The Boolean Hierarchy over Level 1/2 of the Straubing-Therien Hierarchy [article]

Heinz Schmitz, Klaus W. Wagner
1998 arXiv   pre-print
For some fixed alphabet A, a language L of A* is in the class L(1/2) of the Straubing-Therien hierarchy if and only if it can be expressed as a finite union of languages A*aA*bA*...A*cA*, where a,b,...,c are letters. The class L(1) is defined as the boolean closure of L(1/2). It is known that the classes L(1/2) and L(1) are decidable. We give a membership criterion for the single classes of the boolean hierarchy over L(1/2). From this criterion we can conclude that this boolean hierarchy is
more » ... er and that its classes are decidable.In finite model theory the latter implies the decidability of the classes of the boolean hierarchy over the class Sigma(1) of the FO(<)-logic. Moreover we prove a "forbidden-pattern" characterization of L(1) of the type: L is in L(1) if and only if a certain pattern does not appear in the transition graph of a deterministic finite automaton accepting L. We discuss complexity theoretical consequences of our results.
arXiv:cs/9809118v1 fatcat:gez5cgx7efgjhevgwqthywniq4