Theoretical Computer Science ( IF 0.9 ) Pub Date : 2020-11-19 , DOI: 10.1016/j.tcs.2020.11.029 Fumiya Okubo , Takashi Yokomori
We investigate the computing power of the following language operation %: Given two languages over Σ and over Γ with , we consider the language operation . In this case we say that is the -reduction of . This is extended to the language families as follows: . Among many works concerning Dyck-reductions, for the family of recursively enumerable languages , it was shown that (Jantzen & Petersen, 1994) with and that min- (Hirose & Okawa, 1996, and Latteux & Turakainen, 1990), where and min- are the families of linear and minimal linear context-free languages, respectively.
In this paper, we show that each recursively enumerable language L can be represented in the form , for some and a Dyck language D, where () denotes the family of insertion languages (insertion languages where the maximum length of the string to be inserted is 3). We can refine it as , where denotes the Dyck language over binary alphabet. For context-free languages, we show that , where is the family of finite sets. This also derives that with . Further, for regular languages, it is shown that each regular language R can be represented in the form , for some and a finite set . We also present some results which characterize the computability and properties of in the framework of -reduction of .
It is intriguing to note that, from the DNA computing point of view, the notion of L-reduction is naturally motivated by a molecular biological functioning well-known as DNA(RNA) splicing occurring in most eukaryotic genes.
中文翻译:
关于...的计算能力 -减少插入语言
我们研究以下语言操作%的计算能力:给定两种语言 超过Σ和 在Γ上 ,我们考虑语言操作 。在这种情况下,我们说 是个 -减少 。这扩展到语言族,如下所示:。在众多关于Dyck归约的著作中,针对递归可枚举语言家族,结果表明 (Jantzen&Petersen,1994)与 那分钟- (Hirose和Okawa,1996年; Latteux和Turakainen,1990年),其中 和分钟 分别是线性和最小线性上下文无关语言的族。
在本文中,我们证明了每种递归可枚举语言L都可以用以下形式表示:, 对于一些 和戴克语言D,其中 ()表示插入语言的族(插入语言,其中要插入的字符串的最大长度为3)。我们可以将其细化为, 在哪里 用二进制字母表示戴克语言。对于无上下文语言,我们表明, 在哪里 是有限集的族。这也得出 和 。此外,对于常规语言,示出了每种常规语言R可以以以下形式表示:, 对于一些 和一个有限集 。我们还提出了一些表征可计算性和特性的结果 在...的框架内 -减少 。
有趣的是,从DNA计算的角度来看,L-还原的概念是由大多数真核基因中发生的众所周知的分子生物学功能即DNA(RNA)剪接自然激发的。