Insertion–deletion (or ins–del for short) systems are simple models of bio-inspired computing. They are well studied in formal language theory, especially regarding their computational completeness. This concerns the question if all recursively enumerable languages can be generated. This ultimately addresses the question if one can build general-purpose computers rooted in this formalism. The descriptional complexity of an ins–del system is typically measured by its size, a 6-dimensional tuple of non-negative integers \((e,e',e'';d,d',d'')\) where e is the maximum length of the insertion string, \(e'\) (and \(e''\)) is the maximum length of the left (and right) context used for insertion; the last three parameters \(d,d',d''\) are similarly understood for deletion rules. Computational completeness for ins–del systems can even be achieved with rule size (1, 1, 1; 1, 1, 1) but with no rule size strictly smaller than this. This fact has motivated to study ins–del systems in combination with regulation mechanisms. In this context, the six-tuple explained above is called the ID size of a system. Several regulations like graph-control, matrix and semi-conditional have been imposed on ins–del systems. Typically, the computational completeness results are obtained as trade-offs, reducing the ID size, say, to (1, 1, 0; 1, 1, 0) at the expense of increasing other measures of descriptional complexity. In this paper, we study simple semi-conditional ins–del systems, where an ins–del rule can be applied only in the presence or absence of substrings of the derivation string. This brings along two further natural parameters to measure descriptional complexity, namely, the maximum permitting string length p and the maximum forbidden string length f, usually summarized as the degree \(d=(p,f)\). We show that simple semi-conditional ins–del systems of degree (2, 1) and with ID sizes \((1+e,e',e'';1+d,d',d'')\) are computationally complete for any \(e,e',e'',d,d',d''\in \{0,1\}\), with \(e+e'+e''=1\) and \(d+d'+d''=1\). The obtained results complement and improve on the existing results known from the literature. To prove our results, we also show a new normal form for type-0 grammars that appears to be interesting in its own right.

中文翻译：

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简单的半条件插入-删除度数系统（2,1）的计算完整性

插入删除（简称ins-del）系统是受生物启发的计算的简单模型。他们在形式语言理论中得到了很好的研究，尤其是在计算完整性方面。这涉及是否可以生成所有递归可枚举语言的问题。这最终解决了一个问题，即是否有人可以构建植根于这种形式主义的通用计算机。一个的descriptional复杂INS-德尔系统通常通过它的测量尺寸，非负整数的6维元组\（（E，E 'E ''; d，d'，d ''）\） ，其中e是插入字符串的最大长度\（e'\）（和\（e''\））是用于插入的左侧（和右侧）上下文的最大长度；对于删除规则，最后三个参数\（d，d'，d''\）的理解类似。ins-del系统的计算完整性甚至可以通过规则大小（1、1、1、1、1、1）来实现，但没有任何规则的大小可以严格地小于此大小。这一事实促使人们研究与调节机制结合的ins-del系统。在这种情况下，上面解释的六元组称为ID大小一个系统。在ins-del系统上已经强加了一些规则，例如图形控制，矩阵和半条件控制。通常，计算完整性结果是以折衷的方式获得的，从而将ID大小减小为（1，1，0; 1，1，0），但这是以增加描述复杂性的其他措施为代价的。在本文中，我们研究简单的半条件ins-del系统，其中ins-del规则仅可在存在或不存在派生字符串的子字符串的情况下应用。这带来了另外两个自然参数来衡量描述的复杂性，即最大允许字符串长度 p和最大禁止字符串长度 f，通常总结为度\（d =（p，f）\）。我们证明了简单的半条件ins-del系统（2、1）和ID大小为\（（（1 + e，e'，e''; 1 + d，d'，d''）\）是计算完成对于任何\（E，E 'E ''，d，d'，d '' \在\ {0,1 \} \），用\（E + E '+ E''= 1 \）和\（d + d'+ d''= 1 \）。获得的结果补充并改进了文献中已知的现有结果。为了证明我们的结果，我们还显示了一种新型的0型语法范式，其本身似乎很有趣。