### 数学代写|理论计算机代写theoretical computer science代考|Dirac-Based Reduction Techniques for Quantitative Analysis of Discrete-Time Markov Models

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• Statistical Inference 统计推断
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• 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|Introduction

Model checking is a formal approach for verifying quantitative and qualitative properties of computer systems. In this way, the system is modelled by a labelled transition system, and its properties are specified in temporal logic. Because of some stochastic behaviours of these systems, we can use probabilistic model checking to analyse the quantitative property specifications of these systems [1-3]. In this domain, we can use discrete and continuous time Markov Chains to model fully probabilistic systems. Besides, Markov Decision Processes [5] are used to model systems with both probabilistic and non-deterministic behaviours. Probabilistic Computation Tree Logic (PCTL) [1] is used to formally specify the

related system properties. A main part of PCTL properties against MDPs can be verified by computing the extremal reachability probability: The maximal or minimal probability of reaching a given set of goal states. For quantitative parts, numerical computations are needed to calculate these reachability probabilities $[2,6]$. Linear programming $[1,3]$, value iteration and policy iteration are wellknown numerical approaches for computing the optimal reachability probabilities $[2,5]$. PRISM [4] and STORM [7] are examples of probabilistic model checkers that use these numerical methods to compute reachability probabilities.

One of the main challenges of model checking in all variants is the state space explosion problem, i.e., the excessive space requirement to store the states of the model in the memory $[1,9]$. For probabilistic model checking, we have the additional difficulty of solving linear programs. For the feasibility of the algorithms we need efficient heuristics to decrease the running time of these algorithms [3]. A wide range of approaches has been proposed for probabilistic model checking in previous works to tackle these problems. Symbolic model checking [9], compositional verification [10], symmetry reduction for probabilistic systems $[11]$, incremental model construction [12] and statistical model checking [6] have been proposed to reduce the needed space for probabilistic model checking. In addition, several approaches are used to accelerate standard algorithms for probabilistic model checking. SCC-based approaches $[13,14]$ identify strongly connected components (SCCs) of the underlying model and compute reachability probabilities of the states of each component in a right order. Learning based algorithms use the idea of real-time dynamic programming to solve reachability probability problems of MDPs [15]. Prioritization methods focus on finding a good state ordering to update the values of states during iterative computations $[13,14]$. The idea of finding Maximal End Components (MECs) is used in [3] to reduce the number of states of the model. Several techniques are proposed in [17] to reduce the size of a DTMC model. These techniques are used to reduce the model for finite-horizon properties.

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

In this section, we provide an overview of DTMCs and MDPs and reachability properties. We mainly follow the notations of $[1,13]$. Let $S$ be a countable set. A discrete probability distribution on $S$ is a function $P: S \rightarrow[0,1]$ satisfying $\sum_{s \in S} P(s)=1$. We use $\operatorname{Dist}(S)$ as the set of all distributions on $S$. The support of $P$ is defined as the set $S u p p(P)={s \in S \mid P(s)>0}$. A distribution $P$ is Dirac if $S u p p(P)$ has only one member. More details about the proposed definitions in this section and their related probability measures are available in $[1,2,16]$.

Definition 1. A Discrete-time Markov Chain (DTMC) is a tuple $D=(S, \hat{s}$, $\mathbf{P}, R, G$ ) where $S$ is a countable, non-empty set of states, $\hat{s} \in S$ is the initial state, $\mathbf{P}: S \times S \rightarrow[0,1]$ is the probabilistic transition function, $R: S \times S \rightarrow \Re_{\geq 0}$ is a reward function wich assigns to each transition of $P$ a non-negative reward value and $G \subseteq S$ is the set of Goal states.

A DTMC $D$ is called finite if $S$ is finite. For a finite $D$, size $(D)$ is the number of states of $D$ plus the number of transitions of the form $\left(s, s^{\prime}\right) \in S \times S$ with $\mathbf{P}$ $\left(s, s^{\prime}\right)>0$. A path represents a possible execution of $D[2]$ and is a non-empty (finite or infinite) sequence of states $s_{0} s_{1} s_{2} \ldots$ such that $\mathbf{P}\left(s_{i}, s_{i+1}\right)>0$ for all $i \geq 0$. We use $P_{a t h s_{D, s}}$ to denote the set of all paths of $D$ that start in the state $s$ and we use FPaths ${ }{D, s}$ for the subset of finite paths of $P{a t h} s_{D, s}$. We also use Path $s_{D}$ and FPaths $_{D}$ for $\cup_{s \in S} P a t h s_{D, s}$ and $\cup_{s \in S} F P a t h s_{D, s}$ respectively. For a finite path $\pi=s_{0} s_{1} \ldots s_{k}$, the accumulated reward is defined as: $\sum_{i<k} R\left(s_{i}, s_{i+1}\right)$. For an infinite path $\pi=s_{0} s_{1} \ldots$ and for every $j \geq 0$, let $\pi[j]=s_{j}$ denote the $(j+1)$ th state of $\pi$ and $\pi[. . j]$ the $(j+1)$ th prefix of the form $s_{0} s_{1} \ldots s_{j}$ of $\pi$. We use pref $(\pi)$ as the set of all prefixes of $\pi$.

## 数学代写|理论计算机代写theoretical computer science代考|Probability Measure of a Markov Chain

In order to reason about the behaviour of a Markov chain $D$, we need to formally use the cylinder sets of the finite paths of $D[1]$.

