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Summary: Local Correction of Juntas
Noga Alon1
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv
University, Tel Aviv 69978, Israel and Institute for Advanced Study, Princeton, New Jersey
08540, USA.
Amit Weinstein2
Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel.
Abstract
A Boolean function f over n variables is said to be q-locally correctable if, given
a black-box access to a function g which is "close" to an isomorphism f of f,
we can compute f(x) for any x Zn
2 with good probability using q queries to
g.
We observe that any k-junta, that is, any function which depends only on
k of its input variables, is O(2k
)-locally correctable. Moreover, we show that
there are examples where this is essentially best possible, and locally correcting
some k-juntas requires a number of queries which is exponential in k. These
examples, however, are far from being typical, and indeed we prove that for
almost every k-junta, O(k log k) queries suffice.
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