On the Difference Between Finite-State and Pushdown Depth
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.
KeywordsAlgorithmic information theory Kolmogorov complexity Bennett’s logical depth
The authors would like to thank the anonymous referees for their useful comments, specifically to explore how the chosen binary representations of FSTs affects FS-depth.
- 2.Bennett, C.H.: Logical depth and physical complexity. In: Bennett, C. (ed.) The Universal Turing Machine, A Half-Century Survey, pp. 227–257. Oxford University Press, New York (1988)Google Scholar
- 10.Moser, P.: Polynomial depth, highness and lowness for E. Inf. Comput. (2019, accepted) Google Scholar
- 12.Moser, P., Stephan, F.: Depth, highness and DNR degrees. Discrete Math. Theor. Comput. Sci. 19(4) (2017) Google Scholar