C. Glaer and H. Schmitz, Languages of dot-depth 3/2, Proceedings 17th Symposium on Theoretical Aspects of Computer Science, Springer Verlag, Lecture Notes in Computer Sciences 1770, (2000), 555-566. 27  Home/Search   Document Details and Download   Summary   Related Articles   Check

....corresponding to H (see [14] The main open problem about these hierarchies is to decide, given a rational language, whether it belongs to the n th level of a given hierarchy. For the Straubing Th erien and the Brzozowski hierarchies, the problem has been solved positively for n 3 2 [33, 3, 4, 12, 13, 28, 8, 30] and for the group hierarchy, for n 1 [14, 11] It is still open for the other values of n, although some partial results for the levels 2 and 5=2 of the StraubingTh erien hierarchy are known [25, 41, 42, 28, 46, 10] A logical approach is also possible : it amounts to decide whether a rst ....

C. Glaer and H. Schmitz, Languages of dot-depth 3/2, Proceedings 17th Symposium on Theoretical Aspects of Computer Science, Springer Verlag, Lecture Notes in Computer Sciences 1770, (2000), 555-566. 27

....interpretations of this phenomenon. Our Techniques. We give a short preview of the approach. Our e ective characterization of the classes B 1=2 (l) is as follows. Let F be a deterministic nite automaton (DFA) with transition function and accepting some language L. We use a result from [GS00] saying that any word w can be written as w = w 0 u 1 w 1 umwm such that each factor is of bounded length and the u i are idempotent in F (i.e. s; u i ) s; u i u i ) for all states s) For xed F this leads to the notion of so called structured words, which are words from A with a ....

.... um xm and y = x 0 u 1 z 1 u 1 x 1 u 2 z 2 u 2 x 2 um z m um xm : The notions of F chains and alternations in such chains are de ned similar to k chains. Note that F and k are transitive and re exive. It can be shown that every word can be structured. This is a consequence of [GS00, Lemma 1] which is a combinatorial tool that relates lengths of words and loops in automata. Lemma 2. For every DFA F and w 2 A there exists some x 2 A F with w = x. 3.2 Well Partial Orderings We show that the word extensions we consider share a fundamental property: A F ; F ) for given ....

C. Glaer and H. Schmitz. Languages of dot-depth 3/2. In Proceedings 17th Symposium on Theoretical Aspects of Computer Science, volume 1770 of LNCS, pages 555-566. Springer Verlag, 2000.

.... 0 B n 3=2 : Pol(Bn 1 ) for n 0 L n 3=2 : Pol(Ln 1 ) for n 0 By de nition, all these classes are closed under union and it is known, that they are also closed under intersection and under taking residuals [1, 10] Up to now, levels 1=2, 1 and 3=2 of both hierarchies are known to be decidable [11, 7, 1, 10, 6] while the question is open for any other level. Partial results are known for level 2 of the STH which is decidable if a two letter alphabet is considered [14] Supported by the Studienstiftung des Deutschen Volkes. Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Wa ....

....level. Partial results are known for level 2 of the STH which is decidable if a two letter alphabet is considered [14] Supported by the Studienstiftung des Deutschen Volkes. Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Wa 847 4 1. We take up the discussion started in [6] and look at known results of the type L belongs to the class C if and only if the accepting automaton does not have subgraph S in its transition graph . Such a forbidden pattern characterization implies decidablility of the membership problem of C, and even more, it re ects the e ect of language ....

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C. Glaer and H. Schmitz. Languages of dot-depth 3/2. In Proceedings 17th STACS, volume 1770 of LNCS, pages 555-566. Springer Verlag, 2000.

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