Skip to main content
Springer Nature Link
Log in

Boolean Formulas Are Hard to Learn for Most Gate Bases

  • Conference paper
  • First Online:
Algorithmic Learning Theory (ALT 1999)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1720))

Included in the following conference series:

  • 595 Accesses

Abstract

Boolean formulas are known not to be PAC-predictable even with membership queries under some cryptographic assumptions. In this paper, we study the learning complexity of some subclasses of boolean formulas obtained by varying the basis of elementary operations allowed as connectives. This broad family of classes includes, as a particular case, general boolean formulas, by considering the basis given by {AND,OR,NOT}. We completely solve the problem. We prove the following dichotomy theorem: For any set of basic boolean functions, the resulting set of formulas is either polynomially learnable from equivalence queries or membership queries alone or else it is not PAC-predictable even with membership queries under cryptographic assumptions. We identify precisely which sets of basic functions are in which of the two cases. Furthermore, we prove than the learning complexity of formulas over a basis depends only on the absolute expressivity power of the class, ie., the set of functions that can be represented regardless of the size of the representation. In consequence, the same classification holds for the learnability of boolean circuits.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. D. Angluin. Queries and Concept Learning. Machine Learning, 2:319–342, 1988.

    Google Scholar 

  2. D. Angluin, M. Frazier, and L. Pitt. Learning Conjunctions of Horn Clauses. Machine Learning, 9:147–164, 1992.

    MATH  Google Scholar 

  3. D. Angluin, L. Hellerstein, and M. Karpinski. Learning Read-Once Formulas with Queries. Journal of the ACM, 40:185–210, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Angluin and M. Kharitonov. When won’t Membership Queries help. Journal of Computer and System Sciences, 50:336–355, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  5. U. Berggren. Linear Time Deterministic Learning of k-term DNF. In 6th Annual ACM Conference on Computational Learning Theory, COLT’93, pages 37–40, 1993.

    Google Scholar 

  6. N. Creignou. A Dichotomy Theorem for Maximum Generalized Satisfiability Problems. Journal of Computer and System Sciences, 51(3):511–522, 1995.

    Article  MathSciNet  Google Scholar 

  7. N. Creignou and M. Hermann. Complexity of Generalized Satisfiability Counting Problems. Information and Computation, 125:1–12, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  8. V. Dalmau. A Dichotomy Theorem for Learning Quantified Boolean Formulas. Machine Learning, 35(3):207–224, 1999.

    Article  MATH  Google Scholar 

  9. V. Dalmau and P. Jeavons. Learnability of Quantified Formulas. In 4th European Conference on Computational Learning Theory Eurocolt’99, volume 1572 of Lecture Notes in Artificial Intelligence, pages 63–78, Berlin New York, 1999. Springer-Verlag.

    MathSciNet  Google Scholar 

  10. L.M. Goldschlager. A Characterization of Sets of n-Input Gates in Terms of their Computational Power. Technical Report 216, Basser Department of Computer Science, The University of Sidney, 1983.

    Google Scholar 

  11. L.M. Goldschlager and I. Parberry. On the Construction of Parallel Computers from various bases of Boolean Circuits. Theoretical Computer Science, 43:43–58, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  12. P. Jeavons, D. Cohen, and M.C. Cooper. Constraints, Consistency and Closure. Artificial Intelligence, 101:251–265, 1988.

    Article  MathSciNet  Google Scholar 

  13. P. Jeavons, D. Cohen, and M. Gyssens. A Unifying Framework for Tractable Constraints. In 1st International Conference on Principles and Practice of Constraint Programming, CP’95, Cassis (France), September 1995, volume 976 of Lecture Notes in Computer Science, pages 276–291. Springer-Verlag, 1995.

    Google Scholar 

  14. P. Jeavons, D. Cohen, and M. Gyssens. A Test for Tractability. In 2nd International Conference on Principles and Practice of Constraint Programming CP’96, volume 1118 of Lecture Notes in Computer Science, pages 267–281, Berlin/New York, August 1996. Springer-Verlag.

    Google Scholar 

  15. P. Jeavons, D. Cohen, and M. Gyssens. Closure Properties of Constraints. Journal of the ACM, 44(4):527–548, July 1997.

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Jeavons and M. Cooper. Tractable Constraints on Ordered Domains. Artificial Intelligence, 79:327–339, 1996.

    Article  MathSciNet  Google Scholar 

  17. D. Kavvadias and M. Sideri. The Inverse Satisfiability Problem. In 2nd Computing and Combinatorics COCOON’96, volume 1090 of Lecture Notes in Computer Science, pages 250–259. Springer-Verlag, 1996.

    Google Scholar 

  18. Michael Kearns and Leslie Valiant. Cryptographic limitations on learning Boolean formulae and finite automata. Journal of the ACM, 41(1):67–95, January 1994.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Khanna, M. Sudan, and L. Trevisan. Constraint Satisfaction: The Approximability of Minimization Problems. In 12th IEEE Conference on Computational Complexity, 1997.

    Google Scholar 

  20. S. Khanna, M. Sudan, and P. Williamson. A Complete Classification fo the Approximability of Maximation Problems Derived from Boolean Constraint Satisfaction. In 29th Annual ACM Symposium on Theory of Computing, 1997.

    Google Scholar 

  21. E.L. Post. The Two-Valued Iterative Systems of Mathematical Logic, volume 5 of Annals of Mathematics Studies. Princeton, N.J, 1941.

    Google Scholar 

  22. S. Reith and H. Vollmer. The Complexity of Computing Optimal Assignments of Generalized Propositional Formulae. Technical Report TR196, Department of Computer Science, Universität Würzburg, 1999.

    Google Scholar 

  23. S. Reith and K.W. Wagner. The Complexity of Problems Defined by Subclasses of Boolean Functions. Technical Report TR218, Department of Computer Science, Universität Würzburg, 1999.

    Google Scholar 

  24. T.J. Schaefer. The Complexity of Satisfiability Problems. In 10th Annual ACM Symposium on Theory of Computing, pages 216–226, 1978.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament LSI, Universitat Politècnica de Catalunya, Mòdul C5. Jordi Girona Salgado 1-3, Barcelona, 08034, Spain

    Víctor Dalmau

Authors
  1. Víctor Dalmau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, 152-8552, Japan

    Osamu Watanabe

  2. Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo, 169-8050, Japan

    Takashi Yokomori

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dalmau, V. (1999). Boolean Formulas Are Hard to Learn for Most Gate Bases. In: Watanabe, O., Yokomori, T. (eds) Algorithmic Learning Theory. ALT 1999. Lecture Notes in Computer Science(), vol 1720. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46769-6_25

Download citation

  • DOI: https://doi.org/10.1007/3-540-46769-6_25

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66748-3

  • Online ISBN: 978-3-540-46769-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics