Constructing Concise Characteristic Samples for Acceptors of Omega Regular Languages

A characteristic sample for a language $L$ and a learning algorithm $\textbf{L}$ is a finite sample of words $T_L$ labeled by their membership in $L$ such that for any sample $T \supseteq T_L$ consistent with $L$, on input $T$ the learning algorithm $\textbf{L}$ returns a hypothesis equivalent to $L$. Which $\omega$-automata have characteristic sets of polynomial size, and can these sets be constructed in polynomial time? We address these questions here. In brief, non-deterministic omega-automata of any of the common types, in particular B\"uchi, do not have polynomial-size characteristic samples. Families of DFAs (FDFAs) possess polynomial-sized characteristic samples that can be constructed in polynomial time. Since deterministic omega automata of types B\"{u}chi, co-B\"{u}chi, and parity can be polynomially translated to FDFAs it follows that they too possess characteristic samples that can be constructed in polynomial time; however, no corresponding polynomial time learning algorithms is known. For deterministic omega automata that are isomorphic to their right congruence automata, the fully informative languages, polynomial time algorithms for constructing characteristic samples and learning from them is given. The algorithms for constructing characteristic sets in polynomial time for the different omega automata (of types B\"uchi, coB\"uchi, parity, Rabin, Street, or Muller), require deterministic polynomial time algorithms for (1) equivalence of the respective omega automata, and (2) testing membership of the language of the automaton in the informative classes, which we provide.