Block Sensitivity of Minterm-Transitive Functions

Boolean functions with symmetry properties 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 recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Omega(N^{1/3}), and that there exists such a function for which bs(f) = O(N^{3/7}ln N). His example function belongs to a subclass of transitively invariant functions called the minterm-transitive functions (defined in earlier work by Chakraborty). We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy bs(f) = Omega(N^{3/7}). Thus Sun's example function has nearly minimal block sensitivity for this subclass. Second, we give an improved example: a minterm-transitive function for which bs(f) = O(N^{3/7}ln^{1/7}N).Comment: 10 page

On the Sensitivity Complexity of k-Uniform Hypergraph Properties

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k

Block Sensitivity versus Sensitivity

Sensitivity and block sensitivity are useful and well-studied measures of computational complexity, but in spite of their similarities, the largest possible gap between them is still unknown. Rubinstein showed that this gap must be at least quadratic, and Kenyon and Kutin showed that it is at worst exponential, but many strongly suspect that the gap is indeed quadratic, or at worst polynomial. Our work shows that for a large class of functions, which includes Rubinstein\u27s function, the quadratic gap between sensitivity and block sensitivity is the best we can possibly do

Diameter Versus Certificate Complexity of Boolean Functions

In this paper, we introduce a measure of Boolean functions we call diameter, that captures the relationship between certificate complexity and several other measures of Boolean functions. Our measure can be viewed as a variation on alternating number, but while alternating number can be exponentially larger than certificate complexity, we show that diameter is always upper bounded by certificate complexity. We argue that estimating diameter may help to get improved bounds on certificate complexity in terms of sensitivity, and other measures. Previous results due to Lin and Zhang [Krishnamoorthy Dinesh and Jayalal Sarma, 2018] imply that s(f) ? ?(n^{1/3}) for transitive functions with constant alternating number. We improve and extend this bound and prove that s(f) ? ?n for transitive functions with constant alternating number, as well as for transitive functions with constant diameter. {We also show that bs(f) ? ?(n^{3/7}) for transitive functions under the weaker condition that the "minimum" diameter is constant.} Furthermore, we prove that the log-rank conjecture holds for functions of the form f(x ? y) for functions f with diameter bounded above by a polynomial of the logarithm of the Fourier sparsity of the function f

On the Sensitivity Conjecture

The sensitivity of a Boolean function f:{0,1}^n -> {0,1} is the maximal number of neighbors a point in the Boolean hypercube has with different f-value. Roughly speaking, the block sensitivity allows to flip a set of bits (called a block) rather than just one bit, in order to change the value of f. The sensitivity conjecture, posed by Nisan and Szegedy (CC, 1994), states that the block sensitivity, bs(f), is at most polynomial in the sensitivity, s(f), for any Boolean function f. A positive answer to the conjecture will have many consequences, as the block sensitivity is polynomially related to many other complexity measures such as the certificate complexity, the decision tree complexity and the degree. The conjecture is far from being understood, as there is an exponential gap between the known upper and lower bounds relating bs(f) and s(f). We continue a line of work started by Kenyon and Kutin (Inf. Comput., 2004), studying the l-block sensitivity, bs_l(f), where l bounds the size of sensitive blocks. While for bs_2(f) the picture is well understood with almost matching upper and lower bounds, for bs_3(f) it is not. We show that any development in understanding bs_3(f) in terms of s(f) will have great implications on the original question. Namely, we show that either bs(f) is at most sub-exponential in s(f) (which improves the state of the art upper bounds) or that bs_3(f) >= s(f){3-epsilon} for some Boolean functions (which improves the state of the art separations). We generalize the question of bs(f) versus s(f) to bounded functions f:{0,1}^n -> [0,1] and show an analog result to that of Kenyon and Kutin: bs_l(f) = O(s(f))^l. Surprisingly, in this case, the bounds are close to being tight. In particular, we construct a bounded function f:{0,1}^n -> [0, 1] with bs(f) n/log(n) and s(f) = O(log(n)), a clear counterexample to the sensitivity conjecture for bounded functions. Finally, we give a new super-quadratic separation between sensitivity and decision tree complexity by constructing Boolean functions with DT(f) >= s(f)^{2.115}. Prior to this work, only quadratic separations, DT(f) = s(f)^2, were known

