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# 数学代写|动力系统代写Dynamical Systems代考|MAT00011H Linear Temporal Logic

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## 数学代写|动力系统代写Dynamical Systems代考|Linear Temporal Logic

We need to introduce a formal way to construct more complex expressions describing properties of a labeled transition system $(T, l)$, whose truth value can vary with respect to time. For such a purpose, temporal logic was proposed [7]. Temporal logic is a formalism for describing properties of sequences of states as well as tree structures of states. There are many variations of temporal logic, and we will first introduce LTL and then CTL. Their relationship will be demonstrated using examples.

LTL is an extension of propositional logic geared to reasoning about infinite sequences of states. Formulas of LTL are built from a set of atomic propositions, like “it is raining”, and are closed under the application of Boolean connectives, such as conjunction, disjunction, and negation, and temporal operators. In particular, the following temporal operators are used for describing the properties along a specific run of a transition system 1 :

• o (“next state”): It requires that a property holds in the next state of the path. Let’s use $\varphi$ to denote the property of interest, then o $\varphi$ can be illustrated as
which means that the property $\varphi$ holds true at the next state along the path starting from the current state. The rest of states on the path remain unmarked since their labeling won’t change the truth value of the formula $о \varphi$.
• $\diamond($ “eventually”): It is used to assert that a property will hold at some future state on the path. For example, the expression $\diamond \varphi$ can be illustrated as

## 数学代写|动力系统代写Dynamical Systems代考|Syntax

Definition 3.4 LTL formulas are recursively defined from predicates in $\mathcal{P}$ according to the following rules:
(1) true, false, and $p_i$ are LTL formulas for all $p_i \in \mathcal{P}$;
(2) if $\varphi_1$ and $\varphi_2$ are LTL formulas, then $\varphi_1 \wedge \varphi_2$ and $\neg \varphi_1$ are LTL formulas;
(3) if $\varphi_1$ and $\varphi_2$ are LTL formulas, then $\circ \varphi_1$ and $\varphi_1 \sqcup \varphi_2$ are LTL formulas.
Note that we can define other Boolean operators based on $\wedge$ and $\neg$, for example, $\varphi_1 \vee \varphi_2$ is equivalent to $\neg\left(\neg \varphi_1 \wedge \neg \varphi_2\right)$. As another example, the implication $\varphi_1 \Rightarrow$ $\varphi_2$ can be represented equivalently as $\neg \varphi_1 \vee \varphi_2$. Similarly, other temporal operators, such as $\rangle$ and $\square$, can be represented in LTL as well. Let’s consider a couple of examples to illustrate the syntax of LTL formulas.

Example 3.5 A simple safety property, such as “no collision”, can be expressed as an LTL formula $\square \neg$ collision. Similarly, ১finish expresses a simple reachability property.

Example 3.6 The LTL formula $\square \diamond \varphi$ is true for traces (generated from a labeled transition system) that satisfy $\varphi$ infinitely often, e.g., $\square \diamond$ hungry, which is a liveness requirement. This can be understood as follows. For a trace to satisfy $\square \diamond \varphi$, all states along the trace should satisfy $\nabla \varphi$. This is illustrate as follows:

We also know that to satisfy $\diamond \varphi$, the property $\varphi$ must hold true in the finite future along the trace. Therefore, $\varphi$ must hold true for an infinite number of times along the trace. Otherwise, if $\varphi$ no longer hold true after $N$ steps, then $\diamond \varphi$ is violated for all states after $N$ on the trace, which contradicts the fact that $\Delta \varphi$ should hold true for all states on the trace.

Therefore, a fairness requirement, say serve both customer 1 and customer 2 infinitely often, can be expressed as an LTL formula $(\square \diamond$ serve1) $\wedge$ $(\square \diamond$ serve2).

Note that if we change the order of temporal operators, it may mean quite different things.

## 数学代写|动力系统代写Dynamical Systems代考|Linear Temporal Logic

LTL 是命题逻辑的扩展，旨在推理无限的状态序列。LTL 的公式由一组原子命题（如“下雨”) 构建而成，并在布尔连接词（如合 取、析取和取反）和时间运算符的应用下闭合。特别地，以下时间算子用于描述沿过渡系统 1 的特定运行的属性:

O(“下一个状态”)：它要求一个属性保持在路径的下一个状态。让我们使用 $\varphi$ 表示感兴趣的属性，然后 $\circ \varphi$ 可以说明为 这意味差该属性 $\varphi$ 在从当前状态开始的路径上的下一个状态下成立。路径上的其余状态保持末标记，因为它们的标记不会改 变公式的真值蒾F $\varphi$.

$\diamond($ “最终”) : 它用于断言属性将在路径上的某个末来状态下保持。例如，表达式 $\propto \varphi$ 可以表示为

## 数学代写|动力系统代写Dynamical Systems代考|Syntax

（1）真、假和 $p_i$ 是所有的 $L T L$ 公式 $p_i \in \mathcal{P}$;
(2) 如果 $\varphi_1$ 和 $\varphi_2$ 是 LTL 公式，那 $\angle \Delta \varphi_1 \wedge \varphi_2$ 和 $\neg \varphi_1$ 是 LTL 公式；
(3) 如果 $\varphi_1$ 和 $\varphi_2$ 是 LTL 公式，那么/⿱ $\varphi_1$ 和 $\varphi_1 \sqcup \varphi_2$ 是 LTL 公式。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。