On the Hierarchy of Block Deterministic Languages

A regular language is k-lookahead deterministic (resp. k-block deterministic) if it is specified by a k-lookahead deterministic (resp. k-block deterministic) regular expression. These two subclasses of regular languages have been respectively introduced by Han and Wood (k-lookahead determinism) and by Giammarresi et al. (k-block determinism) as a possible extension of one-unambiguous languages defined and characterized by Brüggemann-Klein and Wood. In this paper, we study the hierarchy and the inclusion links of these families. We first show that each kblock deterministic language is the alphabetic image of some one-unambiguous language. Moreover, we show that the conversion from a minimal DFA of a k-block deterministic regular language to a k-block deterministic automaton not only requires state elimination, and that the proof given by Han and Wood of a proper hierarchy in k-block deterministic languages based on this result is erroneous. Despite these results, we show by giving a parameterized family that there is a proper hierarchy in k-block deterministic regular languages. We also prove that there is a proper hierarchy in k-lookahead deterministic regular languages by studying particular properties of unary regular expressions. Finally, using our valid results, we confirm that the family of k-block deterministic regular languages is strictly included into the one of k-lookahead deterministic regular languages by showing that any k-block deterministic unary language is one-unambiguous.