Block Languages and Their Bitmap Representations

In this paper we consider block languages, namely sets of words having the same length, and we propose a new representation for these languages. In particular, given an alphabet of size k and a length l, a block language can be represented by a bitmap of length k(l), where each bit indicates whether the corresponding word, according to the lexicographical order, belongs, or not, to the language (bit equal to 1 or 0, respectively). This representation turns out to be a good tool for the investigation of several properties of block languages, making proofs simpler and reasoning clearer. First, we show how to convert bitmaps into deterministic and nondeterministic finite automata. We then focus on the size of the machines obtained from the conversion and we prove that their size is minimal. Finally, we give an analysis of the maximum number of states sufficient to accept every block language in the deterministic and nondeterministic case.