Known algorithms for learning PDFA can only be shown to run in time polynomial in the so-called distinguishability μ of the target machine, besides the number of states and the usual accuracy and confidence parameters. We show that the dependence on μ is necessary in the worst case for every algorithm whose structure resembles existing ones. As a technical tool, a new variant of Statistical Queries termed L_∞-queries is defined. We show how to simulate L_∞-queries using classical Statistical Queries and show that known PAC algorithms for learning PDFA are in fact statistical query algorithms. Our results include a lower bound: every algorithm to learn PDFA with queries using a reasonable tolerance must make Ω(1/μ^1−c) queries for every c>0. Finally, an adaptive algorithm that PAC-learns w.r.t. another measure of complexity is described. This yields better efficiency in many cases, while retaining the same inevitable worst-case behavior. Our algorithm requires fewer input parameters than previously existing ones, and has a better sample bound.