Learning Expressions over Monoids

Abstract

We study the problem of learning an unknown function rep- resented as an expression over a known finite monoid. As in other areas of computational complexity where programs over algebras have been used, the goal is to relate the computational complexity of the learning problem with the algebraic complexity of the finite monoid. Indeed, our results indicate a close connection between both kinds of complexity. We focus on monoids which are either groups or aperiodic, and on the learn- ing model of exact learning from queries. For a group G, we prove that expressions over G are easily learnable if G is nilpotent and impossible to learn efficiently (under cryptographic assumptions) if G is nonsolvable. We present some partial results for solvable groups, and point out a connection between their efficient learnability and the existence of lower bounds on their computational power in the program model. For aperi- odic monoids, our results seem to indicate that the monoid class known as DA captures exactly learnability of expressions by polynomially many Evaluation queries.

Partially supported by the EC ESPRIT Working group NeuroCOLT2, the EC ESPRIT AlcomFT project, and by FRESCO PB98-0937-C04-04. Part of this research was done while this author visited McGill University, supported by NSERC and FCAR.

Partially supported by NSERC and FCAR grants and by the von Humboldt Foundation.