A Learning Algorithm for Multi-dimensional Trees, or: Learning Beyond Context-Freeness

Abstract

We generalize a learning algorithm by Drewes and Högberg [1] for regular tree languages based on a learning model proposed by Angluin [2] to recognizable tree languages of arbitrarily many dimensions, so-called multi-dimensional trees. Trees over multi-dimensional tree domains have been defined by Rogers [3,4]. However, since the algorithm by Drewes and Högberg relies on classical finite state automata, these structures have to be represented in another form to make them a suitable input for the algorithm: We give a new representation for multi-dimensional trees which establishes them as a direct generalization of classical trees over a partitioned alphabet, and show that with this notation Drewes’ and Högberg’s algorithm is able to learn tree languages of arbitrarily many dimensions. Via the correspondence between trees and string languages (“yield operation”) this is equivalent to the statement that this way even some string language classes beyond context-freeness have become learnable with respect to Angluin’s learning model as well.