On the Sensitivity Conjecture for Read-k Formulas

Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link". Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and functions describing graph properties. In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k.

[1]  David Rubinstein.  Sensitivity vs. block sensitivity of Boolean functions , 1995, Comb..

[2]  György Turán,et al.  The Critical Complexity of Graph Properties , 1984, Information Processing Letters.

[3]  Avishay Tal,et al.  Degree and Sensitivity: tails of two distributions , 2016, Electron. Colloquium Comput. Complex..

[4]  Hans Ulrich Simon.  A Tight Omega(log log n)-Bound on the Time for Parallel RAM's to Compute Nondegenerated Boolean Functions , 1982, Inf. Control..

[5]  Andris Ambainis,et al.  A Tight Lower Bound on Certificate Complexity in Terms of Block Sensitivity and Sensitivity , 2014, MFCS.

[6]  Xiaoming Sun.  Block sensitivity of weakly symmetric functions , 2007, Theor. Comput. Sci..

[7]  Noam Nisan,et al.  CREW PRAMS and decision trees , 1989, STOC '89.

[8]  Hiroki Morizumi.  Sensitivity, Block Sensitivity, and Certificate Complexity of Unate Functions and Read-Once Functions , 2014, IFIP TCS.

[9]  H. Buhrman,et al.  Complexity measures and decision tree complexity: a survey , 2002, Theor. Comput. Sci..

[10]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[11]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[12]  Andris Ambainis,et al.  Tighter Relations between Sensitivity and Other Complexity Measures , 2014, ICALP.

[13]  Avishay Tal.  On the Sensitivity Conjecture , 2016, ICALP.

[14]  Claire Mathieu,et al.  Sensitivity, block sensitivity, and l-block sensitivity of boolean functions , 2004, Inf. Comput..

[15]  Ronald de Wolf,et al.  Quantum lower bounds by polynomials , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[16]  Stephen A. Cook,et al.  Upper and Lower Time Bounds for Parallel Random Access Machines without Simultaneous Writes , 1986, SIAM J. Comput..

[17]  Michael E. Saks,et al.  A New Approach to the Sensitivity Conjecture , 2015, ITCS.

[18]  Nathan Linial,et al.  The Equivalence of Two Problems on the Cube , 1992, J. Comb. Theory, Ser. A.

[19]  Sourav Chakraborty.  On the sensitivity of cyclically-invariant Boolean functions , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[20]  Pooya Hatami,et al.  Variations on the Sensitivity Conjecture , 2011, Theory Comput..

[21]  Meena Boppana.  Lattice Variant of the Sensitivity Conjecture , 2012, Electron. Colloquium Comput. Complex..