Definition 2. The Cylinder set of a finite path $\hat{\pi} \in F P a$ h.s $_{D}$ is defined as $\operatorname{Cyl}(\hat{\pi})=\left{\pi \in\right.$ Paths $\left.{D} \mid \hat{\pi} \in \operatorname{pre} f(\pi)\right}$. The probability measure $\operatorname{Pr}^{D}$ is defined on the cylinder sets as $P r^{D}\left(C y l\left(s{0} \ldots s_{n}\right)\right)=\prod_{0 \leq i<n} \mathbf{P}\left(s_{i}, s_{i+1}\right)$.

Markov Decision Processes (MDPs) are a generalization of DTMCs that are used to model systems that have a combination of probabilistic and non-deterministic behaviour. An MDP is a tuple $M=(S, \hat{s}$, Act, $\delta, R, G)$ where $S, \hat{s}$ and $G$ are the same as for DTMCs, Act is a finite set of actions, $R: S \times A c t \times S \rightarrow \Re_{\geq 0}$ is a reward function, assigns to each transition a non-negative reward value and $\delta: S \times A c t \rightarrow \operatorname{Dist}(S)$ is a probabilistic transition function. For every state $s \in S$ of an MDP $M$ one or more actions of Act are defined as enabled actions. We use $A c t(s)$ for this set and define it as $\operatorname{Act}(s)={\alpha \in A c t \mid \delta(s, \alpha)$ is defined $}$.
For $s \in S$ and $\alpha \in \operatorname{Act}(s)$ we use Post $(s, \alpha)$ for the set of $\alpha$ successors of $s$, Post $(s)$ for all successors of $s$ and Pre $(s)$ for predecessors of $s[1]$ :
\begin{aligned} &\operatorname{Post}(s, \alpha) \doteq\left{s^{\prime} \in S \mid \delta\left(s, \alpha, s^{\prime}\right)>0\right} \ &\operatorname{Post}(s) \doteq \cup_{\alpha \in \operatorname{Act}(s)} \operatorname{Post}(s, \alpha), \end{aligned}
To evaluate the operational behaviour of an MDP $M$ we should consider two steps to take a transition from a state $s \in S$. First, one enabled action $\alpha \in \operatorname{Act}(s)$ is chosen non-deterministically. Second, according to the probability distribution $\delta(s, \alpha)$, a successor state $s^{\prime} \in \operatorname{Post}(s, \alpha)$ is selected randomly. In this case, $\delta(s, \alpha)\left(s^{\prime}\right)$ determines the probability of a transition from $s$ to $s^{\prime}$ by the action $\alpha \in \operatorname{Act}(s)$. We extend the definition of paths for MDPs: A path in an MDP $M$ is a non-empty (finite or infinite) sequence $\pi=s_{0} \stackrel{\alpha_{0}}{\rightarrow} s_{1} \stackrel{\alpha_{1}}{\rightarrow} \ldots$ where $s_{i} \in S$ and $\alpha_{i} \in \operatorname{Act}\left(s_{i}\right)$ and $s_{i+1} \in \operatorname{Post}\left(s_{i}, \alpha_{i}\right)$ for every $i \geq 0$. Similar to the case with DTMCs, for a state $s \in S$, we use Paths $_{M, s}$ to denote the set of all paths of $M$ starting in $s$ and $F$ Paths ${ }_{M, s}$ for all finite paths of it. For reasoning about the probabilistic behaviour of an MDP we use the notion of policy (also called adversary) $[2,3,6]$.

Definition 3. A (deterministic) policy of an MDP $M$ is a function $\sigma$ : FPath $_{M} \rightarrow$ Act that for every finite path $\pi=s_{0} \stackrel{\alpha_{0}}{\rightarrow} s_{1} \stackrel{\alpha_{i}}{\rightarrow} \ldots \stackrel{\alpha_{i-1}}{\rightarrow} s_{i}$ selects an enabled action $\alpha_{i} \in \operatorname{Act}\left(s_{i}\right)$. The policy $\sigma$ is memoryless if it depends only on the last state of the path. In general, a policy is defined as function of $F P a t h_{M}$ to a distribution on Act. However, memoryless and deterministic policies are enough for computing the optimal unbounded reachability probabilities [2]. We use $\mathrm{Pol}{M}$ for the set of all deterministic and memoryless policies of $M$. A policy $\sigma \in \operatorname{Pol}{M}$ resolves all non-deterministic choices in $M$ and induces a DTMC $M^{\sigma}$ for which every state is a finite path of $M$.

## 数学代写|理论计算机代写theoretical computer science代考|Probability Measure of a Markov Chain

\begin{aligned} &\operatorname{Post}(s, \alpha) \doteq\left{s^{\prime} \in S \mid \delta\left(s, \alpha, s^{\prime}\ right)>0\right} \ &\operatorname{Post}(s) \doteq \cup_{\alpha \in \operatorname{Act}(s)} \operatorname{Post}(s, \alpha), \end{对齐}\begin{aligned} &\operatorname{Post}(s, \alpha) \doteq\left{s^{\prime} \in S \mid \delta\left(s, \alpha, s^{\prime}\ right)>0\right} \ &\operatorname{Post}(s) \doteq \cup_{\alpha \in \operatorname{Act}(s)} \operatorname{Post}(s, \alpha), \end{对齐}

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