Local correction of juntas

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 σ of f, we can compute f σ ( x ) for any x ∈ Z 2 n 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 ( 2 k ) -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. ► We consider the number of queries needed to locally correct k-juntas. ► For every input x, we must recover f ( x ) with good probability using few queries to f. ► We observe that for every k-junta, O ( 2 k ) queries suffice for local correction. ► This is best possible as we show some k-juntas require exponential number of queries. ► However, for most k-juntas we provide an algorithm which performs O ( k log k ) queries.