Block sensitivity of minterm-transitive functions |
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Authors: | Andrew Drucker |
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Affiliation: | Department of Electrical Engineering and Computer Science, MIT 32-G630, Cambridge, MA 02139, USA |
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Abstract: | Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures.In a recent work of this type, Sun (2007) 9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f)=Ω(N1/3). He also showed that there exists such a function for which bs(f)=O(N3/7lnN). His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005) 3].We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f)=Ω(N3/7). Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which bs(f)=O(N3/7ln1/7N). |
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Keywords: | Boolean functions Block sensitivity Transitively invariant functions Weakly symmetric functions Minterm-transitive functions |
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