小禾参观计算展。最简单的计算装置没有屏幕，只有纸带、读写头和规则表。它能做什么？At a computing exhibition, He finds a model with only a tape, a head and rules. What could such a simple machine compute?
第 1 步：把步骤写得毫不含糊Step 1: Make every step explicit
先不考虑速度，只看规则。Ignore speed and focus on precise rules.
图灵机是研究计算的数学模型。纸带分成格子，每格有一个符号。读写头一次查看一格。规则根据当前状态与符号，决定写什么、向哪移以及新状态。它不是历史上某台笔记本，而是把机械执行步骤这件事说清楚。A Turing machine is a mathematical model of computation. Its tape has cells containing symbols. A head reads one cell at a time. A rule uses the current state and symbol to choose what to write, where to move and the next state. It is not an old laptop: it precisely describes mechanical steps.
第 2 步：让纸带上的数量加一Step 2: Add one on the tape
只会读写移动，怎样做计算？How can reading, writing and moving calculate?
用连续的竖线表示数量。三条线表示三。规则是看到线就向右，看到空格就写一条线并停机。三就变成四。复杂计算需要更多状态和规则。每一步简单，不代表整个过程一定短。纸带可继续延长，是理论模型的重要理想化。Represent a number with consecutive marks. Three marks mean three. Move right while reading a mark; on a blank, write a mark and halt. Three becomes four. More complex computations need more states and rules. Simple individual steps do not imply a short overall process. The extendable tape is an important idealization.
第 3 步：能计算也有边界Step 3: Computability has limits
有了通用机器，所有问题都能解决吗？Can a universal machine solve every problem?
通用图灵机能按编码模拟其他图灵机。这说明程序也能成为输入。但并非所有数学问题都可由算法解决。例如，没有算法能对任意程序和输入，总能正确判断它是否停机。这里说的是普遍不可能，不是某个具体程序永远无法分析。A universal Turing machine can simulate other Turing machines from encoded descriptions, making a program part of the input. Yet not every mathematical problem has an algorithmic solution. No algorithm correctly decides halting for every program and input. This is a universal limitation, not a claim that a particular program can never be analyzed.
🎮 纸带加一机🎮 Unary increment machine
每次按一步，看读写头怎样移动。在四种初始数量上完成加一。Step through the head’s moves and add one to four starting quantities.
一句话记住它：Remember: 图灵机把计算写成明确的读、写、移动和状态变化。A Turing machine formalizes computation as reading, writing, moving and changing state.
规则由当前状态和符号决定。Rules depend on state and the read symbol.程序可以被编码成输入。Programs can be encoded as input.可计算性与计算速度不同。Computability and speed are different questions.
把握度：模型与停机结论较高。有限显示区只展示纸带的一小段，练习使用一元记数。Confidence: high for the model and halting result. The display shows a finite tape segment and uses unary notation.
AI 辅助制作与资料核对。插图和互动为教学示意。AI-assisted production and source checking. Illustrations and interactions are teaching models.
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