Journal of Symbolic Logic

Degrees of monotone complexity

William C. Calhoun

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Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α ↾ n) ≤ Km(β ↾ n)+c for all n. The monotone degree of α is the set of all β such that α ≤Km β and β ≤Km α. We show the monotone degrees contain an antichain of size 2ℵ₀, a countable dense linear ordering (of degrees of cardinality 2ℵ₀), and a minimal pair.

Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalbán. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We define a real to be (Km,K)-trivial if for some constant c, Km(α ↾ n) ≤ K(n)+c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.

Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

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J. Symbolic Logic, Volume 71, Issue 4 (2006), 1327-1341.

First available in Project Euclid: 20 November 2006

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Calhoun, William C. Degrees of monotone complexity. J. Symbolic Logic 71 (2006), no. 4, 1327--1341. doi:10.2178/jsl/1164060458.

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