Kolmogorov complexities Kmax, Kmin on computable partially ordered sets (0801.0351v1)

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.