On the Hierarchy of Block Deterministic Languages

TL;DR

This paper investigates the hierarchy and relationships between $k$-lookahead and $k$-block deterministic regular languages, correcting previous misconceptions and establishing proper inclusion relations among these language classes.

Contribution

It clarifies the hierarchy of $k$-block and $k$-lookahead deterministic languages, correcting earlier proofs and establishing proper inclusions through new constructions and properties.

Findings

Each $k$-block deterministic language is an alphabetic image of a one-unambiguous language.

The conversion from minimal DFA to $k$-block deterministic automaton involves state elimination.

The hierarchy in $k$-block deterministic languages is proper, and $k$-block is strictly included in $k$-lookahead deterministic languages.

Abstract

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.