# Learnability of exclusive-or expansion based on monotone DNF formulas

## Abstract

The learnability of the class of exclusive-or expansion 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 an 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 bounded depth, i.e., formulas represented as exclusive-or of a constant number of monotone DNF formulas. In spite of seemingly strong restriction of the depth being constant, the class of formulas of bounded 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.

## Preview

Unable to display preview. Download preview PDF.

## References

- [Ang88]Dana Angluin. Queries and concept learning.
*Machine Learning*, 2:319–342, 1988.Google Scholar - [BCKT94]Nader H. Bshouty, Richard Cleve, Sampath Kannan, and Christino Tamon. Oracles and queries that are sufficient for exact learning. In
*Proceedings of the 7th Workshop on Computational Learning Theory*, pages 130–139. Association for Computing Machinery, 1994.Google Scholar - [Bsh93]Nader H. Bshouty. Exact learning via the monotone theory. In
*Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science*, pages 302–311. IEEE, 1993.Google Scholar - [Bsh95]Nader H. Bshouty. Simple learning algorithm using divide and conquer. In
*Proceedings of the 8th Workshop on Computational Learning Theory*, pages 447–453. Association for Computing Machinery, 1995.Google Scholar - [HSW89]David Helmbold, Robert Sloan, and Manfred Warmuth. Learning nested differences of intersection-closed concept classes. In
*Proceedings of the 2nd Workshop on Computational Learning Theory*, pages 41–56, 1989.Google Scholar - [Mar58]A. A. Markov. On the Inversion Complexity of a System of Functions.
*J. ACM*, 5, pages 331–334, 1958.CrossRefGoogle Scholar - [SM94]Yoshifumi Sakai and Akira Maruoka. Learning monotone log-term DNF formulas. In
*Proceedings of the 7th Workshop on Computational Learning Theory*, pages 165–172, 1994.Google Scholar - [STM95]Yoshifumi Sakai, Eiji Takimoto, and Akira Maruoka. Proper learning algorithm for functions of
*k*terms under smooth distributions. In*Proceedings of the 8th Workshop on Computational Learning Theory*, pages 206–213. ACM, 1995.Google Scholar