Learning Weighted Automata over Number Rings, Concretely and Categorically

TL;DR

The paper addresses exact learning of $R$-valued automata, including over number rings, by merging category-theoretic learning with computational algebraic number theory. It develops a generic reduction framework that transfers learning problems across categorical settings and specializes it to $\mathcal{O}_{K}$-weighted automata, yielding a polynomial-time exact learning algorithm under a full representation of $\mathcal{O}_{K}$ and an almost-minimal state bound. The key contributions include a concrete reduction from $K$-weighted to $\mathcal{O}_{K}$-weighted automata, the use of pseudo-bases and pseudo-HNF to manage modules over Dedekind domains, and a complexity analysis linking runtime to the field degree $d$ and discriminant $\Delta_{K}$. A fundamental implication is that state-minimal learning over $\mathcal{O}_{K}$ ties to the principal ideal problem, indicating deep connections between automata learning and algebraic number theory with potential quantum-speedups. Overall, the work advances efficient learning in arithmetic settings and strengthens the bridge between learning theory, category theory, and computational number theory.

Abstract

We develop a generic reduction procedure for active learning problems. Our approach is inspired by a recent polynomial-time reduction of the exact learning problem for weighted automata over integers to that for weighted automata over rationals (Buna-Marginean et al. 2024). Our procedure improves the efficiency of a category-theoretic automata learning algorithm, and poses new questions about the complexity of its implementation when instantiated to concrete categories. As our second main contribution, we address these complexity aspects in the concrete setting of learning weighted automata over number rings, that is, rings of integers in an algebraic number field. Assuming a full representation of a number ring OK, we obtain an exact learning algorithm of OK-weighted automata that runs in polynomial time in the size of the target automaton, the logarithm of the length of the longest counterexample, the degree of the number field, and the logarithm of its discriminant. Our algorithm produces an automaton that has at most one more state than the minimal one, and we prove that doing better requires solving the principal ideal problem, for which the best currently known algorithm is in quantum polynomial time.

Figures (2)

• Figure 1: Two equivalent $\mathbb{Q}\mathopen{}\left(i\sqrt{5}\right)\mathclose{}$-weighted automata.
• Figure 2: An $\mathcal{O}_{K}$-WA encoding the ideal $\mathfrak{a}$.

Theorems & Definitions (81)

• Example 0
• Theorem 1
• Definition 2
• Definition 3
• Remark 4
• Definition 6
• Proposition 8
• Lemma 9
• Remark 10
• Proposition 10