On the complexity of learning a language: An improvement of Block's algorithm

TL;DR

This paper analyzes the complexity of language learning algorithms, showing that an existing method can be exponentially improved from quadratic to near-linear time, significantly reducing the steps needed to learn syntax rules.

Contribution

It provides a mathematical analysis of Block's algorithm and introduces an improved algorithm with lower complexity for learning language syntax rules.

Findings

Block's algorithm has quadratic complexity in the number of rules.

The new algorithm reduces complexity to less than n log n.

The improved algorithm significantly decreases the number of steps for language learning.

Abstract

Language learning is thought to be a highly complex process. One of the hurdles in learning a language is to learn the rules of syntax of the language. Rules of syntax are often ordered in that before one rule can applied one must apply another. It has been thought that to learn the order of n rules one must go through all n! permutations. Thus to learn the order of 27 rules would require 27! steps or 1.08889x10^{28} steps. This number is much greater than the number of seconds since the beginning of the universe! In an insightful analysis the linguist Block ([Block 86], pp. 62-63, p.238) showed that with the assumption of transitivity this vast number of learning steps reduces to a mere 377 steps. We present a mathematical analysis of the complexity of Block's algorithm. The algorithm has a complexity of order n^2 given n rules. In addition, we improve Block's results exponentially, by introducing an algorithm that has complexity of order less than n log n.