Bounded Pushdown dimension vs Lempel Ziv information density

TL;DR

This paper introduces bounded pushdown (BPD) dimension, a new measure of information density in sequences using automata with bounded stack movements, and compares it to Lempel-Ziv compression, revealing significant differences.

Contribution

It defines BPD dimension, establishes its equivalence with BPD compressors, and demonstrates that LZ compression is not universal for BPD compressors, highlighting a separation between finite-state and BPD dimensions.

Findings

BPD dimension is robust and equivalent to BPD compressors.

LZ compression fails on certain sequences that BPD compressors can significantly compress.

There is a strong separation between finite-state and BPD dimension.

Abstract

In this paper we introduce a variant of pushdown dimension called bounded pushdown (BPD) dimension, that measures the density of information contained in a sequence, relative to a BPD automata, i.e. a finite state machine equipped with an extra infinite memory stack, with the additional requirement that every input symbol only allows a bounded number of stack movements. BPD automata are a natural real-time restriction of pushdown automata. We show that BPD dimension is a robust notion by giving an equivalent characterization of BPD dimension in terms of BPD compressors. We then study the relationships between BPD compression, and the standard Lempel-Ziv (LZ) compression algorithm, and show that in contrast to the finite-state compressor case, LZ is not universal for bounded pushdown compressors in a strong sense: we construct a sequence that LZ fails to compress signicantly, but that is compressed by at least a factor 2 by a BPD compressor. As a corollary we obtain a strong separation between finite-state and BPD dimension.