### 数学代写|理论计算机代写theoretical computer science代考| Probabilistic Model Checking

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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|理论计算机代写theoretical computer science代考|Probabilistic Model Checking

The aim of probabilistic model checking is to verify a desired quantitative or qualitative property of the system. A main class of PCTL properties includes reachability probabilities and expected rewards. For DTMCs, a reachability probability is the probability of reaching the set of goal states $G$ and for MDPs, it is the extremal probability of reaching $G$. Reward-based properties are defined as the expected accumulated reward (extremal expected reward in the case of MDPs) until reaching a goal state [1]. To formally reason about reachability probabilities, we need to define a probability measure on the set of paths that reach to some states in $G$. For a state $s_{0} \in S$, let reach ${ }{s{0}}(G)$ be the set of all paths that start from $s_{0}$ and have a state from $G$ :
$$\text { reach }{s{0}}(G) \stackrel{\text { def }}{=}\left{\pi \in P a t h s_{D_{1} s_{0}} \mid \pi[i] \in G \text { for some } i \geq 0\right} .$$
The probability measure on the set $\operatorname{reach}{s{0}}(G)$ is defined as [1]:
$$\operatorname{Pr}^{D}\left(\operatorname{reach}{s{0}}(G)\right)=\sum_{s_{0} s_{1} \ldots s_{n} \in \text { reach } x_{0}}(G) \operatorname{Pr}^{D}\left(\operatorname{Cyl}\left(s_{0} s_{1} \ldots s_{n}\right)\right)$$
For MDPs, reachability probabilities are defined as the extremal probabilities of reaching goal states $[13,16]$ :
$$P r_{\max }^{M}\left(\text { reach }{s{0}}(G)\right) \stackrel{\text { def }}{=} \sup {\sigma \in \operatorname{Pol}{M}} \operatorname{Pr}{\sigma}^{M}\left(\operatorname{reach}{s_{0}}(G)\right)$$
Reachability probability properties are divided into qualitative and quantitative properties. A qualitative property of a probabilistic system means the probability of reaching the set of goal states is either 0 or 1 [1]. Qualitative verification is a method to find the set of states for which this reachability probability is exactly 0 or 1 . We denote these sets by $S^{0}$ and $S^{1}$, respectively. For the case of MDPs, we are interested to find those states for which the maximum or minimum reachability probability is 0 or 1 . These sets are denoted by $S_{\text {min }}^{0}$, $S_{\max }^{0}, S_{\min }^{1}$ and $S_{\max }^{1}$ and can be computed by graph-based methods [2].

## 数学代写|理论计算机代写theoretical computer science代考|Quantitative Properties

In probabilistic model checking, we consider a $\mathrm{PCTL}$ property to be quantitative if the probability of reaching goal states is not exactly 0 or 1 . Verification of quantitative properties usually reduces to solving a linear time equation system (for DTMCs) or solving a Bellman equation for MDPs [3,5]. For an arbitrary state $s \in S$ in a DTMC $D$, let $x_{s}$ be the probability of reaching $G$ from $s$, i.e., $x_{s}=P r^{D}($ reach $s(G))$. The values of $x_{s}$ for all $s \in S$ are obtained as the unique solution of the linear equation system $[1,2]$ :
$$x_{s}=\left{\begin{array}{lll} 0 & \text { if } & s \in S^{0} \ 1 & \text { if } & s \in S^{1} \ \sum_{s^{\prime} \in S} \mathbf{P}\left(s, s^{\prime}\right) \cdot x_{s^{\prime}} & \text { if } & s \in S^{?} \end{array}\right.$$
where $S^{?}=S \backslash\left(S^{0} \cup S^{1}\right)$. A model checker can use any direct method (e.g. Gaussian elimination) or iterative method (e.g. Jacobi, Gauss-Seidel) to compute the solution of this linear equation system.

For the case of MDPs, we consider $x_{s}$ as the maximum (or minimum) probability of reaching $G$, i.e., $x_{s}=P r_{\max }^{M}\left(\right.$ reach $\left.h_{s}(G)\right)$. In this case, the values of $x_{s}$ are obtained as the solution of the Bellman equation system:
where $S_{\max }^{?}=S \backslash\left(S_{\max }^{0} \cup S_{\max }^{1}\right)$. Using $x_{s}$ for the maximal expected accumulated reward, we have $[1,2]$ :
$$x_{s}= \begin{cases}0 & \text { if } s \in G \ \infty & \text { if } s \notin S_{\min }^{1} \ \max {\alpha \in A c t(s)} \sum{s^{\prime} \in S}\left(R\left(s, \alpha, s^{\prime}\right)+\delta(s, \alpha)\left(s^{\prime}\right) \cdot x_{s^{\prime}}\right) & \text { otherwise }\end{cases}$$
Some standard direct algorithms (like Simplex algorithm [11]) are able to compute the precise values for the Bellman equation. The main drawback of direct methods is their scalability that limits them to relatively small models [2]. Iterative methods are other alternatives to approximate the values of $x_{s^{*}}$.

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

Value iteration (VI) and policy iteration (PI) are two standard iterative methods that are used to compute the quantitative properties in probabilistic systems. VI and its Gauss-Seidel extension are widely studied in the previous works $[1,2$, $13,15,16]$. We review PI, which is used in the remaining of this paper.

Policy Iteration. This method iterates over policies in order to find the optimal policy that maximizes reachability probabilities of all states. Starting from an arbitrary policy, it improves policies until reaching no change in them [2]. For each policy, the method uses an iterative method to compute the reachability probability values of quotient DTMCs and updates the value of states of the original MDP. The termination of this method is guaranteed for every finite MDP [2]. Modified policy iteration (MPI) [5] performs a limited number of iterations for each quotient DTMC (100 iterations for example) and updates the policy after doing this number of iterations. Algorithm 1 shows the standard policy iteration to compute $\operatorname{Pr}{\max }^{M}\left(\right.$ reach $\left.{s}(G)\right)$. We consider a policy as a mapping $\sigma$ from states to actions. In lines $7-9$, the algorithm uses a greedy approach to update the policy $\sigma$. More details about this algorithm are available in [2].

## 数学代写|理论计算机代写theoretical computer science代考|Quantitative Properties

$$x_{s}=\左{0 如果 s∈小号0 1 如果 s∈小号1 ∑s′∈小号磷(s,s′)⋅Xs′ 如果 s∈小号?\对。$$

$$x_{s}= \begin{cases}0 & \text { if } s \in G \ \infty & \text { if } s \notin S_{\min }^{1} \ \max {\ alpha \in A ct(s)} \sum {s^{\prime} \in S}\left(R\left(s, \alpha, s^{\prime}\right)+\delta(s, \ alpha)\left(s^{\prime}\right) \cdot x_{s^{\prime}}\right) & \text { else }\end{cases}$$

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## MATLAB代写

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