Kolmogorov complexities Kmax, Kmin on computable partially ordered sets

TL;DR

This paper develops a formal framework for Kolmogorov complexity on computable partially ordered sets, analyzing properties of complexity measures derived from maximum and minimum functions, and characterizing conditions under which classical theorems hold.

Contribution

It introduces a machine-free formalization of Kolmogorov complexity on partially ordered sets and characterizes when enumeration and invariance theorems apply for these complexity measures.

Findings

Max^{X→D}_{PR} satisfies the invariance theorem and leads to a variant of Kolmogorov complexity.

Conditions are characterized under which Max^{X→D}_{Rec} satisfies the enumeration and invariance theorems.

Complex relationships between K^D_{min}, K^D_{max}, and K^D are established, including cases of equality and incomparability.

Abstract

We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes Max^{X\to D}_{PR} and Max^{X\to D}_{Rec} of functions X \to D which are pointwise maximum of partial or total computable sequences of functions where D = (D,<) is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for Max^{X\to D}_{PR}, leading to a variant KD;max of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem (resp. the invariance theorem) also holds for Max^{X\to D}_{Rec} . It turns out that Max^{X\to D}_{Rec} may satisfy the invariance theorem but not the enumeration theorem. Also, when Max^{X\to D}_{Rec} satisfies the invariance theorem then the Kolmogorov complexities associated to Max^{X\to D}_{Rec} and Max^{X\to D}_{PR} are equal (up to a constant). Letting K^D_{min} = K^{D^{rev}}_{max}, where D^{rev} is the reverse order, we prove that either K^D_{min} =_{ct} K^D_{max} =_{ct} K^D (=_{ct} is equality up to a constant) or K^D_{min}, K^D_{max} are <=_{ct} incomparable and <_{ct} K^D and >_{ct} K^{0',D}. We characterize the orders leading to each case. We also show that K^D_{min}, K^D_{max} cannot be both much smaller than K^D at any point. These results are proved in a more general setting with two orders on D, one extending the other.