A $d$-limited automaton is a Turing machine that uses only the cells with the input word (and end-markers) and rewrites symbols only in the first $d$ visits. This model was introduced by T. Hibbard in 1967 and he showed that $d$-limited automata recognize context-free languages for each $d \geq 2$. He also proved that languages recognizable by deterministic $d$-limited automata form a hierarchy and it was shown later by Pighizzini and Pisoni that it begins with deterministic context-free languages (DCFLs) (for $d=2$). As well-known, DCFLs are widely used in practice, especially in compilers since they are linear-time recognizable and have the corresponding CF-grammars subclass (LR$(1)$-grammars). In this paper we present a linear time recognition algorithm for deterministic $d$-limited automata (in the RAM model) which opens an opportunity for their possible practical applications. We also generalize this algorithm to deterministic $d(n)$-limited automata: the extension of deterministic $d$-limited automata, where $d$ is not a constant, but a function depending on the input length $n$.

中文翻译：

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确定性$d$有限自动机的线性时间模拟

$d$ 受限自动机是一种图灵机，它仅使用具有输入单词（和结束标记）的单元，并且仅在第一次 $d$ 访问中重写符号。该模型由 T. Hibbard 于 1967 年提出，他证明 $d$ 有限自动机可以识别每个 $d \geq 2$ 的上下文无关语言。他还证明了由确定性 $d$ 有限自动机可识别的语言形成了一个层次结构，后来 Pighizzini 和 Pisoni 证明它以确定性上下文无关语言 (DCFL) 开始（对于 $d=2$）。众所周知，DCFL 在实践中被广泛使用，特别是在编译器中，因为它们是线性时间可识别的并且具有相应的 CF 语法子类（LR$(1)$-grammars）。在本文中，我们提出了一种用于确定性 $d$ 有限自动机（在 RAM 模型中）的线性时间识别算法，这为其可能的实际应用提供了机会。我们还将该算法推广到确定性 $d(n)$ 有限自动机：确定性 $d$ 有限自动机的扩展，其中 $d$ 不是常数，而是取决于输入长度 $n$ 的函数。