufmann, Detroit (1989)Google Scholar
  37. 37.
    Lien, E.: On the equivalence of database models. J. ACM 29 (2), 333–362 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Link, S.: Consistency enforcement in databases. In: Bertossi, L.E., Katona, G.O.H., Schewe, K., Thalheim, B. (eds.) Semantics in Databases, Second International Workshop, Dagstuhl Castle, Germany, January 7–12, 2001, Revised Papers, Lecture Notes in Computer Science, vol. 2582, pp. 139–159. Springer, Berlin (2003)Google Scholar
  39. 39.
    Link, S.: Charting the completeness frontier of inference systems for multivalued dependencies. Acta Inf. 45 (7–8), 565–591 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Link, S.: Characterizations of multivalued dependency implication over undetermined universes. J. Comput. Syst. Sci. 78 (4), 1026–1044 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Link, S.: Propositional reasoning about saturated conditional probabilistic independence. In: Proceedings of the 19th International Workshop on Logic, Language, Information and Computation (WoLLIC). Lecture Notes in Computer Science, vol. 7456, pp. 257–267. Springer, Buenos Aires (2012)Google Scholar
  42. 44.
    Link, S.: Sound approximate reasoning about saturated conditional probabilistic independence under controlled uncertainty. J. Appl. Log. 11 (3), 309–327 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 45.
    Lohmann, P., Vollmer, H.: Complexity results for modal dependence logic. Stud. Logica 101 (2), 343–366 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 46.
    Naumov, P., Nicholls, B.: R.E. axiomatization of conditional independence. In: Proceedings of the 14th Conference on Theoretical Aspects of Rationality and Knowledge (TARK), pp. 148–155 (2013)Google Scholar
  45. 47.
    Niepert, M., Van Gucht, D., Gyssens, M.: Logical and algorithmic properties of stable conditional independence. Int. J. Approx. Reason. 51 (5), 531–543 (2010)Google Scholar
  46. 48.
    Niepert, M., Gyssens, M., Sayrafi, B., Gucht, D.V.: On the conditional independence implication problem: a lattice-theoretic approach. Artif. Intell. 202, 29–51 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 49.
    Parker, D.S. Jr., Parsaye-Ghomi, K.: Inferences involving embedded multivalued dependencies and transitive dependencies. In: Proceedings of the International Conference on Management of Data (SIGMOD), pp. 52–57. ACM, New York (1980)Google Scholar
  48. 50.
    Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, CA (1988)zbMATHGoogle Scholar
  49. 51.
    Sagiv, Y.: An algorithm for inferring multivalued dependencies with an application to propositional logic. J. ACM 27 (2), 250–262 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 52.
    Sagiv, Y., Delobel, C., Parker, D.S. Jr., Fagin, R.: An equivalence between relational database dependencies and a fragment of propositional logic. J. ACM 28 (3), 435–453 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 53.
    Sagiv, Y., Delobel, C., Parker, D.S. Jr., Fagin, R.: Correction to “An equivalence between relational database dependencies and a fragment of propositional logic”. J. ACM 34 (4), 1016–1018 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 54.
    Schaerf, M., Cadoli, M.: Tractable reasoning via approximation. Artif. Intell. 74, 249–310 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 55.
    Stefanini, F.: Graphical models for eliciting structural information. In: Classification and Data Mining, pp. 139–146. Springer, Berlin (2013)Google Scholar
  54. 56.
    Studený, M.: Conditional independence relations have no finite complete characterization. In: Kubik, S., Visek, J. (eds.) Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes, pp. 377–396. Kluwer, Dordrecht (1992)Google Scholar
  55. 57.
    Thalheim, B.: Dependencies in Relational Databases. Teubner, Stuttgart (1991)CrossRefzbMATHGoogle Scholar
  56. 58.
    Väänänen, J.A.: Dependence Logic - A New Approach to Independence Friendly Logic. London Mathematical Society student texts, vol. 70. Cambridge University Press, Cambridge (2007)Google Scholar
  57. 59.
    Väänänen, J.A., Hodges, W.: Dependence of variables construed as an atomic formula. Ann. Pure Appl. Log. 161 (6), 817–828 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 60.
    Vincent, M.: Semantic foundations of 4NF in relational database design. Acta Inf. 36 (3), 173–213 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 61.
    Wong, S., Butz, C., Wu, D.: On the implication problem for probabilistic conditional independency. Trans. Syst. Man Cybern. Part A Syst. Humans 30 (6), 785–805 (2000)CrossRefGoogle Scholar
  60. 62.
    Wu, M.: The practical need for fourth normal form. In: Proceedings of the Twenty-third Technical Symposium on Computer Science Education (SIGCSE), pp. 19–23. ACM, New York (1992)Google Scholar
  61. 63.
    Zaniolo, C.: Database relations with null values. J. Comput. Syst. Sci. 28 (1), 142–166 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.The University of AucklandAucklandNew Zealand

Personalised recommendations