### 计算机代写|量子计算代写Quantum computing代考|Properties of TIFs and TGIFs

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Properties of TIFs

In this Sect., we prove that all TIFs for the given variable ordering are canonical and unique.

Theorem 3.2. Each TIF $\left{t_{i}\right}, 1 \leq i \leq n$, is canonical, i.e., for any function $\mathrm{F}$ of the same number of variables, there exists one and only one set of coefficients $\left{a_{i}\right}$, such that $F=a_{1} t_{1}+{ }{G F(3)} \ldots++{G F(3)} a_{n} t_{n}$.
Proof. In [52] (and references therein), it was shown that an expansion is canonical iff its terms are linearly independent, that is, none of the terms is equal to a linear combination of other terms (over the algebraic field used). Using this fact, it was proven that IFs over GF(2) are canonical. Using an approach which is analogous to the approach presented in [52], one can therefore prove, by induction on the number of variables, that terms in TIFs over ternary Galois field are linearly independent and thus canonical.

Q.E.D.

## 计算机代写|量子计算代写Quantum computing代考|Properties of TGIFs

It is easy to see that, for different variable orderings, some forms are not repeated while other forms are. For example, Kronecker forms

and GRMs over GF(3) are repeated. Therefore the union of sets of TIFs for all variable orders contains more forms than any of the TIF sets taken separately and less forms than the total sum of all of these TIFs.

Theorem 3.3. Ternary Generalized Inclusive Forms (TGIFs) are canonical with respect to the given variable order.

Proof. The proof is analogous to the one in Theorem 3.2. Q.E.D.
Generalized Inclusive Forms include GRMs and PKROs over GF(3) as can be shown by considering all possible combinations of literals for all possible orders of variables. If we relax the requirement of fixed variable ordering, and allow any ordering of variables in the branches of the tree but do not allow repetitions of variables in the branches, we generate more general family of forms over GF(3).

Definition 3.4. The family of forms, generated by the S/D tree with no fixed ordering of variables, provided that variables are not repeated along the same branches, is called Ternary Free Generalized Inclusive Forms (TFGIFs).

The studies show that it is difficult to trace the relationship between the number of forms that are repeated for $\mathrm{N}>2$ and the number of forms that are not.

## 计算机代写|量子计算代写Quantum computing代考|An Extended Green/Sasao Hierarchy

Here we introduce the extended Green/Sasao hierarchy with a new sub-family for ternary Reed-Muller logic over GF(3). Definitions $3.2$, 3.3, and $3.4$ defined the Ternary Inclusive Forms (TIFs), Ternary Generalized Inclusive Forms (TGIFs), and Ternary Free Generalized Inclusive Forms (TFGIFs), respectively. Analogously to the binary Reed-Muller case, we introduce the following definitions over GF(3).

Definition 3.5. The Decision Tree (DT) that results from applying the Ternary Shannon Expansion (Eq. (2.23)) recursively to a ternary input-ternary output logic function (i.e., all levels in a DT) is called Ternary Shannon Decision Tree (TSDT). The result expression (flattened form) from the TSDT is called Ternary Shannon Expression, which is a canonical expression.

Definition 3.6. The Decision Trees (DTs) that result from applying the Ternary Davio expansions (Eqs. (2.24), (2.25), and (2.26)) recursively to a ternary-input ternary -output logic function (i.e., all levels in a DT) are called: Ternary Zero-Polarity Davio Decision Tree $\left(\right.$ TD $_{0}$ DT), Ternary First-Polarity Davio Decision Tree (TD ${ }{1}$ DT), and Ternary Second-Polarity Davio Decision Tree (TD ${ }{2} \mathrm{DT}$ ), respectively. The resulting expressions (flattened forms) from $\mathrm{TD}{0} \mathrm{DT}, \mathrm{TD}{1} \mathrm{DT}$, and $\mathrm{TD}{2} \mathrm{DT}$ are called: $\mathrm{TD}{0}, \mathrm{TD}{1}$, and $\mathrm{TD}{2}$ expressions, respectively. These expressions are canonical.

Definition 3.7. The Decision Tree (DT) that results from applying any of the Ternary Davio expansions (nodes) for all nodes in each level (variable) in the DT is called Ternary Reed-Muller Decision Tree (TRMDT). The corresponding expression is called Ternary Fixed Polarity Reed-Muller (TFPRM) Expression. This expression is canonical for a given set of polarities.

Definition 3.8. The Decision Tree (DT) that results from using any of the Ternary Shannon $(S)$ or Davio $\left(D_{0}, D_{1}\right.$, or $D_{2}$ ) expansions (Nodes) for all nodes in each level (variable) in the DT (that has fixed order of variables), is called Ternary Kronecker Decision Tree (TKRODT). The resulting expression is called Ternary Kronecker Expression. This expression is canonical.

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## MATLAB代写

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