Languages Generated by Iterated Idempotencies.

The rewrite relation with parameters m and n and with the possible length limit = k or :::; k we denote by w~, =kW~· or ::;kw~ respectively. The idempotency languages generated from a starting word w by the respective operations are wDAlso other special cases of idempotency languages besides duplication have come up in different contexts. The investigations of Ito et al. about insertion and deletion, Le., operations that are also observed in DNA molecules, have established that w5 and w~ both preserve regularity.Our investigations about idempotency relations and languages start out from the case of a uniform length bound. For these relations =kW~ the conditions for confluence are characterized completely. Also the question of regularity is -k n answered for aH the languages w- D 1 are more complicated and belong to the class of context-free languages.For a generallength bound, i.e."for the relations :"::kW~, confluence does not hold so frequently. This complicatedness of the relations results also in more complicated languages, which are often non-regular, as for example the languages WWithout any length bound, idempotency relations have a very complicated structure. Over alphabets of one or two letters we still characterize the conditions for confluence. Over three or more letters, in contrast, only a few cases are solved. We determine the combinations of parameters that result in the regularity of wDIn a second chapter sorne more involved questions are solved for the special case of duplication. First we shed sorne light on the reasons why it is so difficult to determine the context-freeness ofduplication languages. We show that they fulfiH aH pumping properties and that they are very dense. Therefore aH the standard tools to prove non-context-freness do not apply here.The concept of root in Formal Language ·Theory is frequently used to describe the reduction of a word to another one, which is in sorne sense elementary.For example, there are primitive roots, periodicity roots, etc. Elementary in connection with duplication are square-free words, Le., words that do not contain any repetition. Thus we define the duplication root of w to consist of aH the square-free words, from which w can be reached via the relation w~.Besides sorne general observations we prove the decidability of the question, whether the duplication root of a language is finite.Then we devise acode, which is robust under duplication of its code words.This would keep the result of a computation from being destroyed by dupli cations in the code words. We determine the exact conditions, under which infinite such codes exist: over an alphabet of two letters they exist for a length bound of 2, over three letters already for a length bound of 1.Also we apply duplication to entire languages rather than to single words; then it is interesting to determine, whether regular and context-free languages are closed under this operation. We show that the regular languages are closed under uniformly bounded duplication, while they are not closed under duplication with a generallength bound. The context-free languages are closed under both operations.The thesis concludes with a list of open problems related with the thesis' topics

Algorithmic aspects of bio-inspired string operations

We present building blocks for algorithms for the efficient reduction of square factor, i.e. direct repetitions in strings. So the basic problem is this: given a string, compute all strings that can be obtained by reducing factors of the form zz to z. Two types of algorithms are treated: an offline algorithm is one that can compute a data structure on the given string in advance before the actual search for the square begins; in contrast, online algorithms receive all input only at the time when a request is made. For offline algorithms we treat the following problem: Let u and w be two strings such that w is obtained from u by reducing a square factor zz to only z. If we further are given the suffix table of u, how can we derive the suffix table for w without computing it from scratch? As the suffix table plays a key role in online algorithms for the detection of squares in a string, this derivation can make the iterated reduction of squares more efficient. On the other hand, we also show how a suffix array, used for the offline detection of squares, can be adapted to the new string resulting from the deletion of a square. Because the deletion is a very local change, this adaption is more eficient than the computation of the new suffix array from scratch