Regular Languages meet Prefix Sorting

Indexing strings via prefix (or suffix) sorting is, arguably, one of the most successful algorithmic techniques developed in the last decades. Can indexing be extended to languages? The main contribution of this paper is to initiate the study of the sub-class of regular languages accepted by an automaton whose states can be prefix-sorted. Starting from the recent notion of Wheeler graph [Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting to labeled graphs-we investigate the properties of Wheeler languages, that is, regular languages admitting an accepting Wheeler finite automaton. Interestingly, we characterize this family as the natural extension of regular languages endowed with the co-lexicographic ordering: when sorted, the strings belonging to a Wheeler language are partitioned into a finite number of co-lexicographic intervals, each formed by elements from a single Myhill-Nerode equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with $n$ states admits an equivalent Wheeler DFA (WDFA) with at most $2n-1-|\Sigma|$ states that can be computed in $O(n^3)$ time. This is in sharp contrast with general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper superset of the WDFAs, a $O(n\log n)$-time online algorithm to sort acyclic WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By contribution (i), our algorithms can also be used to index any WNFA at the moderate price of doubling the automaton's size. (iii) We provide a minimization theorem that characterizes the smallest WDFA recognizing the same language of any input WDFA. The corresponding constructive algorithm runs in optimal linear time in the acyclic case, and in $O(n\log n)$ time in the general case. (iv) We show how to compute the smallest WDFA equivalent to any acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version with new results (W-MH theorem, linear determinization), added author: Giovanna D'Agostin

Wheeler Languages

The recently introduced class of Wheeler graphs, inspired by the Burrows-Wheeler Transform (BWT) of a given string, admits an efficient index data structure for searching for subpaths with a given path label, and lifts the applicability of the Burrows-Wheeler transform from strings to languages. In this paper we study the regular languages accepted by automata having a Wheeler graph as transition function, and prove results on determination, Myhill_Nerode characterization, decidability, and closure properties for this class of languages

Data Capture and Presentation in the Romanian Online Dialect Atlas

RODA,the Romanian Online Dialect Atlas (Embleton, Uritescu, and Wheeler 2002, 2004,2006, in press), is a two-stage project involving (I) the transfer of data from a hard copy atlas of the Crisana dialect of Romanian (Stan and Uritescu 1996,2003) to an online system for general availability, and (2) the application of innovative statistical methods to the data. Romanian, as the prime exemplar of the eastern Romance languages. has had scholarly attention, including the detailed work of Stan and Britescu (1996,2003) and Uritescu (l984a, 1984b) on the dialects of the Crisana region in north-west Romania. In digitizing this data to make it more broadly accessible, and in successfully digitizing a hardcopy dialect atlas of Finnish (Embleton and Wheeler 1997b, 2000), we encountered several situations worth highlighting to others who may be considering parallel projects

The view from elsewhere: perspectives on ALife Modeling

Many artificial life researchers stress the interdisciplinary character of the field. Against such a backdrop, this report reviews and discusses artificial life, as it is depicted in, and as it interfaces with, adjacent disciplines (in particular, philosophy, biology, and linguistics), and in the light of a specific historical example of interdisciplinary research (namely cybernetics) with which artificial life shares many features. This report grew out of a workshop held at the Sixth European Conference on Artificial Life in Prague and features individual contributions from the workshop's eight speakers, plus a section designed to reflect the debates that took place during the workshop's discussion sessions. The major theme that emerged during these sessions was the identity and status of artificial life as a scientific endeavor

Editorial: Grammar in the face of diversity

The river one dips one’s toes into from one editorial to the next is never the same, as Heraclitus might have observed. Part 1 of this double issue (December, 2005) consisted of eight articles from contributors based in five countries: the United States, England, New Zealand, South Africa and Canada. Part 2 contains six articles and two teacher narratives from the United States (two), Scotland, the Netherlands, Australia (2), Indonesia and Denmark. The inclusion of contributors from European countries outside of the United Kingdom is a reminder that debates over the “grammar” question are not confined to the Anglophonic world. I am grateful to Amos van Gelderen and Anette Wulff for finding time to contribute to a journal, which hitherto has addressed itself to readers in a relatively small range of (officially) English speaking constituencies. I am also grateful to Handoyo Widodo for his contribution, written in the context of English-language teaching in Indonesia

Clustering and Arnoux-Rauzy words

We characterize the clustering of a word under the Burrows-Wheeler transform in terms of the resolution of a bounded number of bispecial factors belonging to the language generated by all its powers. We use this criterion to compute, in every given Arnoux-Rauzy language on three letters, an explicit bound $K$ such that each word of length at least $K$ is not clustering; this bound is sharp for a set of Arnoux-Rauzy languages including the Tribonacci one. In the other direction, we characterize all standard Arnoux-Rauzy clustering words, and all perfectly clustering Arnoux-Rauzy words. We extend some results to episturmian languages, characterizing those which produce infinitely many clustering words, and to larger alphabets