## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 42, Number 3 (1971), 1075-1078.

### On Memory Saved by Randomization

Martin E. Hellman and Thomas M. Cover

#### Abstract

It is known that deterministic automata are generally not optimal in the problem of learning with finite memory. It is natural to ask how much memory is saved by randomization. In this note it is shown that the memory saving is arbitrarily large in the sense that for any memory size $m < \infty$, and $\delta > 0$, there exist problems such that all $m$-state deterministic algorithms have probability of error $P(e) \geqq \frac{1}{2} - \delta$, while the optimal two-state randomized algorithm has $P(e) \leqq \delta$.

#### Article information

**Source**

Ann. Math. Statist., Volume 42, Number 3 (1971), 1075-1078.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177693334

**Digital Object Identifier**

doi:10.1214/aoms/1177693334

**Mathematical Reviews number (MathSciNet)**

MR278853

**Zentralblatt MATH identifier**

0218.62027

**JSTOR**

links.jstor.org

#### Citation

Hellman, Martin E.; Cover, Thomas M. On Memory Saved by Randomization. Ann. Math. Statist. 42 (1971), no. 3, 1075--1078. doi:10.1214/aoms/1177693334. https://projecteuclid.org/euclid.aoms/1177693334