### 数学代写|计算复杂度理论代写Computational complexity theory代考|COMP4500

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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

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Algebraic Criterion

In this section, we provide some tools to prove the elusiveness of a Boolean function. We begin with a simple one.

Theorem 5.10 A Boolean function with an odd number of truth assignments is elusive.

Proof. The constant functions $f \equiv 0$ and $f \equiv 1$ have 0 and $2^{n}$ truth assignments, respectively. Hence, a Boolean function with an odd number of truth assignments must be a nonconstant function. If $f$ has at least two variables and $x_{i}$ is one of them, then the number of truth assignments of $f$ is the sum of those of $\left.f\right|{x{i}=0}$ and $\left.f\right|{x{i}=1}$. Therefore, either $\left.f\right|{x{i}=0}$ or $\left.f\right|{x{i}=1}$ has an odd number of truth assignments and is not a constant function. Thus, for any decision tree computing $f$, tracing the subtrees with an odd number of truth assignments, we will meet all variables in a path from the root to a leaf.
Define
$$p_{f}(k)=\sum_{t \in{0,1}^{n}} f(t) k^{| t t},$$
where $|t|$ is the number of l’s in string $t$ (recall that a Boolean assignment may be viewed as a binary string). It is easy to see that $p_{f}(1)$ is the number of truth assignments for $f$. The following theorem is an extension of Theorem $5.10$.

Theorem 5.11 For a Boolean function $f$ of $n$ variables, $(k+1)^{n-D(f)} \mid p_{f}(k)$.
Proof. We prove this theorem by induction on $D(f)$. First, we note that if $f \equiv 0$, then $p_{f}(k)=0$ and if $f \equiv 1$ then $p_{f}(k)=(k+1)^{n}$ (by the binomial theorem). This means that the theorem holds for $D(f)=0$. Now, consider $f$ with $D(f)>0$ and a decision tree $T$ of depth $D(f)$ computing $f$. Without loss of generality, assume that the root of $T$ is labeled by $x_{1}$. Denote $f_{0}=\left.f\right|{x{1}=0}$ and $f_{1}=\left.f\right|{x{1}=1}$. Then
\begin{aligned} p_{f}(k) &=\sum_{t \in{0,1}^{n}} f(t) k^{|t|} \ &=\sum_{s \in{0,1}^{n-1}} f(0 s) k^{|s|}+\sum_{s \in{0,1}^{n-1}} f(1 s) k^{1+|s|} \ &=p_{f_{0}}(k)+k p_{f_{1}}(k) . \end{aligned}
Note that $D\left(f_{0}\right) \leq D(f)-1$ and $D\left(f_{1}\right) \leq D(f)-1$. By the induction hypothesis, $(k+1)^{n-1-D\left(f_{0}\right)} \mid p_{f_{0}}(k)$ and $(k+1)^{n-1-D\left(f_{1}\right)} \mid p_{f_{1}}(k)$. Thus, $(k+1)^{n-D(f)} \mid p_{f}(k)$
An important corollary is as follows. Denote $\mu(f)=p_{f}(-1)$.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Monotone Graph Properties

In this section, we prove a general lower bound for the decision tree complexity of nontrivial monotone graph properties.

First, let us analyze how to use Theorem $5.13$ to study nontrivial monotone graph properties. Note that every graph property is weakly symmetric and for any nontrivial monotone graph property, the complete graph must have the property and the empty graph must not have the property. Therefore, if we want to use Theorem $5.13$ to prove the elusiveness of a nontrivial monotone graph property, we need to verify only one condition that the number of variables is a prime power. For a graph property, however, the number of variables is the number of possible edges, which equals $n(n-1) / 2$ for $n$ vertices and it is not a prime power for $n>3$. Thus, Theorem $5.13$ cannot be used directly to show elusiveness of graph properties. However, it can be used to establish a little weaker results by finding a partial assignment such that the number of remaining variables becomes a prime power. The following lemmas are derived from this idea.

Lemma $5.16$ If $\mathcal{P}$ is a nontrivial monotone property of graphs of order $n=2^{m}$, then $D(\mathcal{P}) \geq n^{2} / 4$.

