Skip to main content

Cryptographic limitations on learning Boolean formulae and finite automata

  • Chapter
  • First Online:
Machine Learning: From Theory to Applications

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 661))

This short unrefereed paper has appeared previously in Proceedings of the 21st Annual ACM Symposium on Theory of Computing (ACM, 1989), pages 433–444. The full refereed version of this paper will be published in Jounal of the ACM. The research was done while M. Kearns was at Harvard University, and visiting Oxford University and AT&T Bell Laboratories, and while L. Valiant was visiting Oxford University. M. Kearns was supported by an AT&T Bell Laboratories Ph.D. scholarship, and L.Valiant by grants NSF-DCR-86-00379, ONR-N00014-85-K-0445, DAAL03-86-K-0171 and by SERC.

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Adleman, K. Manders, G. Miller. On taking roots in finite fields. Proc. 18th IEEE Symp. on Foundations of Computer Science, 1977, pp. 175–178.

    Google Scholar 

  2. D. Aldous. On the Markov chain simulation method for uniform combinatorial distributions and simulated annealing. U.C. Berkeley Statistics Department, technical report 60, 1986.

    Google Scholar 

  3. W. Alexi, B. Chor, O. Goldreich, C. P. Schnorr. RSA and Rabin functions: Certain parts are as hard as the whole. SIAM J. on Computing, 17(2) 1988, pp. 194–209.

    Google Scholar 

  4. D. Angluin. Lecture notes on the complexity of some problems in number theory. Yale University Computer Science Department, technical report TR-243, 1982.

    Google Scholar 

  5. D. Angluin. Learning regular sets from queries and counterexamples. Inf. and Computation, 75, 1987, pp. 87–106.

    Google Scholar 

  6. D. Angluin. Queries and concept learning. Machine Learning, 2, 1987, pp. 319–342.

    Google Scholar 

  7. D. Angluin, L. G. Valiant. Fast probabilistic algorithms for Hamiltonian circuits and matchings. J. of Computer and Systems Sciences, 18 1979, pp. 155–193.

    Google Scholar 

  8. P. W. Beame, S. A. Cook, H. J. Hoover. Log depth circuits for division and related problems. SIAM J. on Computing, 15 (4), 1986, pp. 994–1003.

    Google Scholar 

  9. G.M. Benedek, A. Itai. Learnability by fixed distributions. Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp. 80–90.

    Google Scholar 

  10. A. Blum. An O(n0.4)-approximation algorithm for 3-coloring. Proc. 21st ACM Symp. on Theory of Computing, 1989.

    Google Scholar 

  11. A. Blum, R. L. Rivest. Training a 3-node neural network is NP-complete. Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp. 9–18.

    Google Scholar 

  12. A. Blumer, A. Ehrenfeucht, D. Haussler, M. Warmuth. Classifying learnable geometric concepts with the Vapnik-Chervonenkis dimension. Proc. of the 18th ACM Symp. on Theory of Computing, 1986, pp. 273–282.

    Google Scholar 

  13. A. Blumer, A. Ehrenfeucht, D. Haussler, M. Warmuth. Occam's razor. Inf. Proc. Letters, 24 1987, pp. 377–380.

    Google Scholar 

  14. A.K. Chandra, L.J. Stockmeyer, U. Vishkin. Constant depth reducibility. SIAM J. on Computing, 13 (2), 1984, pp. 423–432.

    Google Scholar 

  15. H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics, 23, 1952, pp. 493–509.

    Google Scholar 

  16. A. Ehrenfeucht, D. Haussler. Personal Communication.

    Google Scholar 

  17. A. Ehrenfeucht, D. Haussler, M. Kearns. L. G. Valiant. A general lower bound on the number of examples needed for learning. Information and Computation, to appear. Also in Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp. 139–154.

    Google Scholar 

  18. M. Garey, D. Johnson, Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, CA, 1979.

    Google Scholar 

  19. E. M. Gold. Complexity of automaton identification from given data. Inf. and Control, 37, 1978, pp. 302–320.

    Google Scholar 

  20. O. Goldreich, S. Goldwasser, S. Micali. How to construct random functions. J. of the ACM, 33(4) 1986, pp. 792–807.

    Google Scholar 

  21. D. Haussler, M. Kearns, N. Littlestone, M. Warmuth. Equivalence of models for polynomial learnability. Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp 42–55.