On the Sensitivity Conjecture for Disjunctive Normal Forms

The sensitivity conjecture of Nisan and Szegedy [CC\u2794] asks whether for any Boolean function f, the maximum sensitivity s(f), is polynomially related to its block sensitivity bs(f), and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function f admitting the Normalized Block property, bs(f) <= 4 * s(f)^2. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. Recently, Gopalan et al. [ITCS\u2716] showed that every Boolean function f is uniquely specified by its values on a Hamming ball of radius at most 2 * s(f). We extend this result and also construct examples of Boolean functions which provide the matching lower bounds

Energy-Efficient Digital Circuit Design using Threshold Logic Gates

abstract: Improving energy efficiency has always been the prime objective of the custom and automated digital circuit design techniques. As a result, a multitude of methods to reduce power without sacrificing performance have been proposed. However, as the field of design automation has matured over the last few decades, there have been no new automated design techniques, that can provide considerable improvements in circuit power, leakage and area. Although emerging nano-devices are expected to replace the existing MOSFET devices, they are far from being as mature as semiconductor devices and their full potential and promises are many years away from being practical. The research described in this dissertation consists of four main parts. First is a new circuit architecture of a differential threshold logic flipflop called PNAND. The PNAND gate is an edge-triggered multi-input sequential cell whose next state function is a threshold function of its inputs. Second a new approach, called hybridization, that replaces flipflops and parts of their logic cones with PNAND cells is described. The resulting \hybrid circuit, which consists of conventional logic cells and PNANDs, is shown to have significantly less power consumption, smaller area, less standby power and less power variation. Third, a new architecture of a field programmable array, called field programmable threshold logic array (FPTLA), in which the standard lookup table (LUT) is replaced by a PNAND is described. The FPTLA is shown to have as much as 50% lower energy-delay product compared to conventional FPGA using well known FPGA modeling tool called VPR. Fourth, a novel clock skewing technique that makes use of the completion detection feature of the differential mode flipflops is described. This clock skewing method improves the area and power of the ASIC circuits by increasing slack on timing paths. An additional advantage of this method is the elimination of hold time violation on given short paths. Several circuit design methodologies such as retiming and asynchronous circuit design can use the proposed threshold logic gate effectively. Therefore, the use of threshold logic flipflops in conventional design methodologies opens new avenues of research towards more energy-efficient circuits.Dissertation/ThesisDoctoral Dissertation Computer Science 201

Decision Tree Complexity versus Block Sensitivity and Degree

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. It is known that decision tree complexity is bounded above by the cube of block sensitivity, and the cube of polynomial degree. However, the widest separation between decision tree complexity and each of block sensitivity and degree that is witnessed by known Boolean functions is quadratic. In this work, we investigate the tightness of the existing cubic upper bounds. We improve the cubic upper bounds for many interesting classes of Boolean functions. We show that for graph properties and for functions with a constant number of alternations, both of the cubic upper bounds can be improved to quadratic. We define a class of Boolean functions, which we call the zebra functions, that comprises Boolean functions where each monotone path from 0^n to 1^n has an equal number of alternations. This class contains the symmetric and monotone functions as its subclasses. We show that for any zebra function, decision tree complexity is at most the square of block sensitivity, and certificate complexity is at most the square of degree. Finally, we show using a lifting theorem of communication complexity by G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper bound on the decision tree complexity for all functions is in a sense equivalent to the potentially easier task of proving a similar upper bound on communication complexity for each bi-partition of the input variables, for all functions. In particular, this implies that to bound the decision tree complexity it suffices to bound smaller measures like parity decision tree complexity, subcube decision tree complexity and decision tree rank, that are defined in terms of models that can be efficiently simulated by communication protocols