Proof. Let $H_{i}$ be the disjoint union of $2^{m-i}$ copies of the complete graph of order $2^{i}$. Then $H_{0} \subset H_{1} \subset \cdots \subset H_{m}=K_{n}$. Since $\mathcal{P}$ is nontrivial and is monotone, $H_{m}$ has the property $\mathcal{P}$ and $H_{0}$ does not have the property $\mathcal{P}$. Thus, there exists an index $j$ such that $H_{j+1}$ has the property $\mathcal{P}$ and $H_{j}$ does not have the property $\mathcal{P}$. Partition $H_{j}$ into two parts with vertex sets $A$ and $B$, respectively, each containing exactly $2^{m-j-1}$ disjoint copies of the complete graph of order $2^{j}$. Let $K_{A, B}$ be the complete bipartite graph between $A$ and $B$. Then $H_{j+1}$ is a subgraph of $H_{j} \cup K_{A, B}$. So, $H_{j} \cup K_{A, B}$ has the property $\mathcal{P}$. Now, let $f$ be the function on bipartite graphs between $A$ and $B$ such that $f$ has the value 1 at a bipartite graph $G$ between $A$ and $B$ if and only if $H_{j} \cup G$ has the property $\mathcal{P}$. Then $f$ is a nontrivial monotone weakly symmetric function with $|A| \cdot|B|\left(=2^{2 m-2}\right)$ variables. By Theorem $5.13, D(\mathcal{P}) \geq D(f)=2^{2 m-2}=n^{2} / 4$.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Topological Criterion

In this section, we introduce a powerful tool to study the elusiveness of monotone Boolean functions. We start with some concepts in topology.
A triangle is a two-dimensional polygon with the minimum number of vertices. A tetrahedron is a three-dimensional polytope with the minimum number of vertices. They are the simplest polytopes with respect to the specific dimensions. They are both called simplexes. The concept of simplexes is a generalization of the notions of triangles and tetrahedrons. In general, a simplex is a polytope with the minimum number of vertices among all polytopes with certain dimension. For example, a point is a zero-dimensional simplex and a straight line segment is a one-dimensional simplex. The convex hull of linearly independent $n+1$ points in a Euclidean space is an $n$-dimensional simplex.

A face of a simplex $S$ is a simplex whose vertex set is a subset of vertices of $S$. A geometric simplicial complex is a family $\Gamma$ of simplexes satisfying the following conditions:
(a) For $S \in \Gamma$, every face of $S$ is in $\Gamma$.
(b) For $S, S^{\prime} \in \Gamma, S \cap S^{\prime}$ is a face for both $S$ and $S^{\prime}$.
(c) For $S, S^{\prime} \in \Gamma, S \cap S^{\prime}$ is also a simplex in $\Gamma$.
In Figure $5.5$, there are three examples; first two are not geometric simplicial complexes, the last one is.

Consider a set $X$ and a family $\Delta$ of subsets of $X . \Delta$ is called an (abstract) simplicial complex if for any $A$ in $\Delta$ every subset of $A$ also belongs to $\Delta$. Each member of $\Delta$ is called a face of $\Delta$. The dimension of a face $A$ is $|A|-1$. Any face of dimension 0 is called a vertex. For example, consider a set $X={a, b, c, d}$. The following family is a simplicial complex on $X$ :
\begin{aligned} \Delta=&{\emptyset,{a},{b},{c},{d},{a, b},{b, c},\ &{c, d},{d, a},{a, c},{a, b, c},{a, c, d}} \end{aligned}
The set ${a, b, c}$ is a face of dimension 2 and the empty set $\emptyset$ is a face of dimension $-1$.

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Algebraic Criterion

pF(ķ)=∑吨∈0,1nF(吨)ķ|吨吨,

## 数学代写|计算复杂度理论代写Computational complexity theory代考|Topological Criterion

(a) 对于小号∈Γ, 每一张脸小号在Γ.
(b) 为小号,小号′∈Γ,小号∩小号′是一张脸小号和小号′.
(c) 为小号,小号′∈Γ,小号∩小号′也是一个单纯形Γ.

\begin{aligned} \Delta=&{\emptyset,{a},{b},{c},{d},{a, b},{b, c},\ &{c, d},{ d, a},{a, c},{a, b, c},{a, c, d}} \end{对齐}\begin{aligned} \Delta=&{\emptyset,{a},{b},{c},{d},{a, b},{b, c},\ &{c, d},{ d, a},{a, c},{a, b, c},{a, c, d}} \end{对齐}

## 有限元方法代写

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## MATLAB代写

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