\n\n## Table of contents\n\n## Overview\n\n
\n\n
Disclaimer: This page is specific to the search of BB(5), which has been completed as of July 2nd 2024.
\n\nWith the Busy Beaver Challenge we want to decide the halting problem of all 5-state Turing machines (from all-0 tape). That way we will learn BB(5), the 5
th busy beaver value. See
Story.\n\nIn order to achieve this goal we need to analyse the behavior of every single 5-state Turing machine. We quickly run into a problem: there are roughly 16 trillion 5-state Turing machines (
to be exact).\n\nThankfully most of this space is not _useful_ to us and only a fraction needs to be studied in order to find BB(5). This is for instance because there are ways to permute the states (aside from the start state) of a machine and 2 ways to permute the move directions which does not change behavior hence only one of these 48 machines needs to be studied.\n\nHence, we aim at _sparsely_ enumerating the space of 5-state Turing machines: that is trying to enumerate the least amount of machines that are necessary to study in order to find BB(5).\n\n\n\n### Phase 1, phase 2\n\nThe method that we present to sparsely enumerate the space of 5-state Turing machines and analyse their behavior is fundamentally inspired by [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html) with some notable differences that we will outline. The first difference is that our method is divided into two successive and independent phases:\n\n1. **Phase 1: seed database.** Sparsely enumerate the space of 5-state Turing machines and mark as **undecided** any machine that exceeded the set [time or space limits](#time-and-space-limits). Undecided machines are put in the [seed database](#seed-database) which _seeds_ the Busy Beaver Challenge.\n\n2. **Phase 2: deciders.** Write independent [deciders](#deciders), i.e. programs that will decide the behavior of families of machines in the seed database.\n\n\n**Phase 1** was completed in December 2021:\n\n- it enumerated 125,479,953 Turing machines in 30 hours (splitting the task among several computers in parallel). See these [metrics](#metrics) for more.\n- it marked **88,664,064** machines as undecided and they are stored in the [seed database](#seed-database). We refer to undecided 5-state machines thanks to their index in the seed database (e.g. Machine #7,410,754).\n\nAlthough **Phase 1** of the project was completed, it needs to be reproduced independently in order to confirm its results and increase trust. See Contribute.\n\n**Phase 2** started in January 2022 and you are invited to write your own deciders for the remaining (or yet-unknown) families and to reproduce or verify existing ones! See Contribute.\n\n#### Currently applied deciders\n\nCurrently applied deciders are [listed on the forum](https://discuss.bbchallenge.org/t/currently-applied-deciders/32/3) and you are invited to Contribute.\n\n\n\n### Why two phases?\n\nPhase 2's deciders could be integrated into the enumeration algorithm of phase 1 in order to mark a lot less than 88,664,064 machines as undecided to begin with. This is the approach taken by [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html).\n\nHowever we strongly advocate for our model where responsibility is split into two independent phases, that is because:\n\n1. Splitting responsibilities yields shorter and easier to verify/test/debug code for both phases. In particular, it is very important to establish trust in the seed database of undecided machines hence the simpler the code that generates it the better.\n\n2. Some deciders require a lot more resources than others and might only be relevant to a very small and targetted subset of machines. Hence we don't want to execute them on all enumerated machines which would considerably slow down the enumeration process.\n\nOur approach provides modularity and hopefully facilitates reproducibility, peer reviewing, and trust in the final result.\n\n\n\n## Seed database\n\nThe Busy Beaver Challenge is based on a [downloadable](#download) seed database of 88,664,064 undecided 5-state machines which was constructed during [phase 1](#phase-1-phase-2) of the project, completed in December 2021. You are more than invited to reproduce this result, see Contribute.\n\nThe code to construct the seed database is available at [https://github.com/bbchallenge/bbchallenge-seed](https://github.com/bbchallenge/bbchallenge-seed). This code is open source and was built with readability and concision in mind: it \"only\" consists of 675 lines of Go and 105 lines of C and is unit tested. See our reproducibility and verifiability statement.