数数很容易
Counting is Easy
摘要：
对于任何固定的$k$，一个非常简单的单磁带图灵机可以实时的模拟$k$独立的计数器。非正式地，计数器是一个存储单元，它维护一个单一的整数（最初为0），递增它，递减它，或报告其符号（正，负或零）的命令。任何以计数器的形式响应每个连续命令的自动机都被称为模拟计数器。(Only对于一个信号询问是感兴趣的反应，当然。零是唯一真正的问题，因为模拟器可以很容易地使用零检测来跟踪有限状态控制中的正性和负性。在本文中，我们描述了一个非常简单的实时模拟，基于五个简单的重写规则，任何固定数量的独立计数器$k$。在一个只有一个二进制工作带的图灵机上，模拟实时的运行，在每一步处理一个任意的计数器命令。对于任何指定的$ε> 0$，对于第一个$n$命令，模拟所使用的空间可以保持为$(k+ε) \log_2 n$位。
英文摘要：
For any fixed $k$, a remarkably simple single-tape Turing machine can simulate $k$ independent counters in real time. Informally, a counter is a storage unit that maintains a single integer (initially 0), incrementing it, decrementing it, or reporting its sign (positive, negative, or zero) on command. Any automaton that responds to each successive command as a counter would is said to simulate a counter. (Only for a sign inquiry is the response of interest, of course. And zeroness is the only real issue, since a simulator can readily use zero detection to keep track of positivity and negativity in finite-state control. In this paper we describe a remarkably simple real-time simulation, based on just five simple rewriting rules, of any fixed number $k$ of independent counters. On a Turing machine with a single, binary work tape, the simulation runs in real time, handling an arbitrary counter command at each step. The space used by the simulation can be held to $(k+ε) \log_2 n$ bits for the first $n$ commands, for any specified $ε> 0$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
