Testing linear-invariant function isomorphism
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We make a step towards characterizing the boolean functions to which isomorphism can be efficiently tested. Specifically, we prove that isomorphism to any boolean function on $\{0, 1\}^n$ with a polynomial number of distinct permutations can be tested with a number of queries that is independent of~$n$. We also show some partial results in the converse direction, and discuss related problems: testing isomorphism up to linear transformations, and testing isomorphism against a uniform (hyper)graph that is given in advance. Our results regarding the latter topic generalize a theorem of Fischer (SICOMP 2005), and in the process we also provide a simpler proof of his original result which avoids the use of Szemer\'edi's regularity lemma.