The complexity of computing maximal word functions

Abstract

Maximal word functions occur in data retrival applications and have connections with ranking problems i.e. with problems related to data compression. Compute the maximal word function of a string x with respect to a language L ⊆ Σ* consists in determining the lexicographically greatest word belonging lo L, which is smaller than or equal to x. In this paper we investigate the complexity of computing maximal word functions, establishing sharp boundaries between classes of languages for which the maximal word function can be efficiently computed and classes of languages for which such a problem seems to be difficult to solve. For example, we will show that the maximal word function for the class of context free languages is in AC2 while for the class of languages accepted by 2-way pushdown automata the maximal word function can be efficiently (polynomially) computed if and only if P = NP. We will also show connections with ranking, detector and constructor functions as introduced in [14]. This paper is a continuation of the study enterprised by many authors ([6], [10], [13] and [14]) with the intention of identifying properties others than membership which are easily computable for certain class of languages.