On the Difference Between Finite-State and Pushdown Depth

Abstract

This paper expands upon existing and introduces new formulations of Bennett’s logical depth. A new notion based on pushdown compressors is developed. A pushdown deep sequence is constructed. The separation of (previously published) finite-state based and pushdown based depth is shown. The previously published finite state depth notion is extended to an almost everywhere (a.e.) version. An a.e. finite-state deep sequence is shown to exist along with a sequence that is infinitely often (i.o.) but not a.e. finite-state deep. For both finite-state and pushdown, easy and random sequences with respect to each notion are shown to be non-deep, and that a slow growth law holds for pushdown depth.