Castor Ministerialis
Christian Hercher
TL;DR
Castor Ministerialis investigates a variant of Busy Beavers where the tape must be blank at both the start and the end (a so-called civil servant beaver). It develops normalization, symmetry-breaking, and halting-detection techniques to prune the enormous search space, obtaining a definitive value for $n=5$ and analyzing a six-state candidate that halts after $438{,}120$ steps. The work also extends the framework to larger alphabets, providing partial maxima and a comparative table that shows alphabet size can have a larger impact than simply adding states. Overall, the study advances understanding of computation under fully blank final-state constraints and offers practical methods for exhaustive TM search in combinatorially large spaces.
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.