  The model of learning with \emph{local membership queries} interpolates
between the PAC model and the membership queries model by allowing the learner
to query the label of any example that is similar to an example in the training
set. This model, recently proposed and studied by Awasthi, Feldman and Kanade,
aims to facilitate practical use of membership queries.
  We continue this line of work, proving both positive and negative results in
the {\em distribution free} setting. We restrict to the boolean cube $\{-1,
1\}^n$, and say that a query is $q$-local if it is of a hamming distance $\le
q$ from some training example. On the positive side, we show that $1$-local
queries already give an additional strength, and allow to learn a certain type
of DNF formulas. On the negative side, we show that even
$\left(n^{0.99}\right)$-local queries cannot help to learn various classes
including Automata, DNFs and more. Likewise, $q$-local queries for any constant
$q$ cannot help to learn Juntas, Decision Trees, Sparse Polynomials and more.
Moreover, for these classes, an algorithm that uses
$\left(\log^{0.99}(n)\right)$-local queries would lead to a breakthrough in the
best known running times.
