Nearly tight bounds for testing function isomorphism

We study the problem of testing isomorphism (equivalence up to relabelling of the variables) of two Boolean functions <i>f,g</i>: {0, 1}^<i>n</i> → {0, 1}. Our main focus is on the most studied case, where one of the functions is given (explicitly) and the other function may be queried. We prove that for every <i>k</i> ≤ <i>n</i>, the worst-case query complexity of testing isomorphism to a given <i>k</i>-junta is Ω(<i>k</i>) and <i>O</i>(<i>k</i> log <i>k</i>). Consequently, the query complexity of testing function isomorphism is θ(<i>n</i>). Prior to this work, only lower bounds of Ω(log <i>k</i>) queries were known, for limited ranges of <i>k</i>, proved by Fischer et al. (FOCS 2002), Blais and O'Donnell (CCC 2010), and recently by Alon and Blais (RANDOM 2010). The nearly tight <i>O</i>(<i>k</i> log <i>k</i>) upper bound improves on the <i>Õ</i>(<i>k</i>⁴) upper bound from Fischer et al. (FOCS 2002). Extending the lower bound proof, we also show polynomial query-complexity lower bounds for the problems of testing whether a function can be computed by a circuit of size ≤ <i>s</i>, and testing whether the Fourier degree of a function is ≤ <i>d</i>. This answers questions posed by Diakonikolas et al. (FOCS 2007). We also address two closely related problems -- 1. Testing isomorphism to a <i>k</i>-junta with one-sided error: we prove that for any 1 < <i>k</i> < <i>n</i> − 1, the query complexity is Ω(log (^<i>n</i>_<i>k</i>)), which is almost optimal. This lower bound is a consequence of a proof that the query complexity of testing, with one-sided error, whether a function is a <i>k</i>-parity is Θ(log (^<i>n</i>_<i>k</i>). 2. Testing isomorphism between two unknown functions that can be queried: we prove that the query complexity in this setting is Ω(√2^<i>n</i>) and <i>O</i>(√2^<i>n</i><i>n</i> log <i>n</i>).