In this paper, we study active learning algorithms for weighted automata over a semiring. We show that a variant of Angluin’s seminal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathtt {L}^{\!\star }$$\end{document}L⋆ algorithm works when the semiring is a principal ideal domain, but not for general semirings such as the natural numbers.