State Complexity of Binary Coded Regular Languages

Abstract

For the given non-unary input alphabet $\varSigma$, a maximal prefix code h mapping strings over $\varSigma$ to binary strings, and an optimal deterministic finite automaton (DFA) $\mathcal {A}$ with n states recognizing a language $\mathcal {L}$ over $\varSigma$, we consider the problem of how many states we need for an automaton $\mathcal {A}'$ that decides membership in $h(\mathcal {L})$, the binary coded version of $\mathcal {L}$. Namely, $\mathcal {A}'$ accepts binary inputs belonging to $h(\mathcal {L})$ and rejects binary inputs belonging to $h(\mathcal {L}^{\scriptscriptstyle \mathrm {C}})$, where $\mathcal {L}^{\scriptscriptstyle \mathrm {C}}$ is the complement of $\mathcal {L}$. The outcome on inputs that are not valid binary codes for any string in $\varSigma ^{*}$ can be arbitrary: $\mathcal {A}'$ may accept, reject, or halt in a “don’t care” state. We show that any optimal deterministic don’t care finite automaton (dcDFA) $\mathcal {A}'$ solving this promise problem uses at most $(\Vert {\varSigma }\Vert -1){\cdot }n$ states but at least n states. We also show that, for each non-unary input alphabet $\varSigma$, there exists a maximal binary prefix code h such that, for each $n\ge 2$ and for each N in range from n to $(\Vert {\varSigma }\Vert -1){\cdot }n$, there exists a language $\mathcal {L}$ over $\varSigma$ such that the optimal DFA recognizing $\mathcal {L}$ uses exactly n states and any optimal dcDFA for solving the above promise problem uses exactly N states. Thus, we have the complete state hierarchy for deciding membership in the binary coded version of $\mathcal {L}$, with no magic numbers in between the lower and upper bounds.

Supported by the Slovak grant contract VEGA 1/0177/21.