\n\nThis is to be compared to the unpublished ≈8000 lines of C reported by [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html#Enumeration) or the ≈6000 uncommented lines of Pascal of [https://skelet.ludost.net/bb/nreg.html](https://skelet.ludost.net/bb/nreg.html) and justifies our clear separation between [phase 1 and phase 2](#phase-1-phase-2) in this project.\n\nThe main aim of the Busy Beaver Challenge is to decide every machine in the seed database.\n\n\n\n### Construction\n\nThe algorithm that we implement to sparsely enumerate the space of 5-state Turing machines is a variation of [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html#Enumeration) but the core idea is the same.\n\nThe algorithm recursively constructs the tree of 5-state Turing machines starting from the following common ancestor:\n\n\n
\n\n| | 0 | 1 |\n| --- | --- | --- |\n| A | 1RB | --- |\n| B | --- | --- |\n| C | --- | --- |\n| D | --- | --- |\n| E | --- | --- |\n\n
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\n
\nBB_SPACE(n) = \n
Maximum number of memory cells visited by a halting Turing machine with n states starting from all-0 memory tape
\n
\nBB_SPACE(5) = 12,289\n
\n\nWhich is the number of memory cells visited by the 5-state busy beaver time champion.\n\nIt turns that BB_SPACE(5) is a much more practical cut-off to use in the enumeration algorithm than BB(5) as many more machines will visit more than 12,289 memory cells before they exceed 47,176,870 time steps.\n\nNote that if our conjecture is false, i.e. if BB_SPACE(5) > 12,289, the true BB_SPACE winner is necessarily in the seed database and should hopefully be discovered through the effort of deciding the database. Same if BB(5) > 47,176,870.\n\n\n\n#### Metrics\n\nThe enumeration algorithm was run in December 2021 and here are some metrics about the enumerated space of 5-state Turing machines:\n\n| | # machines | # machines | # machines |\n| ---------------------------------------- | ----------- | ---------- | ---------- |\n| total halting (undefined transition met) | | 34,104,723 |\n| total non-halting (using BB(4)) | | 2,711,166 |\n| total pruned | | 944,579 |\n| total undecided (time limit) | | | 14,322,029 |\n| total undecided (space limit) | | | 74,342,035 |\n| total undecided | | 88,664,064 | |\n| total enumerated | 126,424,532 | |\n\n\n\n### Download\n\n\n\n#### Mirrors\n\nThe seed database of 88,664,064 undecided 5-state machines is available for download at:\n\n- http://docs.bbchallenge.org/all_5_states_undecided_machines_with_global_header.zip\n- [ipfs://QmcgucgLRjAQAjU41w6HR7GJbcte3F14gv9oXcf8uZ8aFM](ipfs://QmcgucgLRjAQAjU41w6HR7GJbcte3F14gv9oXcf8uZ8aFM)\n\nThe zipped database is 243M and approx 2G unzipped, each machine is encoded on 30 bytes and the first 30 bytes consist of a reserved header, see [format](#format).\n\nDatabase shasum:\n\n1. zipped: `2576b647185063db2aa3dc2f5622908e99f3cd40`\n2. unzipped: `e57063afefd900fa629cfefb40731fd083d90b5e`\n\nYou are welcome to host the database on your own mirror (as long as preserving shasum), see\nContribute.\n\n\n\n### Format\n\nThe database is a binary file where each machine is described on 30 bytes. It starts with a 30-byte reserved **header** which currently contains the following information (first 13 bytes):\n\n1. `14,322,029`: number of undecided machines that exceeded the 47M steps time limit. 4-byte big endian integer\n2. `74,342,035`: number of undecided machines that exceeded the 12k cells space limit. 4-byte big endian integer\n3. `88,664,064`: total number of undecided machines. 4-byte big endian integer\n4. `1`: a boolean indicating that the machines were lexicographically sorted. The first 14,322,029 undecided machines (47M time limit exceeded) were lexicographically sorted independently of the next 74,342,035 undecided machines (12k space limit exceeded). 1-byte boolean\n\nThen, each machine is encoded on 30 bytes. First come the `14,322,029` machines that exceeded the time limit and then the `74,342,035` machines that exceeded the space limit, see [time and space limits](#time-and-space-limits). These two sets of machines are both lexicographically sorted.\n\nThe 30-byte encoding of a 5-state Turing machine is better understood with an example, for instance with machine [#7,103,458](https://bbchallenge.org/7103458) of the database:\n\n\n
\n\n| | 0 | 1 |\n| --- | --- | --- |\n| A | 1RB | 0LD |\n| B | 0LC | 1LE |\n| C | 1LD | 1LC |\n| D | 0RA | --- |\n| E | 1RB | 1RE |\n\n
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