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I specifically mean language families that admit arbitrarily long strings -- not conjunctions over n bits or decision lists or any other "simple" language contained in {0,1}^n.

I am asking about "automata-theoretic" regular languages as opposed to "logic-theoretic" ones: something like the piecewise testable languages, start-height-zero languages, locally testable languages, that sort of thing. The relevant complexity parameter n is the size of the minimal accepting DFA. So, succinctly put: is there an interesting family of n-state DFAs that is known to be efficiently PAC learnable?

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There is a recent result on the polynomial pac-learnability of semilinear sets at LICS 2010: Parikh Images of Regular Languages: Complexity and Applications. I guess this is not what you are looking for.

You should also have a look to the paper of Clark and Thollard: PAC-learnability of Probabilistic Deterministic Finite State Automata

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    $\begingroup$ Right, I am familiar with Clark and Thollard's paper -- but they make distribution assumptions so it's not true PAC... $\endgroup$ – Aryeh Oct 14 '10 at 7:59
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This paper gives a good hint about PAC-learning result for piecewise languages: Learning Linearly Separable Languages

The work of Clark & Thollard was refined by Castro & Gavalda in a way that can fit what you are looking for : Towards feasible PAC-learning probabilistic deterministic finite automata

And this work is a good answer of the initial question: On the Learnability of Shuffle Ideals. One of the authors is likely to be the same person who formerly asked the question here, but I found this page by working on that problem and have just found this paper: it might help other to have this reference.

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    $\begingroup$ My guess is that @Aryeh is aware of at least two of these papers :) $\endgroup$ – Lev Reyzin Mar 19 '13 at 2:29
  • $\begingroup$ Indeed, I vaguely recall co-authoring #1 and #3... None of these give positive PAC results of the type I asked about. In #1, we require a margin, which is a distribution-dependent quantity. In #3, we give strong negative results. $\endgroup$ – Aryeh Mar 23 '13 at 21:37

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