Improved Bounds for Functions Related to Busy Beavers

  title={Improved Bounds for Functions Related to Busy Beavers},
  author={Amir M. Ben-Amram and Holger Petersen},
  journal={Theory of Computing Systems},
Abstract. Consider Turing machines that use a tape infinite in both directions, with the tape alphabet {0,1} . Rado's busy beaver function, ones(n), is the maximum number of 1's such a machine, with n states, started on a blank (all-zero) tape, may leave on its tape when it halts. The function ones(n) is non-computable; in fact, it grows faster than any computable function. Other functions with a similar nature can be defined also. All involve machines of n states, started on a blank tape. The… 
A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory
This paper presents an explicit description of a 7,910-state Turing machine Z with 1 tape and a 2-symbol alphabet that cannot be proved to run forever in ZFC (even though it presumably does), and gives the first known upper bound on the highest provable Busy Beaver number in Z FC.
Busy beaver machines and the observant otter heuristic (or how to tame dreadful dragons)
The Busy Beaver Competition: a historical survey
The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.
The Busy Beaver Frontier
  • S. Aaronson
  • Computer Science
    Electron. Colloquium Comput. Complex.
  • 2020
A personal view of the BB function 58 years after its introduction is emphasized, emphasizing lesser-known insights, recent progress, and especially favorite open problems.
Appendix A Formal ( In ) Computability and Randomness
  • Computer Science
  • 2018
A very brief survey of the consequences of Turing’s work, and a hint on the fact that almost all elements of the continuum, and, in particular, almost all reals, are incomputable, head on to modern, algorithmic, definitions of randomness, and of random reals.
Busy Beaver Scores and Alphabet Size
The Busy Beaver Game introduced by Rado (1962) generalized to non-binary alphabets is investigated and it is shown that an increase of the alphabet by a factor of three admits an increased of activity.
Bibliography on the Busy Beaver Problem
In 1981 the editor of this bibliography was attracted to the Busy Beaver Problem by a reference in the Dutch translation of Ogilvy’s Tomorrow’s Math [Ogi72]. Tracing back in time one finds of course


A lower bound on Rado's sigma function for binary Turing machines
In this note we show how to construct some simply-configured N-state binary Turing machines that will start on a blank tape and eventually halt after printing a very large number of ones. The number
Busy beaver competition and Collatz-like problems
It is proved that the halting problem for Turing machines amounts to a well-known problem of number theory, that of the behavior of the repeated iteration of Collatz-like functions, that is functions defined by cases according to congruence classes.
Attacking the Busy Beaver 5
A new approach to the computation of Σ (5) is presented, together with preliminary results, especially Σ(5)≥4098, which includes techniques to reduce the number of inspected Turing machines, to accelerate simulation of Turing machines and to decide nontermination of Turing Machines.
Computer Studies of Turing Machine Problems
This paper solves a problem relating to Turing machines arising in connection with the Busy Beaver logical game with the help of a computer program, and the values of two very well-defined positive integers are determined to b~ 6 and 21 respectively.
Formal languages and their relation to automata
  • J. Hopcroft, J. Ullman
  • Computer Science
    Addison-Wesley series in computer science and information processing
  • 1969
The theory of formal languages as a coherent theory is presented and its relationship to automata theory is made explicit, including the Turing machine and certain advanced topics in language theory.
On non-computable functions
The construction of non-computable functions used in this paper is based on the principle that a finite, non-empty set of non-negative integers has a largest element. Also, this principle is used
Graph based crossover – a case study with the Busy Beaver problem
Experimental results show that, in this domain, the new graph-based operator provides a clear advantage over two-point crossover.
Introduction to Automata Theory, Languages and Computation
An Introduction to Automata Theory
Great Aunt Eugenia and other automata Sundry machines Implementing finite automata Implementation and realization Behavioural equivalence, SP partitions and reduced machines Parallel and serial
Crossover, Macromutationand, and Population-Based Search
A simple method for testing the usefulness of crossover for a particular problem is presented, which makes it possible to identify situations in which crossover is apparently useful but is in fact producing gains that are only equal to those that can be obtained with macromutation and no population.