Reductions for Learning via Queries
William Gasarch∗ and Geoffrey R. Hird†
1
Introduction
In [5, 6] the following question was considered: how much can an inductive inference machine learn if it is augmented with the ability to ask questions (about the recursive function it is trying to learn)? The questions must be in some query language L; so L is one of the parameters of learning. The other parameters of learning are type of question allowed (e.g., bounding the number of alternations) and the type of learning being used (e.g., the number of mindchanges may be bounded). The following questions were considered: 1. For certain values of the parameters, what classes of recursive functions could be learned? A key theme that emerges is that the less expressive (or loosely) the more decidable a query language is, the fewer classes of recursive functions can be learned using it. Of particular interest is when REC (the set of all recursive functions) can be EX-learned with queries to L; which is written REC ∈ QEX[L]. In [5, 6] it was shown that, for some query languages L, REC ∈ / QEX[L], while for others REC ∈ QEX[L]. 2. How do the resulting learning paradigms compare to each other? In [7] it was shown that, for several query languages L, there were sets of recursive functions that could be learned with no queries (just receiving data) and n + 1 mindchanges, that could not be learned with queries to L and n mindchanges; this is written EXn+1 − QEXn [L] 6= ∅. As a corollary we obtain QEXn [L] ⊂ QEXn+1 [L]. ∗
Dept. of Computer Science and Institute for Advanced Studies, University of Maryland, College Park, MD 20742. Supported in part by NSF grants CCR-8803641 and CCR9020079 (
[email protected]). † Odyssey Research Associates, 301 Dates Drive, Ithaca NY 14850, U.S.A. (
[email protected]).
In [6] ω-reg sets were used to obtain REC ∈ / QEX[S, < ] (queries can use symbols < and S where S stands for the Successor function). In this case the machinery was already in the literature. In [5] the machinery of k-good sets was developed to help obtain REC ∈ / QEX[+,
