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, ie 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 significantly, 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

/pdf/bounded-pushdown-dimension-vs-lempel-ziv-information-density-1og9f33q4a.pdf

Bounded Pushdown Dimension vs Lempel Ziv Information Density

The pressing need for eficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation The celebrated Lempel-Ziv algorithm LZ78 [10] was introduced as a general purpose compression algorithm that outperforms finite-state compressors on all sequences We compare the performance of the Lempel-Ziv algorithm with that of the pushdown compressors, or compression algorithms that can be implemented with a pushdown transducer This comparison is made without any a priori assumption on the data's source and considering the asymptotic compression ratio for infinite sequences We prove that Lempel-Ziv is incomparable with pushdown compressors

Pushdown Compression

This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon and Moser, notions of finite-state-depth and pushdown-depth were examined and compared. These were based on finite-state transducers and information lossless pushdown compressors respectively. Unfortunately a full separation between the two notions was not established. This paper introduces a new formulation of pushdown-depth based on restricting how fast a pushdown compressor's stack can grow. This improved formulation allows us to do a full comparison by demonstrating the existence of sequences with high finite-state-depth and low pushdown-depth, and vice-versa. A new notion based on the Lempel-Ziv `78 algorithm is also introduced. Its difference from finite-state-depth is shown by demonstrating the existence of a Lempel-Ziv deep sequence that is not finite-state deep and vice versa. Lempel-Ziv-depth's difference from pushdown-depth is shown by building sequences that have a pushdown-depth of roughly 1/2 but low Lempel-Ziv depth, and a sequence with high Lempel-Ziv depth but low pushdown-depth. Properties of all three notions are also discussed and proved.

pdf/pushdown-and-lempel-ziv-depth-s8rh4iq531.pdf

Pushdown and Lempel-Ziv Depth.

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 significantly, 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.

Bounded Pushdown dimension vs Lempel Ziv information density.

This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon and Moser, notions of finite-state-depth and pushdown-depth were examined and compared. These were based on finite-state transducers and information lossless pushdown compressors respectively. Unfortunately a full separation between the two notions was not established. This paper introduces a new formulation of pushdown-depth based on restricting how fast a pushdown compressor's stack can grow. This improved formulation allows us to do a full comparison by demonstrating the existence of sequences with high finite-state-depth and low pushdown-depth, and vice-versa. A new notion based on the Lempel-Ziv `78 algorithm is also introduced. Its difference from finite-state-depth is shown by demonstrating the existence of a Lempel-Ziv deep sequence that is not finite-state deep and vice versa. Lempel-Ziv-depth's difference from pushdown-depth is shown by building sequences that have a pushdown-depth of roughly 1/2 but low Lempel-Ziv depth, and a sequence with high Lempel-Ziv depth but low pushdown-depth. Properties of all three notions are also discussed and proved. 

Pushdown and Lempel-Ziv depth

Pushdown dimension

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

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