### 计算机代写|量子计算代写Quantum computing代考|New Multiple-Valued S/D Trees

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 计算机代写|量子计算代写Quantum computing代考|Canonical Galois Field Sum-Of-Product Forms

Economical and highly testable implementations of Boolean functions $[99,198,199,204,217]$, based on Reed-Muller (ANDEXOR) logic, play an important role in logic synthesis and circuit design. AND-EXOR circuits include canonical forms (i.e., expansions that are unique representations of a Boolean function). Several large families of canonical forms: Fixed Polarity ReedMuller (FPRM) forms, Generalized Reed-Muller (GRM) forms, Kronecker (KRO) forms, and Pseudo-Kronecker (PSDKRO) forms, referred to as the Green/Sasao hierarchy, have been described $[4,9]$. Because canonical families have higher testability and some other properties desirable for efficient synthesis, especially of some classes of functions, they are widely investigated. A similar ternary version of the binary Green/Sasao hierarchy was developed in [4]. This new hierarchy will find applications in minimizing Galois field Sum-Of-Product (GFSOP) expressions (i.e., expressions that are in the sum-of-product form which uses the additions and multiplications of arbitrary radix Galois field that was introduced in Chapt. 2), creation of new forms, decision diagrams, and regular structures (Such new structures will be discussed in details in Appendix D.)

The state-of-the-art minimizers of Exclusive Sum-Of-Product (ESOP) expressions $[80,85,114,157,214,234,235,242]$ (i.e., expressions that are in the sum-of-product form which uses the addition and multiplication of Galois field of radix two that was introduced in Figs. 2.1a and 2.1b, respectively) are based on heuristics and give the exact solution only for functions with a small number of variables. The formulation for finding the exact ESOP was given in [52], but all known exact algorithms can deliver solutions for not all but only certain functions of more than five variables. Because GFSOP minimization is even more difficult, it is

important to investigate structural properties and the counts of their canonical subfamilies.

Recently, two families of binary canonical Reed-Muller forms, called Inclusive Forms (IFs) and Generalized Inclusive Forms (GIFs) have been proposed [52]. The second family was the first to include all minimum ESOPs (binary GFSOPs). In this Chapt., we propose, as analogous to the binary case, two general families of canonical ternary Reed-Muller forms, called Ternary Inclusive Forms (TIFs), and their generalization, Ternary Generalized Inclusive Forms (TGIFs). The second family includes minimum GFSOPs over ternary Galois field GF(3). One of the basic motivations in this work is the application of these TIFs and TGIFs to find the minimum GFSOP for multiple-valued inputs multiplevalued outputs for reversible logic synthesis using, for instance, reversible cascades in Chapt. 8 , a problem that has not yet been solved.

## 计算机代写|量子计算代写Quantum computing代考|Green/Sasao Hierarchy of Binary Canonical Forms

The Green/Sasao hierarchy of families of canonical forms and corresponding decision diagrams is based on three generic expansions, Shannon, positive Davio, and negative Davio expansions. This includes [217]: Shannon Decision Trees and Diagrams, Positive Davio Decision Trees and Diagrams, Negative Davio Decision Trees and Diagarms, Fixed Polarity Reed-Muller Decision Trees and Diagrams, Kronecker Decision Trees and Diagrams, Pseudo Reed-Muller Decision Trees and Diagrams, pseudo Kronecker Decision Trees and Diagrams, and LinearlyIndependent Decision Trees and Diagrams. A set-theoretic relationship between families of canonical forms over $\mathrm{GF}(2)$ was proposed and extended in [52] by introducing binary IF, GIF, and FGIF forms. Figure $3.1$ illustrates the set-theoretic relationship between families of canonical forms over GF(2).

Analogously to the Green/Sasao hierarchy of binary ReedMuller families of spectral transforms over GF(2) that is shown in Fig. 3.1, we will introduce the extended Green/Sasao hierarchy of spectral transforms, with a new sub-family, for ternary Reed-Muller logic over GF(3) in Sect. 3.5.

## 计算机代写|量子计算代写Quantum computing代考|Binary S/D Trees and their Inclusive Forms

Two general families of DDs were introduced in [52]. These families are based on the Shannon expansion and the Generalized Davio expansion, and are produced using the S/D Trees. These families are called the Inclusive Forms (IFs) and the Generalized Inclusive Forms (GIFs), respectively. It was proven [52] that these forms include a minimum ESOP. The expansions over $\mathrm{GF}(2)$ are shown in Fig. 3.2, where Fig. 3.2d shows the new expansion, which is based on binary Davio expansions, called generalized Davio (D) expansion that generates the negative and positive Davio expansions as special cases.

The S/D trees for IFs of two variables of order ${a, b}$, and the S/D trees for IFs of two variables of order ${b, a}$ were fully illustrated [52]. The set of Generalized Inclusive Forms (GIFs) for two variables is the union of the two sets of Inclusive Forms (IFs). The total number of the GIFs is equal to:
$$# G I F=2 \cdot\left(# I F_{a, b}\right)-#\left(I F_{a, b} \cap I F_{b, a}\right) .$$
Thus for two variables:
\begin{aligned} &# I F_{a, b}=1+2+2+4+4+8+8+16=45, \ &# I F_{b, a}=1+2+2+4+4+8+8+16=45, \ &# G I F=2 \cdot(45)-(1+4+4+16)=65 . \end{aligned}
Properties and experimental results of the binary Inclusive Forms and the binary Generalized Inclusive Forms were investigated [52], where it was proven that GIFs include a minimum ESOP.

## 计算机代写|量子计算代写Quantum computing代考|Binary S/D Trees and their Inclusive Forms

[52] 中介绍了两个一般的 DD 家族。这些族基于香农扩展和广义戴维奥扩展，并使用 S/D 树生成。这些族分别称为包容形式 (IF) 和广义包容形式 (GIF)。已证明 [52] 这些表格包括最低 ESOP。展开超过GF(2)如图 3.2 所示，其中图 3.2d 显示了新的扩展，它基于二进制 Davio 扩展，称为广义 Davio (D) 扩展，它生成负和正 Davio 扩展作为特殊情况。

# G I F=2 \cdot\left(# I F_{a, b}\right)-#\left(I F_{a, b} \cap I F_{b, a}\right) 。# G I F=2 \cdot\left(# I F_{a, b}\right)-#\left(I F_{a, b} \cap I F_{b, a}\right) 。

\begin{对齐} I F_{a, b}=1+2+2+4+4+8+8+16=45, \ I F_{b, a}=1+2+2+ 4+4+8+8+16=45, \ G I F=2 \cdot(45)-(1+4+4+16)=65 。\end{对齐}\begin{对齐} I F_{a, b}=1+2+2+4+4+8+8+16=45, \ I F_{b, a}=1+2+2+ 4+4+8+8+16=45, \ G I F=2 \cdot(45)-(1+4+4+16)=65 。\end{对齐}

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## MATLAB代写

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