Project/Area Number  10440034 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Kyoto Sangyo University 
Principal Investigator 
ITO Masami Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (50065843)

CoInvestigator(Kenkyūbuntansha) 
ISHIDA Hisashi Kyoto Sangyo University, Mathematics, Professor, 理学部, 教授 (10103714)
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)
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)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥7,800,000 (Direct Cost : ¥7,800,000)
Fiscal Year 2000 : ¥2,300,000 (Direct Cost : ¥2,300,000)
Fiscal Year 1999 : ¥2,800,000 (Direct Cost : ¥2,800,000)
Fiscal Year 1998 : ¥2,700,000 (Direct Cost : ¥2,700,000)

Keywords  primitive word / contextfree language / regular language / decidability problem / root(L) / deg(L) / palindrome / Petri net / 原始語 / 文脈自由言語 / 正規言語 / 決定問題 / 回文 / ペトリネット / ペトリネット言語 / プレフィックス・コード / CPN言語 / 非決定性有向オートマトン / 稠密文脈自由言語 / 星型積 / 形式言語 / 非決定問題 / オートマトン / シャッフル演算 
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 contextfree. 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 contextfree) 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 contextfree language L ⊆ X^+, it is undecidable whether root(L) is regular (or contextfree). (4) For a given regular language L ⊆ X^+, it is decidable whether L ⊆ Q(X) holds. (5) For a given contextfree 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 contextfree 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 contextfree 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 contextfree 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 coderelated Petri net languages etc.
