Query Learning of Advice and Nominal Automata
Kevin Zhou
TL;DR
The paper addresses query-learning of automata for two extensions of DFAs: advice DFAs and nominal DFAs. It adopts the Chase & Freitag dimension-based framework, leveraging Littlestone dimension and consistency dimension to obtain upper bounds on EQ+MQ query complexity. For advice DFAs, it proves the first known upper bound $O(n^3 m k \log n)$ for languages over an alphabet of size $k$ restricted to length-$m$ strings. For nominal DFAs, it derives a bound $O(n^{O(k)}/k^k)$ for fixed $G$-alphabet $A$, improving previous factorial-type dependencies and removing reliance on the longest counterexample, by analyzing nominal sets, orbits, and product structures. The work also outlines potential generalizations to other symmetries and to $\,\omega$-languages, offering structural insights even without explicit algorithmic guarantees.
Abstract
Learning automata by queries is a long-studied area initiated by Angluin in 1987 with the introduction of the $L^*$ algorithm to learn regular languages, with a large body of work afterwards on many different variations and generalizations of DFAs. Recently, Chase and Freitag introduced a novel approach to proving query learning bounds by computing combinatorial complexity measures for the classes in question, which they applied to the setting of DFAs to obtain qualitatively different results compared to the $L^*$ algorithm. Using this approach, we prove new query learning bounds for two generalizations of DFAs. The first setting is that of advice DFAs, which are DFAs augmented with an advice string that informs the DFA's transition behavior at each step. For advice DFAs, we give the first known upper bounds for query complexity. The second setting is that of nominal DFAs, which generalize DFAs to infinite alphabets which admit some structure via symmetries. For nominal DFAs, we make qualitative improvements over prior results.