### The learnability of exclusive-or expansions based on
monotone DNF formulas.

**Authors: Eiji Takimoto, Yoshifumi Sakai and Akira Maruoka**.

Email: t2@eceit.tohoku.ac.jp

**Source: ***Theoretical Computer Science* Vol. 241,
No. 1-2, 2000, 37-50.

**Abstract.**
The learnability of the class of exclusive-or expansions based on
monotone DNF formulas is investigated. The class consists of the
formulas of the form
*f* = *f*_{1} ^{ ... }
*f*_{d}, where *f*_{1} > ^{ ... } > *f*_{d} are monotone DNF
formulas. It is shown that any Boolean function can be represented as a
formula in this class, and that the representation in the simplest form is
unique. Learning algorithms that learn such formulas using various
queries are presented: An algorithm with subset and superset queries and
one with membership and equivalence queries are given. The former can
learn any formula in the class, while the latter is proved to learn formulas
of constant depth, i.e., formulas represented as exclusive-or of a constant
number of monotone DNF formulas. In spite of the seemingly strong
restriction of the depth being constant, the class of formulas of constant
depth includes functions with very high complexity in terms of DNF and
CNF representations, so the latter algorithm could learn Boolean
functions significantly complex otherwise represented.

©Copyright 2000 Elsevier Science