Let R_eps denote randomized query complexity for error probability eps, and R:=R_{1/3}. In this work we investigate whether a perfect composition theorem R(f o g^n)=Omega(R(f).R(g)) holds for a relation f in {0,1}^n * S and a total inner function g:{0,1}^m \to {0, 1}.

Let D^(prod) denote the maximum distributional query ... more >>>

For any Boolean functions $f$ and $g$, the question whether $\text{R}(f\circ g) = \tilde{\Theta}(\text{R}(f) \cdot \text{R}(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{\text{deg}}(f\circ g) = \tilde{\Theta}(\widetilde{\text{deg}}(f)\cdot\widetilde{\text{deg}}(g))$. These questions are two of the most important and ... more >>>

A classic result of Nisan [SICOMP '91] states that the deterministic decision tree depth complexity of every total Boolean function is at most the cube of its randomized decision tree depth complexity. The question whether randomness helps in significantly reducing the size of decision trees appears not to have been ... more >>>

We show that the deterministic decision tree complexity of a (partial) function or relation $f$ lifts to the deterministic parity decision tree (PDT) size complexity of the composed function/relation $f \circ g$ as long as the gadget $g$ satisfies a property that we call stifling. We observe that several simple ... more >>>

Relations between the decision tree complexity and various other complexity measures of Boolean functions is a thriving topic of research in computational complexity. While decision tree complexity is long known to be polynomially related with many other measures, the optimal exponents of many of these relations are not known. It ... more >>>

We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let $IP$ denote Inner Product on ... more >>>

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Our contributions are as follows. Let $f : \mathbb{F}_2^n \to \{-1, 1\}$ be a Boolean function with Fourier support ... more >>>

Let $\R(\cdot)$ stand for the bounded-error randomized query complexity. We show that for any relation $f \subseteq \{0,1\}^n \times \mathcal{S}$ and partial Boolean function $g \subseteq \{0,1\}^n \times \{0,1\}$, $\R_{1/3}(f \circ g^n) = \Omega(\R_{4/9}(f) \cdot \sqrt{\R_{1/3}(g)})$. Independently of us, Gavinsky, Lee and Santha \cite{newcomp} proved this result. By an example ... more >>>

Let $f$ be a Boolean function on $n$-bits, and $\mathsf{IP}$ the inner-product function on $2b$ bits. Let $f^{\mathsf{IP}}:=f \circ \mathsf{IP}^n$ be the two party function obtained by composing $f$ with $\mathsf{IP}$. In this work we bound the one-way communication complexity of $f^{\IP}$ in terms of the non-adaptive query complexity of ... more >>>

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is ... more >>>

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $\R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. ... more >>>

The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently
been used to prove separation results among various measures of
complexity such as deterministic, randomized and quantum query
complexities, exact and approximate polynomial degrees, etc. In
particular, the widest possible (quadratic) separations between
deterministic and zero-error ... more >>>

We show that there exists a Boolean function $F$ which observes the following separations among deterministic query complexity $(D(F))$, randomized zero error query complexity $(R_0(F))$
and randomized one-sided error query complexity $(R_1(F))$: $R_1(F) = \widetilde{O}(\sqrt{D(F)})$ and $R_0(F)=\widetilde{O}(D(F))^{3/4}$. This refutes the conjecture made by
Saks and Wigderson that for any Boolean ... more >>>

We prove that the Fourier dimension of any Boolean function with
Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof
method yields an improved bound of $\widetilde{O}(\sqrt{s})$
assuming a conjecture of Tsang~\etal~\cite{tsang}, that for every
Boolean function of sparsity $s$ there is an affine subspace of
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