Block sensitivity of minterm-transitive functions

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 bl...

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Bibliographic Details
Published in:Theoretical computer science Vol. 412; no. 41; pp. 5796 - 5801
Main Author: Drucker, Andrew
Format: Journal Article
Language:English
Published: Oxford Elsevier B.V 23.09.2011
Elsevier
Subjects:
Applied sciences
Block sensitivity
Blocking
Boolean functions
Calculus of variations and optimal control
Combinatorics. Ordered structures
Complexity theory
Computer science; control theory; systems
Exact sciences and technology
Group theory
Group theory and generalizations
Invariants
Mathematical analysis
Mathematical models
Mathematics
Minterm-transitive functions
Miscellaneous
Order, lattices, ordered algebraic structures
Permutations
Sciences and techniques of general use
Sun
Theoretical computing
Transitively invariant functions
Weakly symmetric functions
Minterm-transitive functions
Weakly symmetric functions
Boolean functions
Block sensitivity
Transitively invariant functions
Lower bound
Invariant
Computer theory
Boolean function
Symmetric function
Complexity measure
Research
Transitive group
Permutation group
Symmetry
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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 b s ( f ) = Ω ( N 1 / 3 ) . He also showed that there exists such a function for which b s ( f ) = O ( N 3 / 7 ln N ) . 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 b s ( f ) = Ω ( N 3 / 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 b s ( f ) = O ( N 3 / 7 ln 1 / 7 N ) .
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ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2011.06.025