  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 prefixes of infinite sequences
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 type result. 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 that there exists a
normal pebble-deep sequence even though there is no normal finite-state-deep
sequence. We then show that there exists a sequence which has pebble-depth
level of roughly $1/2$ and Lempel-Ziv-depth level of roughly $0$. Finally we
show the existence of a sequence which has a pebble-depth level of roughly $1$
and a pushdown-depth level of roughly $1/2$.
