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.