Busy Beavers and Kolmogorov Complexity
نویسنده
چکیده
The idea to find the “maximal number that can be named” can be traced back to Archimedes (see his Psammit [1]). From the viewpoint of computation theory the natural question is “which number can be described by at most n bits”? This question led to the definition of the so-called “busy beaver” numbers (introduced by T. Rado). In this note we consider different versions of the busy beaver-like notions defined in terms of Kolmogorov complexity. We show that these versions differ depending on the version of complexity used (plain, prefix, or a priori complexities) and find out how these notions are related, providing matching lower and upper bounds.
منابع مشابه
Busy beavers gone wild
When considering a class of Turing machines that halt when started from a blank tape, busy beavers for that class, as coined by Tibor Radó, are those Turing machines which eventually halt after the maximum number of steps (or after producing the maximum number of non-blank symbols). Finding for a class of Turing machines the busy beaver champion (for any of the two competitions) is of course un...
متن کاملAttacking the Busy Beaver 5
Since T. Rado in 1962 defined the busy beaver game several approaches have used computer technology to search busy beavers or even compute special values of Σ. Σ(5) is not yet known. A new approach to the computation of Σ(5) is presented, together with preliminary results, especially Σ(5)≥4098. This approach includes techniques to reduce the number of inspected Turing machines, to accelerate si...
متن کاملOptimal probabilistic polynomial time compression and the Slepian-Wolf theorem: tighter version and simple proofs
We give simplify the proofs of the 2 results in Marius Zimand’s paper Kolmogorov complexity version of Slepian-Wolf coding, proceedings of STOC 2017, p22–32. The first is a universal polynomial time compression algorithm: on input ε > 0, a number k and a string x, computes in polynomial time with probability 1 − ε a program of length k+O(log(|x|/ε)) that outputs x, provided that there exists su...
متن کاملCalculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines
Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths...
متن کاملOn the logical depth function
We investigate the logical depth function of finite strings. For the function associated with string x with argument b and value d we have that d is the least running time of the computation of an element of the set of b-incompressible programs for x. For a given string we consider the possible gap between two values of this function if the arguments differ by just a constant. We show that ther...
متن کامل