Pushdown and Lempel-Ziv Depth
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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 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.
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