Computability of the Zero-Error Capacity with Kolmogorov Oracle

Abstract

The zero-error capacity of a discrete classical channel was first defined by Shannon as the least upper bound of rates for which one transmits information with zero probability of error. The problem of finding the zero-error capacity C 0 , which assigns a capacity to each channel as a function, was reformulated in terms of graph theory as a function Θ, which assigns a value to each simple graph. This paper studies the computability of the zero-error capacity. For the computability, the concept of a Turing machine and a Kolmogorov oracle is used. It is unknown if the zero-error capacity is computable in general. We show that in general the zero-error capacity is semi-computable with the help of a Kolmogorov Oracle. Furthermore, we show that C 0 and Θ are computable functions if and only if there is a computable sequence of computable functions of upper bounds, i.e. the converse exist in the sense of information theory, which point-wise converges to C 0 or Θ. Finally, we examine Zuiddam’s characterization of C 0 and Θ in terms of algorithmic computability.