A two-dimensional antidictionary automaton for a toric surface

The antidictionary is the set of all minimal forbidden words (MFWs) which never appear in a given input word, and it is useful for data compression, sequence comparison, and so on. In the one-dimensional case, the antidictionaries of periodic sequences are well studied; indeed, a tight upper bound on the length of an MFW and an antidictionary automaton from the antidictionary which can accept any subword of the periodic sequence are presented. On the other hand, in the two-dimensional case, for a 2D array of symbols on a toric surface, which corresponds to a periodic sequence in the one-dimensional case, tight upper bounds on the height (width) of an MFW have not known. In this paper, we provide a tight upper bound on the height (width) of an MFW for the array. Moreover, we present an antidictionary automaton which can accept any subword of the array.

I. Introduction

Given an input word, a minimal forbidden word (MFW) is a word that never appear in the input word while all of its proper subwords appear in the input word. The antidictionary is the set of all MFWs for the input word [1], [2]. Antidictionaries have been applied to universal two-pass antidictionary coding algorithms [3], [4] and a method of sequence comparison using antidictionaries [5]. The coding algorithms encode a periodic sequence, which is obtained by concatenating the first symbol to the last symbol of an input word, instead of input word. For a periodic sequence of an input word, a tight upper bound on the length of an MFW is shown [3]–[5]. In [3], [4], an antidictionary automaton (AD-automaton) which can accept any subword of the periodic sequence is presented. Moreover, techniques for encoding the antidictionary of a periodic sequence have been proposed. The techniques are based on a technique of a lossless data compression algorithm, called compression via subword enumeration [6], [7].