All Classical Adversary Methods Are Equivalent for Total Functions

We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions and are equal to the fractional block sensitivity fbs( f ). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between fbs( f ) and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than √ n ⋅ bs( f ), where n is the number of variables and bs( f ) is the block sensitivity. Then, we exhibit a partial function f that matches this upper bound, fbs( f ) = Ω (√ n ⋅ bs( f )).

Sensitivities and block sensitivities of elementary symmetric Boolean functions

Boolean functions have important applications in molecular regulatory networks, engineering, cryptography, information technology, and computer science. Symmetric Boolean functions have received a lot of attention in several decades. Sensitivity and block sensitivity are important complexity measures of Boolean functions. In this paper, we study the sensitivity of elementary symmetric Boolean functions and obtain many explicit formulas. We also obtain a formula for the block sensitivity of symmetric Boolean functions and discuss its applications in elementary symmetric Boolean functions.

Unitary signings and induced subgraphs of Cayley graphs of $\mathbb{Z}_2^{n}$

Boolean functions play an important role in many different areas of computer science. The _local sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of coordinates whose flip changes the value of $f(x)$, i.e., the number of i's such that $f(x)\not=f(x+e_i)$, where $e_i$ is the $i$-th unit vector. The _sensitivity_ of a Boolean function is its maximum local sensitivity. In other words, the sensitivity measures the robustness of a Boolean function with respect to a perturbation of its input. Another notion that measures the robustness is block sensitivity. The _local block sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of disjoint subsets $I$ of $\{1,..,n\}$ such that flipping the coordinates indexed by $I$ changes the value of $f(x)$, and the _block sensitivity_ of $f$ is its maximum local block sensitivity. Since the local block sensitivity is at least the local sensitivity for any input $x$, the block sensitivity of $f$ is at least the sensitivity of $f$. The next example demonstrates that the block sensitivity of a Boolean function is not linearly bounded by its sensitivity. Fix an integer $k\ge 2$ and define a Boolean function $f:\{0,1\}^{2k^2}\to\{0,1\}$ as follows: the coordinates of $x\in\{0,1\}^{2k^2}$ are split into $k$ blocks of size $2k$ each and $f(x)=1$ if and only if at least one of the blocks contains exactly two entries equal to one and these entries are consecutive. While the sensitivity of the function $f$ is $2k$, its block sensitivity is $k^2$. The Sensitivity Conjecture, made by Nisan and Szegedy in 1992, asserts that the block sensitivity of a Boolean function is polynomially bounded by its sensivity. The example above shows that the degree of such a polynomial must be at least two. The Sensitivity Conjecture has been recently proven by Huang in [Annals of Mathematics 190 (2019), 949-955](https://doi.org/10.4007/annals.2019.190.3.6). He proved the following combinatorial statement that implies the conjecture (with the degree of the polynomial equal to four): any subset of more than half of the vertices of the $n$-dimensional cube $\{0,1\}^n$ induces a subgraph that contains a vertex with degree at least $\sqrt{n}$. The present article extends this result as follows: every Cayley graph with the vertex set $\{0,1\}^n$ and any generating set of size $d$ (the vertex set is viewed as a vector space over the binary field) satisfies that any subset of more than half of its vertices induces a subgraph that contains a vertex of degree at least $\sqrt{d}$. In particular, when the generating set consists of the $n$ unit vectors, the Cayley graph is the $n$-dimensional hypercube.

Accuracy of Intraoperative Frozen Section in Diagnosing Malignancy of Ovarian Neoplasm

Objective: To evaluate the accuracy of frozen section for ovarian neoplasm in our hospital. Method: A retrospective evaluation was conducted on medical records of patients with ovarian neoplasms who underwent a frozen section laparotomy between the years 2008 and 2013 at Dr. Cipto Mangunkusumo Hospital. Records with incomplete data on frozen section or paraffin block report were excluded. Criteria for frozen section laparotomy in our facility was based on a malignancy score of equal to or more than 6. Frozen section reports were compared to paraffin block report based on benign, borderline, or malignant cases. Result: From 139 patients with ovarian neoplasm, only 91 patients fulfilled the inclusion and exclusion criteria. Frozen section examination revealed benign cases was 15.4%, borderline cases was 15.4%, and malignant cases was 69.2%. Based on histopathological type, clear cell cystoadenocarcinoma was the most commonly observed histotype (19.8%). The sensitivity of frozen section for benign, borderline, and malignancy cases respectively was 81.8%, 76.9%, 91.0%. The specificity of frozen section for benign, borderline, and malignancy case respectively was 93.8%, 94.8%, 91.6%. Conclusion: We found that the accuracy of intraoperative frozen section in our facility is adequate to diagnose ovarian neoplasm and can be used to assist in determining the extent of surgical management. [Indones J Obstet Gynecol 2015; 3: 161-164] Keywords: frozen section, ovarian neoplasm, paraffin block, sensitivity, specitificity