Lower Bounds for Testing Function Isomorphism

We prove new lower bounds in the area of property testing of boolean functions. Specifically, we study the problem of testing whether a boolean function $f$ is isomorphic to a fixed function $g$ (i.e., is equal to $g$ up to permutation of the input variables). The analogous problem for testing graphs was solved by Fischer in 2005. The setting of boolean functions, however, appears to be more difficult, and no progress has been made since the initial study of the problem by Fischer et al. in 2004. Our first result shows that any non-adaptive algorithm for testing isomorphism to a function that ``strongly'' depends on $k$ variables requires $\log k - O(1)$ queries (assuming $k/n$ is bounded away from 1). This lower bound affirms and strengthens a conjecture appearing in the 2004 work of Fischer et al. Its proof relies on total variation bounds between hypergeometric distributions which may be of independent interest. Our second result concerns the simplest interesting case not covered by our first result: non-adaptively testing isomorphism to the Majority function on $k$ variables. Here we show that $\Omega(k^{1/12})$ queries are necessary (again assuming $k/n$ is bounded away from 1). The proof of this result relies on recently developed multidimensional invariance principle tools.