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The Implication Problem for Functional Dependencies and Variants of Marginal Distribution Equivalences

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Foundations of Information and Knowledge Systems (FoIKS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS))

Abstract

We study functional dependencies together with two different probabilistic dependency notions: unary marginal identity and unary marginal distribution equivalence. A unary marginal identity states that two variables x and y are identically distributed. A unary marginal distribution equivalence states that the multiset consisting of the marginal probabilities of all the values for variable x is the same as the corresponding multiset for y. We present a sound and complete axiomatization and a polynomial-time algorithm for the implication problem for the class of these dependencies, and show that this class has Armstrong relations.

The author was supported by the Finnish Academy of Science and Letters (the Vilho, Yrjö and Kalle Väisälä Foundation) and by grant 345634 of the Academy of Finland.

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Notes

  1. 1.

    Our axiomatization is obviously only for the atomic level of these logics.

  2. 2.

    Note that a probabilistic team is actually not a relation but a probability distribution. Therefore, to be exact, instead of Armstrong relations, we should speak of Armstrong models, which is a more general notion introduced in [9]. In our setting, the Armstrong models we construct are uniform distributions over a relation, so each model is determined by a relation, and it suffices to speak of Armstrong relations.

  3. 3.

    This presentation of the algorithm is based on [7].

  4. 4.

    Sometimes the tuples of variables are assumed to be disjoint which disallows the probabilistic conditional independencies of this form. In the context of team semantics, such an assuption is usually not made.

References

  1. Abramsky, S., Puljujärvi, J., Väänänen, J.: Team semantics and independence notions in quantum physics (2021)

    Google Scholar 

  2. Albert, R., Grädel, E.: Unifying hidden-variable problems from quantum mechanics by logics of dependence and independence. Ann. Pure Appl. Logic 103088 (2022)

    Google Scholar 

  3. Armstrong, W.W.: Dependency structures of data base relationships. In: Proceedings of IFIP World Computer Congress, pp. 580–583 (1974)

    Google Scholar 

  4. Beeri, C., Bernstein, P.A.: Computational problems related to the design of normal form relational schemas. ACM Trans. Database Syst. 4(1), 30–59 (1979)

    Article  Google Scholar 

  5. Beeri, C., Dowd, M., Fagin, R., Statman, R.: On the structure of armstrong relations for functional dependencies. J. ACM 31(1), 30–46 (1984)

    Article  MathSciNet  Google Scholar 

  6. Chandra, A.K., Vardi, M.Y.: The implication problem for functional and inclusion dependencies is undecidable. SIAM J. Comput. 14(3), 671–677 (1985)

    Article  MathSciNet  Google Scholar 

  7. Cosmadakis, S.S., Kanellakis, P.C., Vardi, M.Y.: Polynomial-time implication problems for unary inclusion dependencies. J. ACM 37(1), 15–46 (1990)

    Article  MathSciNet  Google Scholar 

  8. Durand, A., Hannula, M., Kontinen, J., Meier, A., Virtema, J.: Approximation and dependence via multiteam semantics. Ann. Math. Artif. Intell. 83(3–4), 297–320 (2018)

    Article  MathSciNet  Google Scholar 

  9. Fagin, R.: Horn clauses and database dependencies. J. ACM 29, 952–985 (1982)

    Article  MathSciNet  Google Scholar 

  10. Fagin, R., Vardi, M.Y.: Armstrong databases for functional and inclusion dependencies. Inf. Process. Lett. 16(1), 13–19 (1983)

    Article  MathSciNet  Google Scholar 

  11. Galliani, P.: Inclusion and exclusion dependencies in team semantics: on some logics of imperfect information. Ann. Pure Appl. Logic 163(1), 68–84 (2012)

    Article  MathSciNet  Google Scholar 

  12. Geiger, D., Paz, A., Pearl, J.: Axioms and algorithms for inferences involving probabilistic independence. Inf. Comput. 91(1), 128–141 (1991)

    Article  MathSciNet  Google Scholar 

  13. Grädel, E., Väänänen, J.: Dependence and independence. Stud. Logica. 101(2), 399–410 (2013)

    Article  MathSciNet  Google Scholar 

  14. Hannula, M., Hirvonen, Å., Kontinen, J., Kulikov, V., Virtema, J.: Facets of distribution identities in probabilistic team semantics. In: Calimeri, F., Leone, N., Manna, M. (eds.) JELIA 2019. LNCS (LNAI), vol. 11468, pp. 304–320. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-19570-0_20

    Chapter  MATH  Google Scholar 

  15. Hannula, M., Link, S.: On the interaction of functional and inclusion dependencies with independence atoms. In: Pei, J., Manolopoulos, Y., Sadiq, S., Li, J. (eds.) DASFAA 2018. LNCS, vol. 10828, pp. 353–369. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-91458-9_21

    Chapter  Google Scholar 

  16. Hannula, M., Virtema, J.: Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals (2021)

    Google Scholar 

  17. Hodges, W.: Compositional Semantics for a Language of Imperfect Information. J. Interest Group Pure Appl. Logics 5(4), 539–563 (1997)

    MathSciNet  MATH  Google Scholar 

  18. Langeveldt, W.-D., Link, S.: Empirical evidence for the usefulness of armstrong relations in the acquisition of meaningful functional dependencies. Inf. Syst. 35(3), 352–374 (2010)

    Article  Google Scholar 

  19. Mannila, H., Räihä, K.-J.: Design by example: an application of armstrong relations. J. Comput. Syst. Sci. 33(2), 126–141 (1986)

    Article  MathSciNet  Google Scholar 

  20. Reingold, O.: Undirected connectivity in log-space. J. ACM 55(4), 1–24 (2008)

    Article  MathSciNet  Google Scholar 

  21. Studený, M.: Conditional independence relations have no finite complete characterization, pp. 377–396. Kluwer (1992)

    Google Scholar 

  22. Studený, M.: Semigraphoids are two-antecedental approximations of stochastic conditional independence models. In: Uncertainty in Artificial Intelligence, Proceedings of the Tenth Conference, pp. 546–552 (1994)

    Google Scholar 

  23. Tarjan, R.: Depth first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)

    Article  MathSciNet  Google Scholar 

  24. Väänänen, J.: Dependence Logic. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

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Acknowledgements

I would like to thank the anonymous referees for valuable comments, and Miika Hannula and Juha Kontinen for useful discussions and advice.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Minna Hirvonen

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  1. Minna Hirvonen
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Correspondence to Minna Hirvonen .

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  1. Université d'Artois and CNRS, Lens, France

    Ivan Varzinczak

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Hirvonen, M. (2022). The Implication Problem for Functional Dependencies and Variants of Marginal Distribution Equivalences. In: Varzinczak, I. (eds) Foundations of Information and Knowledge Systems. FoIKS 2022. Lecture Notes in Computer Science. Springer, Cham. https://doi.org/10.1007/978-3-031-11321-5_8

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  • DOI: https://doi.org/10.1007/978-3-031-11321-5_8

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-031-11321-5

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