Learning unary automata

• Authors: Gregor Gramlich;Ralf Herrmann
• Venue: Journal of Automata, Languages and Combinatorics
• Year: 2007

We determine the complexity of learning problems for unary regular languages. We begin by investigating the minimum consistent dfa (resp. nfa) problem which is known not to be efficiently approximable within any polynomial, unless P = NP. For the case of unary dfa's, we exhibit an efficient algorithm. On the other hand, we show the intractability of the unary minimum consistent nfa problem but provide an efficient quadratic approximation for its optimization version. The VC dimension for the class of languages accepted by unary dfa's with at most n states is computed as n+logn±Θ(log logn), which (together with the efficient solution for the consistency problem) yields an efficient PAC learning algorithm for this class. We also show that there are no efficient PAC learning algorithms for the class of languages accepted by unary nfa's with at most n states, unless every problem in NP is solvable by a quasipolynomial time Monte-Carlo algorithm. Here we assume that nfa's with few states serve as hypotheses. In the context of learning with equivalence queries, we consider the number of counterexamples required to learn a unary regular language that is accepted by a dfa with at most n states. When submitting dfa's with at most nd states (d≤n) as queries, we show the upper bound O(n2/d) and the lower bound Ω(n2.In d)/(d.(In n)2)). If only prime cycle lengths ≤ n are allowed as queries, we prove that Θ(n2/In n) counterexamples are necessary and sufficient.