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## 数学代写|离散数学作业代写discrete mathematics代考|Finite-State Machines

The neurophysiologists Warren McCulloch and Walter Pitts published early work on finite-state automata in 1943. They were interested in modelling the thought process for humans and machines. Moore and Mealy developed this work further in the mid-1950s, and their finite-state machines are referred to as the ‘Mealy machine’ and the ‘Moore machine’. The Mealy machine determines its outputs through the current state and the input, whereas the output of Moore’s machine is based upon the current state alone.

Definition $7.1$ (Finite-State Machine) A finite-state machine (FSM) is an abstract mathematical machine that consists of a finite number of states. It includes a start state $\mathrm{q}_0$ in which the machine is in initially; a finite set of states Q; an input alphabet $\Sigma$; a state transition function $\delta$ and a set of final accepting states $\mathrm{F}$ (where $F \subseteq Q$ ).
The state transition function $\delta$ takes the current state and an input symbol, and returns the next state. That is, the transition function is of the form:
$$\delta: \mathrm{Q} \times \Sigma \rightarrow \mathrm{Q}$$
The transition function provides rules that define the action of the machine for each input symbol, and its definition may be extended to provide output as well as a transition of the state. State diagrams are used to represent finite-state machines, and each state accepts a finite number of inputs. A finite-state machine (Fig. 7.1) may be deterministic or non-deterministic, and a deterministic machine changes to exactly (or at most) $^1$ one state for each input transition, whereas a non-deterministic machine may have a choice of states to move for a particular input symbol.

Finite-state automata can compute only very primitive functions, and so they are not adequate as a model for computing. There are more powerful automata such as the Turing machine that is essentially a finite automaton with a potentially infinite storage (memory). Anything that is computable by a Turing machine.

A finite-state machine can model a system that has a finite number of states, and a finite number of inputs/events that can trigger transitions between states. The behaviour of the system at a point in time is determined from the current state and input, with behaviour defined for the possible input to that state. The system starts in a particular initial state.

## 数学代写|离散数学作业代写discrete mathematics代考|Pushdown Automata

A pushdown automaton (PDA) is essentially a finite-state machine with a stack, and it includes three components namely an input tape; a control unit and a potentially infinite stack. The stack head scans the top symbol of the stack, and two operations (push or pop) may be performed on the stack. The push operation adds a new symbol to the top of the stack, whereas the pop operation reads and removes an element from the top of the stack (Fig. 7.4).

A push down automaton may remember a potentially infinite amount of information, whereas a finite-state automaton remembers only a finite amount of information. A PDA also differs from a FSM in that it may use the top of the stack to decide on which transition to take, and it may manipulate the stack as part of the performance of a transition. The input and current state determine the transition in a finite-state machine, and a FSM has no stack to work with.

A pushdown automaton is defined formally as a 7-tuple $\left(\Sigma, \mathrm{Q}, \Gamma, \delta, q_0, \mathrm{Z}, F\right)$. The set $\Sigma$ is a finite set which is called the input alphabet; the set $Q$ is a finite set of states; $\Gamma$ is the set of stack symbols; $\delta$ is the transition function which maps $Q \times{\Sigma \cup{\varepsilon}}^3 \times \Gamma$ into finite subsets of $Q \times \Gamma^{* 2} ; q_0$ is the initial state; $Z$ is the initial stack top symbol on the stack (i.e. $Z \in \Gamma$ ) and $F$ is the set of accepting states (i.e. $F \subseteq Q$ ).

Figure $7.5$ shows a transition from state $q_1$ to $q_2$, which is labelled as $a$, $b \rightarrow c$. This means that at if the input symbol $a$ occurs in state $q_1$, and the symbol on the top of the stack is $b$, then $b$ is popped from the stack and $c$ is pushed onto the stack. The new state is $q_2$.

In general, a pushdown automaton has several transitions for a given input symbol, and so pushdown automata are mainly non-deterministic. If a pushdown automaton has at most one transition for the same combination of state, input symbol and top of stack symbol it is said to be a deterministic PDA (DPDA). The set of strings (or language) accepted by a pushdown automaton $M$ is denoted $L(M)$.
The class of languages accepted by pushdown automata is the context free languages, and every context free grammar can be transformed into an equivalent non-deterministic pushdown automaton. Chapter 12 has more detailed information on the classification of languages.

# 离散数学代写

## 数学代写|离散数学作业代写discrete mathematics代考|有限状态机

$$\delta: \mathrm{Q} \times \Sigma \rightarrow \mathrm{Q}$$

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## 有限元方法代写

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师