Concatenation Hierarchies and Forbidden Patterns

We make the following progress on the dot–depth problem: (1) We introduce classes CB n and CL n of starfree languages defined via forbidden patterns in finite automata. It is shown for all n 0 that CB n (CL n) contains level n + 1=2 of the dot–depth hierarchy (Straubing–Thérien hierarchy, resp.). Since we prove that CB n and CL n have decidable membership problems, this yields a lower bound algorithm for the dot–depth of a given language. (2) We prove many structural similarities between our hierarchies fCB ng and fCL ng, and the mentioned concatenation hierarchies. Both show the same inclusion structure and can be separated by the same languages. Moreover, we see that our pattern classes are not too large, since CL n does not capture level n + 1=2 of the dot–depth hierarchy. We establish an effective conjecture for the dot– depth problem, namely that CB n and CL n , and the respective levels of concatenation hierarchies in fact coincide. It is known from literature that this is true for n = 0; 1. (3) We prove the decidability of level 5=2 of the Straubing–Thérien hierarchy in case of a two– letter alphabet — in exactly the way as predicted by our conjecture. To our knowledge, no results concerning the decidability of this class (and of any level 5=2 of these concatenation hierarchy) were previously known. The result is based on a more general upward translation of decidability, connecting concatenation hierarchies and our pattern hierarchies.