Degrees of monotone complexity

Levin and Schnorr (independently) introduced the monotone complexity, Km(), of a binary string . We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering Km on 2 by Km iff there is a constant c such that Km( n) Km( n) + c for all n. The monotone degree of is the set of all such that Km and Km . We show the monotone degrees contain an antichain of size 20 , a countable dense linear ordering (of degrees of cardinality 20 ), and a minimal pair. Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the Kdegrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalb

• Type: Journal
• Year: 2006
• Where: JSYML
• Authors: William C. Calhoun
• Keywords: Countable Dense Linear, JSYML 2006, Monotone Complexity, Monotone Degrees