Triviality and Minimality in the Degrees of Monotone Complexity

Monotone complexity, Km, is a variant of Kolmogorov complexity that was introduced independently by Levin and Schnorr. The relative randomness of reals may be defined via monotone complexity. Equivalence classes of reals under monotone complexity are the Km-degrees, similar to the K-degrees defined via prefix-free complexity. A real α is Km-trivial if Km(α ↾ n) Km(n). Here, an argument by Stephan is strengthened to show that each Turing degree d ≥ 0′ contains a Km-trivial real. In contrast, all K-trivial reals are Δ02 (Chaitin) and low, since they are low for random (Hirschfeldt and Nies). A non-decreasing, function f:ω → ω is defined to be computably infinitesimal if it is dominated by every computable, non-decreasing, unbounded function. It is shown that the monotone complexity of any Km-trivial real is computably infinitesimal. If a Km-minimal real exists, it is Km-trivial. The operation ⊗ horizontally stretches the complexity graph of a real α by a strictly increasing computable function f. Areal α is invariant under computable stretching if α ⊗f ≡ Kmα for any such f. It is shown that any Km-minimal real is invariant under computable stretching.