State Complexity of Binary Coded Regular Languages

• Published in: Descriptional Complexity of Formal Systems Vol. 13439; pp. 72 - 84
• Main Authors: Geffert, Viliam, Pališínová, Dominika, Szabari, Alexander
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2022; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: don’t care automata; finite automata; prefix codes; promise problems; state complexity
• ISBN: 9783031132568; 3031132564
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-031-13257-5_6

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