Learning from Positive and Negative Examples: Dichotomies and Parameterized Algorithms

Jonas Lingg, Mateus de Oliveira Oliveira, Petra Wolf

Abstract: We take a closer look on the complexity landscape of one of the most fundamental and well-studied problems in computational learning theory: the problem of learning a finite automaton A consistent with a set $$P$$ of positive examples and with a set $$N$$ of negative examples. By consistency, we mean that A accepts all strings in $$P$$ and rejects all strings in $$N$$ . It is well known that this problem is NP-hard when parameterized only by the number of states of the automaton. Therefore, our analysis takes a more refined parameterization: we consider the number k of states in A, the size $$|\varSigma |$$ of the alphabet, the maximum size l of a string in $$P\cup N$$ , and the number $$c=|P\cup N|$$ of strings in both sets. First, we prove several Pvs. NP-hard dichotomy results for these parameters when the learned automaton is drawn from different classes of finite automata. One of our dichotomy results closes a gap for the general DFA consistency problem, as here, for fixed alphabet size, the NP-hardness proofs in the literature have some issues. Interestingly, our NP-hardness results hold even for severely restricted classes of automata, such as partially-ordered automata and permutation automata. On the other hand, we provide parameterized algorithms for several combinations of parameters and show that most of them are optimal under the exponential time hypothesis.