Castor Ministerialis

TL;DR

This paper explores a variant of the Busy Beavers problem, focusing on Turing machines that start and remain on the blank tape, providing definitive bounds for machines with up to five states and analyzing candidates with six states.

Contribution

It introduces a new variant of the Busy Beavers problem and offers definitive results for machines with up to five states, including behavior analysis of six-state candidates.

Findings

Bounded the runtime of machines with up to five states under the new variant.

Analyzed the behavior of a six-states candidate.

Provided insights into generalizations for m-symbol alphabets.

Abstract

The famous problem of Busy Beavers can be stated as the question on how long a $n$-state Turing machine (using a 2-symbol alphabet or -- in a generalization -- a $m$-symbol alphabet) can run if it is started on the blank tape before it holds. Thus, not halting Turing machines are excluded. For up to four states the answer to this question is well-known. Recently, it could be verified that the widely assumed candidate for five states is in fact the champion. And there is progress in searching for good candidates with six or more states. We investigate a variant of this problem: Additionally to the requirement that the Turing machines have to start from the blank tape we only consider such Turing machines that hold on the blank tape, too. For this variant we give definitive answers on how long such a Turing machine with up to five states can run, analyze the behavior of a six-states candidate and give some findings on the generalization of Turing-machines with $m$-symbol alphabet.