Nearly Tight Bounds for Testing Function Isomorphism

We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove the following: (1) For most functions $f:\{0,1\}^n \to \{0,1\}$, the query complexity of testing isomorphism to $f$ is $\Omega(n)$. Moreover, the query complexity of testing isomorphism to most $k$-juntas $f:\{0,1\}^n \to \{0,1\}$ is $\Omega(k)$. (2) Isomorphism to any $k$-junta $f:\{0,1\}^n \to \{0,1\}$ can be tested with $O(k \log k)$ queries. (3) For some $k$-juntas $f:\{0,1\}^n \to \{0,1\}$, testing isomorphism to $f$ with one-sided error requires $\Omega(k\log(n/k))$ queries. In particular, testing whether $f:\{0,1\}^n \to \{0,1\}$ is a $k$-parity with one-sided error requires $\Omega(k\log(n/k))$ queries. (4) The query complexity of testing isomorphism between two unknown functions $f,g:\{0,1\}^n \to \{0,1\}$ is $\widetilde{\Theta}(2^{n/2})$. These bounds are tight up to logarithmic factors, and they significantly strengthen the bounds proved by Fischer, Kindler, Ron, Safra, and Samorodnitsky [J. Comput. System Sci., 68 (2004), pp. 753--787] and Blais and O'Donnell [Proceedings of the IEEE Conference on Computational Complexity, 2010, pp. 235--246]. We also obtain results closely related to isomorphism testing, answering a question posed by Diakonikolas, Lee, Matulef, Onak, Rubinfeld, Servedio, and Wan [Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007, pp. 549--558]: testing whether a function $f:\{0,1\}^n \to \{0,1\}$ can be computed by a circuit of size $\le s$ requires $s^{\Omega(1)}$ queries. All of our lower bounds apply to general (adaptive) testers.