Pushdown dimension

TL;DR

This paper introduces the concept of pushdown dimension, showing it can be strictly lower than finite-state dimension for certain sequences, thus highlighting the greater computational power of pushdown automata.

Contribution

It establishes a formal relationship between pushdown and finite-state dimensions and demonstrates sequences where pushdown dimension is significantly lower, revealing new insights into automata complexity.

Findings

Pushdown dimension is bounded above by finite-state dimension.

Existence of sequences with pushdown dimension at most half their finite-state dimension.

Pushdown automata are strictly more powerful than finite automata in this context.

Abstract

This paper develops the theory of pushdown dimension and explores its relationship with finite-state dimension. Pushdown dimension is trivially bounded above by finite-state dimension for all sequences, since a pushdown gambler can simulate any finite-state gambler. We show that for every rational 0 < d < 1, there exists a sequence with finite-state dimension d whose pushdown dimension is at most d/2. This establishes a quantitative analogue of the well-known fact that pushdown automata decide strictly more languages than finite automata.