Query learning of derived $\omega$-tree languages in polynomial time

TL;DR

This paper introduces the first polynomial time algorithm for learning certain classes of infinite tree languages derived from weak regular omega-word languages, using a reduction to known omega-word learning algorithms.

Contribution

It provides a novel polynomial time reduction from learning derived omega-tree languages to learning omega-word languages, enabling efficient learning of complex infinite tree languages.

Findings

First polynomial time algorithm for learning derived omega-tree languages

Reduction from tree languages to omega-word languages for learning

Polynomial time implementation of subset queries for various acceptors

Abstract

We present the first polynomial time algorithm to learn nontrivial classes of languages of infinite trees. Specifically, our algorithm uses membership and equivalence queries to learn classes of $\omega$-tree languages derived from weak regular $\omega$-word languages in polynomial time. The method is a general polynomial time reduction of learning a class of derived $\omega$-tree languages to learning the underlying class of $\omega$-word languages, for any class of $\omega$-word languages recognized by a deterministic B\"{u}chi acceptor. Our reduction, combined with the polynomial time learning algorithm of Maler and Pnueli [1995] for the class of weak regular $\omega$-word languages yields the main result. We also show that subset queries that return counterexamples can be implemented in polynomial time using subset queries that return no counterexamples for deterministic or non-deterministic finite word acceptors, and deterministic or non-deterministic B\"{u}chi $\omega$-word acceptors. A previous claim of an algorithm to learn regular $\omega$-trees due to Jayasrirani, Begam and Thomas [2008] is unfortunately incorrect, as shown in Angluin [2016].