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

Partially-commutative context-free languages

The paper is about a class of languages that extends context-free languages (CFL) and is stable under shuffle. Specifically, we investigate the class of partially-commutative context-free languages (PCCFL), where non-terminal symbols are commutative according to a binary independence relation, very much like in trace theory. The class has been recently proposed as a robust class subsuming CFL and commutative CFL. This paper surveys properties of PCCFL. We identify a natural corresponding automaton model: stateless multi-pushdown automata. We show stability of the class under natural operations, including homomorphic images and shuffle. Finally, we relate expressiveness of PCCFL to two other relevant classes: CFL extended with shuffle and trace-closures of CFL. Among technical contributions of the paper are pumping lemmas, as an elegant completion of known pumping properties of regular languages, CFL and commutative CFL.Comment: In Proceedings EXPRESS/SOS 2012, arXiv:1208.244

On Infinite Words Determined by Indexed Languages

We characterize the infinite words determined by indexed languages. An infinite language $L$ determines an infinite word $\alpha$ if every string in $L$ is a prefix of $\alpha$. If $L$ is regular or context-free, it is known that $\alpha$ must be ultimately periodic. We show that if $L$ is an indexed language, then $\alpha$ is a morphic word, i.e., $\alpha$ can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. To obtain this result, we prove a new pumping lemma for the indexed languages, which may be of independent interest.Comment: Full version of paper accepted for publication at MFCS 201