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. © 2012 Elsevier B.V. All rights reserved.
Balle, B., Castro, J., & Gavaldà, R. (2013). Learning probabilistic automata: A study in state distinguishability. In Theoretical Computer Science (Vol. 473, pp. 46–60). https://doi.org/10.1016/j.tcs.2012.10.009