On the learnability of shuffle ideals

  • Authors:

  • Dana Angluin;James Aspnes;Aryeh Kontorovich

  • Affiliations:

  • Department of Computer Science, Yale University, New Haven, CT;Department of Computer Science, Yale University, New Haven, CT;Department of Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel

  • Venue:
  • ALT'12 Proceedings of the 23rd international conference on Algorithmic Learning Theory
  • Year:
  • 2012

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Abstract

Although PAC learning unrestricted regular languages is long known to be a very difficult problem, one might suppose the existence (and even an abundance) of natural efficiently learnable sub-families. When our literature search for a natural efficiently learnable regular family came up empty, we proposed the shuffle ideals as a prime candidate. A shuffle ideal generated by a string u is simply the collection of all strings containing u as a (discontiguous) subsequence. This fundamental language family is of theoretical interest in its own right and also provides the building blocks for other important language families. Somewhat surprisingly, we discovered that even a class as simple as the shuffle ideals is not properly PAC learnable, unless RP=NP. In the positive direction, we give an efficient algorithm for properly learning shuffle ideals in the statistical query (and therefore also PAC) model under the uniform distribution.