NDLI: Nearly Tight Bounds for Testing Function Isomorphism

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Alon, Noga García-Soriano, David Chakraborty, Sourav Blais, Eric Matsliah, Arie
Copyright Year	2013
Abstract	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.
Starting Page	459
Ending Page	493
Page Count	35
File Format	PDF
ISSN	00975397
DOI	10.1137/110832677
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	42
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2013-03-12
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems Boolean functions property testing isomorphism
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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