Automata, Formal Language and Computation Theory

A =（A，X）是可指示的，如果存在一个字μ X-D1 * Y-D1这样说|A-D-1A-D1|（1）在这种情况下，可以清楚地界定方向的概念。在另一方，非确定性自动数据的指示性可以以几种非确定性的方式确定。我们提供了以下三个指示性注意事项：A word u ∈ X-D1 * X-D1对X说：（1） a D1-directing word of a D1-directable automation A if （c ∈ A） （a ∈A）（au ― D1A ― D1 ={c}），（2） a D2 directing of a D2-directable automaton A if（A， B，A）（auイイD1AイエD1 = buイD1AイエD1），（3） a D3-directing word of a D3-directable automaton A if （？c？A） （？a？A） （c？au？D1A？D1）。By D？D2i？D2（A） we denote the set of all Di-directing word of A。Notice that the above three notions are equivalent for deterministic automata，I.e.D21（A） = D23（A）。在这种情况下，我们将这些语言定义为D（A），我们将定义以下语言类别：D2 D2 = {D（A）|A ：一个决定 ... More istic directable automation}，椭圆D2 nd（i）D2 ={Di（A）|A = a nonderministic d-directable automaton } D2cnd（i）D2 ={Di（A）|A ：一个完整的非确定性可分方向自动机}（I= 1，2，3），然后每一个以上的语言类别都成为常规语言的一个子集，更确切地说，这些类别都是在包容性关系下固定较低的晶格，现在我们认为所有非确定性可互换自动机的类别。By Fundy D2 D D2，Fundy D2 nd（I）D2，Fundy D2cnd（I）D2，我们拒绝语言类的对应，然后这些类将构成一个线性排序集，用于确定可定向自动数据A = （A， X）类|A（A）|= n，Cerny下面的概念是众所周知的：让A = （A，X）成为一个决定性的可直接自动化|A（A）|= n.由P（A），我们拒绝最小数{|u（u）|||A-D-1A-D1| = 1}。让d（n） = max{p（A）|A ：一个具有决定性的可直接自动机|A（A）|=n}。Then d（n） = （n - 1）イイD12エD1。We have dealt with corresponding problems for the class of nondeterministic directable automata。与B一起使用。Irma。《Moreover》，在研究阶段期间，与M. Katsuura和Z. Esik一起完成了与研究有关的如此令人愉快的操作。与M. Katsuura和P. Domosi的研究有关的所有原始单词也都是如此。Less（低）

A deterministic automaton A =(A,X) is said to be directable if there exists a word μ∈ XィイD1*ィエD1 such that |AuィイD1AィエD1| = 1.In this case, the notion of directability can be uniquely defined. On the other hand, the directability of nondeterministic automata can be difined in several nonequivalent ways. We provide the following three notions of directability.A word u ∈ XィイD1*ィエD1 over X is said to be :(1) a D1 -directing word of a D1 -directable automation A if (∃c ∈ A) (∀a ∈A) (auィイD1AィエD1 ={c}),(2) a D2 directing word of a D2 -directable automaton A if (∀a, b∈ A) (auィイD1AィエD1 = buィイD1AィエD1),(3) a D3-directing word of a D3 -directable automaton A if (∃c ∈ A) (∀a∈A) (c∈ auィイD1AィエD1).By DィイD2iィエD2(A) we denote the set of all Di-directing word of A. Notice that the above three notions are equivalent for deterministic automata, I.e.DィイD21ィエD2(A) = DィイD23ィエD2(A). In this case, we denote these languages as D(A), we define the following classes of languages : ∠ィイD2DィエD2 = {D(A)|A : a determin … More istic directable automation}, ∠ィイD2nd(i)ィエD2 ={Di(A) | A : a nondeterministic Di-directable automaton } ∠ィイD2cnd(i)ィエD2 ={Di(A) | A : a complete nondeterministic Di-directable automaton }(I= 1,2,3).Then each of the above classes of languages becomes a subclass of regular languages, More exactly, these classes constitute a lower lattice under inclusion relation, Now we consider the class of all nondeterministic commutative automata. By ∠ィイD2DィエD2, ∠ィイD2nd(I)ィエD2, ∠ィイD2cnd(I)ィエD2, we denote corresponding classes of commutative languages, then these classes constitute a linear ordered set, For the class of deterministic directable automata A = (A, X) with |A| = n, the following Cerny's conjecture is well-known :Let A = (A,X) be a deterministic directable automation with |A| = n.By P(A), we denote the number min {|u| | |AuィイD1AィエD1| = 1}. let d(n) = max{p(A) | A : a deterministic directable automaton with |A| =n}. Then d(n) = (n - 1)ィイD12ィエD1.We have dealt with corresponding problems for the class of nondeterministic directable automata.The above research has been done mainly with B. Irma. Moreover, during the research term, the research related so shuffle operations on posets and languages has been done with M.Katsura and Z.Esik. The research related to the set of all primitive words has been also done with M.Katsura and P.Domosi. Less