Pushdown and Lempel-Ziv depth

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