DeepAI AI Chat
Log In Sign Up

Plain stopping time and conditional complexities revisited

by   Mikhail Andreev, et al.

In this paper we analyze the notion of "stopping time complexity", informally defined as the amount of information needed to specify when to stop while reading an infinite sequence. This notion was introduced by Vovk and Pavlovic (2016). It turns out that plain stopping time complexity of a binary string x could be equivalently defined as (a) the minimal plain complexity of a Turing machine that stops after reading x on a one-directional input tape; (b) the minimal plain complexity of an algorithm that enumerates a prefix-free set containing x; (c) the conditional complexity C(x|x*) where x in the condition is understood as a prefix of an infinite binary sequence while the first x is understood as a terminated binary string; (d) as a minimal upper semicomputable function K such that each binary sequence has at most 2^n prefixes z such that K(z)<n; (e) as C^X(x) where C^X(z) is plain Kolmogorov complexity of z relative to oracle X and the maximum is taken over all extensions X of x. We also show that some of these equivalent definitions become non-equivalent in the more general setting where the condition y and the object x may differ. We also answer an open question from Chernov, Hutter and Schmidhuber.


page 1

page 2

page 3

page 4


A New Algebraic Approach for String Reconstruction from Substring Compositions

We consider the problem of binary string reconstruction from the multise...

NYTRO: When Subsampling Meets Early Stopping

Early stopping is a well known approach to reduce the time complexity fo...

On information content in certain objects

The fine approach to measure information dependence is based on the tota...

Verifying Time Complexity of Binary Search using Dafny

Formal software verification techniques are widely used to specify and p...

Approximations of Kolmogorov Complexity

In this paper we show that the approximating the Kolmogorov complexity o...

Information Kernels

Given a set X of finite strings, one interesting question to ask is whet...