Descriptive complexity of computable sequences revisited Abstract   The purpose of this paper is to answer two questions left open in Durand et al. (2001)  2 . Namely, we consider the following two complexities of an infinite computable 01sequence xcexb1:      C      0    xe2x80xb2      (  xcexb1  )   , defined as the minimal length of a program with oracle 0xe2x80xb2 that prints xcexb1, and      M    xe2x88x9e    (  xcexb1  )   , defined as      lim      sup      C  (    xcexb1    1  :  n      n  )   , where      xcexb1    1  :  n      denotes the lengthn prefix of xcexb1 and    C  (  x    y  )    stands for conditional Kolmogorov complexity. We show that      C      0    xe2x80xb2      (  xcexb1  )  xe2xa9xbd    M    xe2x88x9e    (  xcexb1  )    O  (  1  )    and      M    xe2x88x9e    (  xcexb1  )    is not bounded by any computable function of      C      0    xe2x80xb2      (  xcexb1  )   , even on the domain of computable sequences.

Quantitative continuity and computable analysis in Coq We give a number of formal proofs of theorems from the field of computable analysis. Many of our results specify executable algorithms that work on infinite inputs by means of operating on finite approximations and are proven correct in the sense of computable analysis. The development is done in the proof assistant Coq and heavily relies on the Incone library for information theoretic continuity. This library is developed by one of the authors and the paper can be used as an introduction to the library as it describes many of its most important features in detail. While the ability to have full executability in a formal development of mathematical statements about real numbers and the like is not a feature that is unique to the Incone library, its original contribution is to adhere to the conventions of computable analysis to provide a general purpose interface for algorithmic reasoning on continuous structures. The results that provide complete computational content include that the algebraic operations and the efficient limit operator on the reals are computable, that certain countably infinite products are isomorphic to spaces of functions, compatibility of the enumeration representation of subsets of natural numbers with the abstract definition of the space of open subsets of the natural numbers, and that continuous realizability implies sequential continuity. We also formalize proofs of noncomputational results that support the correctness of our definitions. These include that the information theoretic notion of continuity used in the library is equivalent to the metric notion of continuity on Baire space, a complete comparison of the different concepts of continuity that arise from metric and representedspace structures and the discontinuity of the unrestricted limit operator on the real numbers and the task of selecting an element of a closed subset of the natural numbers.