半简单拼接系统的描述复杂性
Descriptional Complexity of Semi-Simple Splicing Systems
摘要：
剪接系统是Tom Head在1987年引入的生成机制，用于模拟DNA重组的生物过程。拼接系统的计算引擎是“拼接操作”，即由一组“拼接规则”$r = (α_1, α_2 ; α_3, α_4)$定义的剪切和粘贴二进制字符串操作，其中$α_1, α_2, α_3, α_4$是字母表$Σ$上的单词。对于两个字符串$x = x_1 α_1 α_2 x_2$和$y = y_1 α_3 α_4 y_2$，应用拼接规则$r$将生成字符串$z = x_1 α_1 α_4 y_2$。在本文中，我们专注于一种特殊类型的拼接系统，称为$(i, j)$半简单拼接系统，$i = 1,2$和$j = 3, 4$，其中所有的拼接规则具有的属性，在位置$i$和$j$的两个字符串是单字母，而其他两个字符串是空的。由这样的系统生成的语言由从称为“公理集”的初始集合开始获得的词的集合组成，通过迭代地将拼接规则应用于公理集中的字符串以及中间产生的字符串。我们考虑半简单的拼接系统的公理集是一个正规的语言，并调查的最小的确定性有限自动机的大小，他们产生的语言识别这样的系统的复杂性。
英文摘要：
Splicing systems are generative mechanisms introduced by Tom Head in 1987 to model the biological process of DNA recombination. The computational engine of a splicing system is the "splicing operation", a cut-and-paste binary string operation defined by a set of "splicing rules" $r = (α_1, α_2 ; α_3, α_4)$ where $α_1, α_2, α_3, α_4$ are words over an alphabet $Σ$. For two strings $x = x_1 α_1 α_2 x_2$ and $y = y_1 α_3 α_4 y_2$, applying the splicing rule $r$ produces the string $z = x_1 α_1 α_4 y_2$. In this paper we focus on a particular type of splicing systems, called $(i, j)$ semi-simple splicing systems, $i = 1,2$ and $j = 3, 4$, wherein all splicing rules have the property that the two strings in positions $i$ and $j$ are singleton letters, while the other two strings are empty. The language generated by such a system consists of the set of words that are obtained starting from an initial set called "axiom set", by iteratively applying the splicing rules to strings in the axiom set as well as to intermediately produced strings. We consider semi-simple splicing systems where the axiom set is a regular language, and investigate the descriptional complexity of such systems in terms of the size of the minimal deterministic finite automata that recognize the languages they generate.
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