### 数学代写|理论计算机代写theoretical computer science代考| Improving Iterative Methods for Quantitative

statistics-lab™ 为您的留学生涯保驾护航 在代写理论计算机theoretical computer science方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写理论计算机theoretical computer science代写方面经验极为丰富，各种代写理论计算机theoretical computer science相关的作业也就用不着说。

• 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|理论计算机代写theoretical computer science代考|Reachability Probabilities

We use a heuristic to improve the performance of iterative methods for DTMCs. It considers transitions with Dirac distributions. For this class of transitions, the reachability probability values of the source and destination states of the transition are the same. The idea of our heuristic is to avoid useless updates in order to reduce the total number of updates before termination. We use it to improve the performance of policy iteration method for computing reachability probabilities in MDPs. In this case, Dirac transitions are considered to reduce the number of updates in the computations of each quotient DTMCs. We apply this technique on the SCC-based method to improve the performance of this approach. Although the idea of considering Dirac transitions has been used as a reduction technique in [8], it has been only proposed for the case of deterministic Dirac transitions. The main contribution of our work is to use this heuristic for improving the MPI method for both reachability probabilities and expected rewards to cover nondeterministic action selections. To the bet of our knowledge, none of state of the art model checkers support this technique.

## 数学代写|理论计算机代写theoretical computer science代考|Using Transitions with Dirac Probability

In our work, we consider transitions with Dirac distribution (which we call Dirac transitions) to avoid redundant updates of state values. This type of transitions is also used in statistical model checking [6] but we use it to accelerate iterative methods. For a DTMC D, a Dirac transition is a pair of states $s$ and $s^{\prime}$ with $\mathbf{P}\left(s, s^{\prime}\right)=1$. Based on the definition, for this pair of states, we have $\operatorname{Pr}^{D}\left(\right.$ reach $\left.{s}(G)\right)=\operatorname{Pr}^{D}\left(\right.$ reach $\left.{s^{\prime}}(G)\right)$. As a result, we can ignore this transition and postpone the update of the value of $s$ until the convergence of the value of $s^{\prime}$. Our approach updates the value of $s$ only once and avoids redundant updates. However, there may be some incoming transitions to $s$ that need the value of $s$ in every iteration. Consider Fig. 1 and suppose $s$ is the target state of a transition of the form $(t, s)$. In this case the value of $s$ affects the value of $t$ and we should modify the incoming transitions to $s$ to point to $s^{\prime}$. Algorithm 2 shows details of our approach to remove Dirac transitions from a DTMC $D$ and reduce it to smaller DTMC $D^{\prime}$. The main idea of this algorithm is to partition the state space of $D$ to Dirac-based classes. A Dirac-based class is the set of states that are connected by Dirac transitions. The algorithm uses the $D B C$ array to partition $S^{?}$ to the related classes. Consider for example a sequence of states of the form $s_{i}, s_{j}, \ldots, s_{k-1}, s_{k}$ where there exists Dirac transitions between each two consecutive states and the outgoing transition of $s_{k}$ is not a Dirac one or $s_{k}$ is one of $s_{i}, \ldots, s_{k-1}$. We put these states into one class and set their $D B C$ to $s_{k}$. The reachability probability of all states of this class are equal to the reachability probability of $s_{k}$. Formally, for each state $s_{i} \in S$, we define $D B C\left[s_{i}\right]$ as the index of the last state in the sequence of states that are connected with Dirac transitions. For every state $s \in S$ we have $x_{s}=x_{D B C[s]}$. Algorithm 2 initiates the values of this array in lines 3-5. For every state $s_{i} \in S$ that $D B C\left[s_{i}\right]$ is not determined previously, the algorithm calls the Find_Dirac_Index function to determine its $D B C$. After determining the $D B C$ value of all states of $S$, the algorithm creates the reduced DTMC $D^{\prime}$ (lines $9-13$ ). The states of $D^{\prime}$ are those states $s_{i} \in S$ for which $D B C\left[s_{i}\right]=i$. According to the definition of $S^{1}$, for each state $s \in G$ and each $s^{\prime} \in S$ if $D B C\left[s^{\prime}\right]=D B C[s]$ then we have $s^{\prime} \in G$.

## 数学代写|理论计算机代写theoretical computer science代考|Improving Iterative Methods for Computing Reachability

The Dirac-based reduction technique can be also used for MDPs to avoid redundant updates. However, it can not be directly applied for states with multiple enabled actions. To cover this case and as a novelty of our approach, we use the MPI method. We apply our heuristic for every quotient DTMC to reduce the number of states that should be updated. We propose our it as an extension of SCC-based techniques. We use MPI to compute reachability values of the states of each SCC and apply our Dirac-based heuristic to accelerate the iterative computations of each quotient DTMC. Algorithm 5 presents the overall idea of our method for accelerating SCC-based methods for MDPs. The correctness of this approach relies on the correctness of SCC decomposition for MDPs [13] and the correctness of our Dirac-based DTMC reduction method (Lemma 1 ).

The standard iterative methods can be used to approximate the Bellman equation for extremal expected rewards. In this case, the initial vector of values is set to zero for all states. Value iteration (or policy iteration) should also consider the defined reward of each action in the update of values of each state. More details about iterative methods for expected rewards are available in $[1,2,16]$. To use our heuristic for expected rewards, we use the fact that for every Dirac transition of the form $\left(s_{i}, s_{j}\right)$ we have $x_{s_{i}}=x_{s_{j}}+R\left(x_{i}\right)$ where $x_{i}$ and $x_{j}$ are the expected values for these two states and $R\left(x_{i}\right)$ is the reward for $x_{i}$. In this case, an iterative method does not need to update the value of $x_{i}$ in every iteration. Similar to the case for the reachability probabilities, the incoming transitions to $x_{i}$ should be modified to point to $x_{j}$. In addition the reward of a modified transition is modified by adding the reward of the related Dirac transition.

## 有限元方法代写

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## MATLAB代写

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