    Google Scholar 

  22. D. Haussler, N. Littlestone, M. Warmuth. Predicting 0,1-functions on randomly drawn points. Proc. of the 29th IEEE Symp. on Foundations of Computer Science, 1988, pp. 100–109.

    Google Scholar 

  23. O. Ibarra, T. Jiang. Learning regular languages from counterexamples. First Workshop on Computational Learning Theory, 1988, pp. 371–385.

    Google Scholar 

  24. S. Judd. Learning in neural networks. Proc. of the 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp 2–8.

    Google Scholar 

  25. M. Kearns, M. Li, L. Pitt, L.G. Valiant. On the learnability of Boolean formulae. Proc. of the 19th ACM Symp. on Theory of Computing, 1987, pp. 285–295.

    Google Scholar 

  26. E. Kranakis. Primality and cryptography. John Wiley and Sons, 1986.

    Google Scholar 

  27. L. Levin. One-way functions and pseudorandom generators. Proc. of the 17th ACM Symp. on Theory of Computing, 1985, pp. 363–365.

    Google Scholar 

  28. M. Li, U. Vazirani. On the learnability of finite automata. First Workshop on Computational Learning Theory, 1988, pp. 359–370.

    Google Scholar 

  29. L. Pitt, M. K. Warmuth. Reductions among prediction problems: on the difficulty of predicting automata. Proc. 3d Conference on Structure in Complexity Theory, 1988, pp. 60–69.

    Google Scholar 

  30. L. Pitt, M. K. Warmuth. The Minimum DFA Consistency Problem Cannot be Approximated Within any Polynomial. Proc. 21st ACM Symp. on Theory of Computing, 1989.

    Google Scholar 

  31. L. Pitt, L. G. Valiant. Computational limitations on learning from examples. J. of the ACM, 35(4), 1988, pp. 965–984.

    Google Scholar 

  32. M. O. Rabin. Digital signatures and public key functions as intractable as factorization. M.I.T. Laboratory for Computer Science technical report TM-212, 1979.

    Google Scholar 

  33. J. Reif. On threshold circuits and polynomial computation. Proc. 2nd IEEE Conference on Structure in Complexity Theory, 1987.

    Google Scholar 

  34. R. Rivest, A. Shamir, L. Adleman. A method for obtaining digital signatures and public key cryptosystems. Comm. of the ACM, 21(2) 1978, pp. 120–126.

    Google Scholar 

  35. L. G. Valiant. A theory of the learnable. Comm. of the ACM, 27(11) 1984, pp. 1134–1142.

    Google Scholar 

  36. L. G. Valiant. Learning disjunctions of conjunctions. Proc. 9th International Joint Conference on Artificial Intelligence, 1985, pp. 560–566.

    Google Scholar 

  37. L. G. Valiant. Functionality in neural nets. Proc. 1988 Workshop on Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pp. 28–39.

    Google Scholar 

  38. A. Wigderson. A new approximate graph coloring algorithm. Proc. 14th ACM Symp. on Theory of Computing, 1982, pp. 325–329.

    Google Scholar 

  39. A. C. Yao. Theory and application of trapdoor functions. Proc. 23rd IEEE Symp. on the Foundations of Computer Science, 1982, 80–91.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Stephen José Hanson Werner Remmele Ronald L. Rivest

Rights and permissions

Reprints and permissions

Copyright information

© 1993 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Kearns, M.J., Valiant, L.G. (1993). Cryptographic limitations on learning Boolean formulae and finite automata. In: Hanson, S.J., Remmele, W., Rivest, R.L. (eds) Machine Learning: From Theory to Applications. Lecture Notes in Computer Science, vol 661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56483-7_21

Download citation

  • DOI: https://doi.org/10.1007/3-540-56483-7_21

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-56483-6

  • Online ISBN: 978-3-540-47568-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics