**Authors: Andris Ambainis, Sanjay Jain and Arun Sharma**

**Source: ***Theoretical Computer Science* Vol. **220**, No. 2,
1999, 323 - 343.

**Abstract.**
The approach of ordinal mind change complexity, introduced by Freivalds and Smith,
uses (notations for) constructive ordinals to bound the number of mind changes
made by a learning machine. This approach provides a measure of the extent to
which a learning machine has to keep revising its estimate of the number of
mind changes it will make before converging to a correct hypothesis for
languages in the class being learned. Recently, this notion, which also
yields a measure for the difficulty of learning
a class of languages, has been used to analyze the learnability of rich
concept classes.

The present paper further investigates the utility of ordinal mind change
complexity.
It is shown that for identification from both positive and negative data and
*n* >= 1, the
ordinal mind change complexity of the class of languages formed by unions of up
to *n*+1 pattern languages is only
\omega x_{O} *notn*(*n*) (where *notn*(*n*)
is a notation for *n*, \omega is a
notation for the least limit ordinal and x_{O} represents ordinal multiplication).
This result nicely extends an observation of Lange and Zeugmann that
pattern languages can be identified from both positive and negative data
with 0 mind changes.

Existence of an ordinal mind change bound for a class of learnable languages can be seen as an indication of its learning "tractability". Conditions are investigated under which a class has an ordinal mind change bound for identification from positive data. It is shown that an indexed family of languages has an ordinal mind change bound if it has finite elasticity and can be identified by a conservative machine. It is also shown that the requirement of conservative identification can be sacrificed for the purely topological requirement of M-finite thickness. Interaction between identification by monotonic strategies and existence of ordinal mind change bound is also investigated.

©Copyright 1999, Elsevier Science.