Automata, Formal Language and Computation Theory

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

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