On the Determinization Blowup for Finite Automata Recognizing Equal-Length Languages

Abstract

Motivated by the application to image compression (K. Čulík II, J. Kari, “Image compression using weighted finite automata”, Computers & Graphics, 1993), the paper considers finite automata representing formal languages with all strings of the same length, and investigates relative succinctness of representation by deterministic and nondeterministic finite automata (DFA, NFA). It is shown that an $n$-state NFA recognizing a language of strings of length $\ell$ over a $k$-symbol alphabet can be transformed to a DFA with at most $\ell \cdot k^{\sqrt{\frac{2}{\log _2 k}n + 3\ell + 3}} = 2^{O(\sqrt{n})}$ states. At the same time, for every $k$-symbol alphabet with $k \geqslant 2$, and for every $n \geqslant 1$, there exists an $n$-state NFA recognizing an equal-length language, which requires a DFA with at least $k^{\sqrt{\frac{n}{k-1}} - 2} = 2^{\Omega (\sqrt{n})}$ states.

Supported by the Academy of Finland under grant 257857.