On the Representation and State Complexity of Block Languages

TL;DR

The paper addresses block languages, finite languages consisting of words of a fixed length, by introducing a bitmap representation of size $k^\\ell$ that marks membership in lexicographic order. It establishes that bitmaps can be converted to minimal DFAs in polynomial time, while constructing minimal NFAs from bitmaps is NP-complete, linking automata-theoretic complexity to bitmap descriptions. It derives maximal deterministic and nondeterministic state complexities for block languages and analyzes the operational state complexity of key Boolean and related operations, revealing both reuse of finite-language bounds and improvements in certain block-language settings. The work also provides explicit methods to move between bitmap and automaton representations, enabling efficient computation of quotients and bitmaps and offering insight into the structural differences between block and general finite languages. Overall, it creates a cohesive framework connecting bitmap encodings, minimal automata, and state-complexity questions for block languages with implications for theory and potential applications in coding and data representation.

Abstract

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 $\ell$, a block language can be represented by a bitmap of length $k^\ell$, 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). First, we show how to convert bitmaps into deterministic and nondeterministic finite automata, and we prove that the machines are minimal. Then, we give an analysis of the maximum number of states sufficient to accept every block language in the deterministic and nondeterministic case. Finally, we study the deterministic and nondeterministic state complexity of several operations on these languages. Being a subclass of finite languages, the upper bounds of operational state complexity known for finite languages apply for block languages as well. However, in several cases, smaller values were found.