| Co-Investigator(Kenkyū-buntansha) |
ISHIDA Hisashi Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (10103714)
MURASE Atsushi Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (40157772)
KATSURA Masashi Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (80065870)
WASHIHARA Masako Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (40065800)
YASUGI Mariko Kyoto Sangyo University, Computer Science, Professor, 理学部, 教授 (90022277)
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| Research Abstract |
A word u is said to be primitive if u cannot be represented as the power of another word. By Q(X) we denote the set of all primitive words over X.It is conjectured that Q(X) is not context-free. However, this conjecture is still open. In our research, we investigate some decidability problems concerning Q(X) and its related languages. Let u ∈ X^+. If u = v^l for a positive integer l and a primitive word v, then we denote root (u) = v. For a language L ⊆ X^+, we define root(L) = ∪_<u∈L> root(u). Then we have the following results : (1) For a given regular (or context-free) language L ⊆ X^+, it is decidable whether root(L) is finite. (2) For a given regular language L ⊆ X^+, it is decidable whether root(L) is regular. (3) For a given context-free language L ⊆ X^+, it is undecidable whether root(L) is regular (or context-free). (4) For a given regular language L ⊆ X^+, it is decidable whether L ⊆ Q(X) holds. (5) For a given context-free language L ⊆ X^+, it is undecidable whether L ⊆ Q(X) holds.
Let L ⊆ X^+. Then, by deg(L) we mean the set {i : q ∈ Q(X), q^l ∈ L}. Then we have the following results : (1) For a given regular language L ⊆ X^+, it is decidable whether deg(L) is finite. (2) For a given context-free language L ⊆ X^+, it is undecidable whether deg(L) is finite.
A language L ⊆ X^+ is said to be palindromic if all words in L are palindromes. It is known that there is no dense palindromic regular language contained in Q(X). For the case of context-free palindromic languages, we have the same result and moreover we can prove that deg(L) is infinite (more exactly, aperiodic) if L ⊆ X^+ is a dense palindromic context-free language.
For the period between April 1, 1998 and March 31, 2001, we have also investigated deterministic and/or nondeterministic directable automata and related languages, some kinds of code-related Petri net languages etc.
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