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 Br\"uggemann-Klein and Wood. In this paper, we study the hierarchy and the inclusion links of these families. We first show that each $k$-block 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

Generalizations of 1-deterministic regular languages

We examine two generalizations of 1-deterministic regular languages that are used for the content models of DTDs in XML. They are k-lookahead determinism and k-block-determinism. The k-lookahead determinism uses the first k symbols w(1)w(2)...w(k) of the current input string as lookahead to process the first symbol w(1). On the other hand, the k-block-determinism takes k w(1)w(2)...w(k) as lookahead and process the whole k symbols. We show that there is a hierarchy in k-lookahead determinism and there is a proper hierarchy in k-block-determinism. Moreover, we prove that k-block-deterministic regular languages are a proper subfamily of deterministic k-lookahead regular languages. (C) 2008 Elsevier Inc. All rights reserved