arXiv subjects

Zhiyu Xia

Publications and source records attributed to Zhiyu Xia.

3 recordsLinked to original sources

arXiv2020

On the Degree of Boolean Functions as Polynomials over $\mathbb{Z}_m$

Polynomial representations of Boolean functions over various rings such as $\mathbb{Z}$ and $\mathbb{Z}_m$ have been studied since Minsky and Papert (1969). From then on, they have been employed in a large variety of fields including communication complexity, circuit complexity, learning theory, coding theory and so on. For any integer $m\ge2$, each Boolean function has a unique multilinear polynomial representation over ring $\mathbb Z_m$. The degree of such polynomial is called modulo-$m$ degree, denoted as $\mathrm{deg}_m(\cdot)$. In this paper, we investigate the lower bound of modulo-$m$ degree of Boolean functions. When $m=p^k$ ($k\ge 1$) for some prime $p$, we give a tight lower bound that $\mathrm{deg}_m(f)\geq k(p-1)$ for any non-degenerated function $f:\{0,1\}^n\to\{0,1\}$, provided that $n$ is sufficient large. When $m$ contains two different prime factors $p$ and $q$, we give a nearly optimal lower bound for any symmetric function $f:\{0,1\}^n\to\{0,1\}$ that $\mathrm{deg}_m(f) \geq \frac{n}{2+\frac{1}{p-1}+\frac{1}{q-1}}$.

cs.CC↗

arXiv2019

Structured decomposition for reversible Boolean functions

Reversible Boolean function is a one-to-one function which maps $n$-bit input to $n$-bit output. Reversible logic synthesis has been widely studied due to its relationship with low-energy computation as well as quantum computation. In this work, we give a structured decomposition for even reversible Boolean functions (RBF). Specifically, for $n\geq 6$, any even $n$-bit RBF can be decomposed to $7$ blocks of $(n-1)$-bit RBF, where $7$ is a constant independent of $n$; and the positions of those blocks have large degree of freedom. Moreover, if the $(n-1)$-bit RBFs are required to be even as well, we show for $n\geq 10$, $n$-bit RBF can be decomposed to $10$ even $(n-1)$-bit RBFs. For simplicity, we say our decomposition has block depth $7$ and even block depth $10$. Our result improves Selinger's work in block depth model, by reducing the constant from $9$ to $7$; and from $13$ to $10$ when the blocks are limited to be even. We emphasize that our setting is a bit different from Selinger's. In Selinger's constructive proof, each block is one of two specific positions and thus the decomposition has an alternating structure. We relax this restriction and allow each block to act on arbitrary $(n-1)$ bits. This relaxation keeps the block structure and provides more candidates when choosing positions of blocks.

cs.ET↗

arXiv2019

On the Relationship between Energy Complexity and other Boolean Function Measures

In this work we investigate into energy complexity, a Boolean function measure related to circuit complexity. Given a circuit $\mathcal{C}$ over the standard basis $\{\vee_2,\wedge_2,\neg\}$, the energy complexity of $\mathcal{C}$, denoted by $\mathrm{EC}(\mathcal{C})$, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function $f$, denoted by $\mathrm{EC}(f)$, is the minimum of $\mathrm{EC}(\mathcal{C})$ over all circuits $\mathcal{C}$ computing $f$. This concept has attracted lots of attention in literature. Recently, Dinesh, Otiv, and Sarma [COCOON'18] gave $\mathrm{EC}(f)$ an upper bound in terms of the decision tree complexity, $\mathrm{EC}(f)=O(\mathrm{D}(f)^3)$. They also showed that $\mathrm{EC}(f)\leq 3n-1$, where $n$ is the input size. Recall that the minimum size of circuit to compute $f$ could be as large as $2^n/n$. We improve their upper bounds by showing that $\mathrm{EC}(f)\leq\min\{\frac12\mathrm{D}(f)^2+O(\mathrm{D}(f)),n+2\mathrm{D}(f)-2\}$. For the lower bound, Dinesh, Otiv, and Sarma defined positive sensitivity, a complexity measure denoted by $\mathrm{psens}(f)$, and showed that $\mathrm{EC}(f)\ge\frac{1}{3}\mathrm{psens}(f)$. They asked whether $\mathrm{EC}(f)$ can also be lower bounded by a polynomial of $\mathrm{D}(f)$. In this paper we affirm it by proving $\mathrm{EC}(f)=Ω(\sqrt{\mathrm{D}(f)})$. For non-degenerated functions with input size $n$, we give another lower bound $\mathrm{EC}(f)=Ω(\log{n})$. All these three lower bounds are incomparable to each other. Besides, we also examine the energy complexity of $\mathtt{OR}$ functions and $\mathtt{ADDRESS}$ functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question asking for a non-trivial lower bounds for the energy complexity of $\mathtt{OR}$ functions.

cs.CC↗