Polylog depth, highness and lowness for E

Moser, Philippe

doi:10.1016/j.ic.2019.104483

Cited by 6 publications

(2 citation statements)

References 15 publications

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“…Due to the uncomputabilty of Kolmogorov complexity, several researchers have attempted to adapt Bennett's notion to lower complexity levels. While variations have been based on computable notions [14], more feasible notions based on polynomial time computations [1,17,18] have been studied, including both finite-state transducers and lossless pushdown compressors [7,12]. Similarly to randomness, there is no absolute notion of logical depth.…”

Section: Introductionmentioning

confidence: 99%

Pushdown and Lempel-Ziv Depth

Jordon¹,

Moser²

2020

Preprint

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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 finitestate 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.

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Section: Introductionmentioning

confidence: 99%

Pushdown and Lempel-Ziv Depth

Jordon¹,

Moser²

2020

Preprint

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“…While Bennett's original notion was based on uncomputable Kolmogorov complexity and interacts nicely with several aspects of computability theory [10,25], researchers have studied more feasible notions at lower complexity levels. These include computable notions [18], notions based on polynomial time computations [1,23,24], and notions based on classes of transducers [9,16].…”

Section: Introductionmentioning

confidence: 99%

Pebble-Depth

Jordon¹,

Moser²

2020

Preprint

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In this paper we introduce a new formulation of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of strings from the perspective of finite-state transducers and pebble transducers. Our notion of pebble-depth satisfies the three fundamental properties of depth: i.e. easy sequences and random sequences are not deep, and the existence of a slow growth law. We also compare pebble-depth to other depth notions based on finite-state transducers, pushdown compressors and the Lempel-Ziv 78 compression algorithm. We first demonstrate how there exists a normal pebble-deep sequence even though there is no normal finite-statedeep sequence. We next build a sequence which has a pebble-depth level of roughly 1, a pushdown-depth level of roughly 1/2 and a finite-state-depth level of roughly 0. We then build a sequence which has pebble-depth level of roughly 1/2 and Lempel-Ziv-depth level of roughly 0.

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On the Difference Between Finite-State and Pushdown Depth

Jordon¹,

Moser²

2020

SOFSEM 2020: Theory and Practice of Computer Science

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Polylog depth, highness and lowness for E

Cited by 6 publications

References 15 publications

Pushdown and Lempel-Ziv Depth

Pushdown and Lempel-Ziv Depth

Pebble-Depth

On the Difference Between Finite-State and Pushdown Depth

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