LNCS 2950 - Aspects of Molecular Computing (Frontmatter Pages)

On Languages of Cyclic Words
Manfred Kudlek
Fachbereich Informatik, Universität Hamburg
Vogt-Kölln-Str. 30, D-22527 Hamburg, Germany
kudlek@informatik.uni-hamburg.de
Abstract. Languages of cyclic words and their relation to classical word
languages are considered. To this aim, an associative and commutative
operation on cyclic words is introduced.
1 Introduction
Related to DNA and splicing operations (see [1,2,4,11], etc.) it is of interest to
investigate characterizations of languages of cyclic words.
Languages of cyclic words can be generated in various ways. One possibility
is to consider languages of classical type, like regular, linear, orcontext-free
languages, and then take the collection of all equivalence classes of words with
respect to cyclic permutation from such languages. Another possibility is to consider
languages of cyclic words defined by rational, linear, andalgebraic systems
of equations via least fixed point, with respect to an underlying associative operation
on the monoid of equivalence classes. If the operation is also commutative,
the classes of rational, linear, andalgebraic languages coincide [10]. In the case
of catenation as underlying operation for words such least fixed points give regular,
linear, andcontext-free languages, respectively. A third way is to define
languages of cyclic words by the algebraic closure under some (not necessarily
associative) operation. A fourth way to generate languages of cyclic words is
given by rewriting systems analogous to classical grammars for words, as right
linear, linear, context-free, monotone, andarbitrary grammars [9].
For all notions not defined here we refer to [5,12].
An associative and commutative operation on cyclic words is introduced below.
It is shown that the first two ways of defining languages of cyclic words do
not coincide. It is also shown that the classical classes of regular and context-free
languages are closed under cyclic permutation, but the class of linear languages
is not.
2 Definitions
Let V be an alphabet. λ denotes the empty word, |x| the length of a word, and
|x|a the number of symbols a ∈ V in x.
Furthermore, let REG, LIN, CF, CS, andRE denote the classes of regular,
linear, context-free, context-sensitive, andrecursively enumerable languages,
respectively.
N. Jonoska et al. (Eds.): Molecular Computing (Head Festschrift), LNCS 2950, pp. 278–288, 2004.
c○ Springer-Verlag Berlin Heidelberg 2004