The complexity of exactly learning algebraic concepts

Abstract

In this paper, we investigate the complexity of exact-learning classes of permutation groups and linear spaces over finite fields. These concept classes can be seen as special subclasses of the concept class of circuits. It turns out that in Angluin's exact-learning model these concept classes are learnable with equivalence queries but not learnable with membership queries, indeed not even exact learnable in a probabilistic sense.

More importantly, we raise the question whether the entire power of equivalence queries is needed to learn these concept classes. In order to answer this question we introduce the notion of a teaching assistant. Intuitively speaking, the exact learning model comprising of the teacher and learner, is enhanced to include an intermediate teaching assistant with which the learning algorithm communicates. Building on ideas from complexity theory, we quantify the complexity of teaching assistants for learning permutation groups and linear spaces. As our main theorems, we show:

• The representation class Sym of permutation groups are exactly learnable with an LWPP-assistant.

• The class of linear spaces over finite fields are exactly learnable with an SPP-assistant.

Since LWPP and SPP are low complexity classes, this means that these concept classes are of low complexity. As a negative result, we show that if 3-CNFs are exactly learnable with an LWPP-assistant (SPP-assistant), then NP $\subseteq$LWPP (respectively NP $\subseteq$SPP).