\n\n## Table of contents\n\n\n## Goal\n\n
\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\nThe goal of the Busy Beaver Challenge (bbchallenge for short) is to collaboratively prove or disprove the following conjecture [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html) [[Aaronson, 2020]](https://www.scottaaronson.com/papers/bb.pdf):\n\n
\nBB(5) = 47,176,870\n
\n\nThis conjecture says that if a 5-state 2-symbol [Turing machine](#turing-machines) runs for more than 47,176,870 steps without halting then it will never halt (starting from all-0 memory tape).\n\nThe conjecture, explicitly formulated in [[Aaronson, 2020]](https://www.scottaaronson.com/papers/bb.pdf), is based on the pioneering work of [[Marxen and Buntrock, 1990]](http://turbotm.de/~heiner/BB/mabu90.html) who discovered the current
5-state busy beaver champion, a machine with 5 states that halts after 47,176,870 steps:\n\n
\n
\n\n| | 0 | 1 |\n| --- | --- | --- |\n| A | 1RB | 1LC |\n| B | 1RC | 1RB |\n| C | 1RD | 0LE |\n| D | 1LA | 1LD |\n| E | --- | 0LA |\n\n
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\n\nAchieving the goal of the Busy Beaver Challenge implies studying **88,664,064 Turing machines** and decide whether they halt or not, see
Method.\n\n
You can help!\n\n
\n\n## Turing machines\n\nThe introduction of Turing machines by Alan Turing in 1936 is arguably one of the founding events of computer science, [[Turing, 1936]](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf).\n\nTuring machines can be thought as a primitive computer architecture providing (i) a runtime environment consisting of a moving read/write head over a memory tape divided in adjacent memory cells and (ii) a programming language allowing to control the behavior of the head depending on the content of the memory cell where it is currently at.\n\nThe Turing machines we work with have one bi-infinite memory tape where each cell is either containing a 0 or a 1.\n\nThe programmer specifies the code of the machine in a table with 2 columns and
q rows. Rows are called **states**. Here is the code of the current
5-state busy beaver champion:\n\n
\n
\n\n| | 0 | 1 |\n| --- | ------------------------------------------------------------- | --- |\n| A | 1RB | 1LC |\n| B | 1RC | 1RB |\n| C | 1RD | 0LE |\n| D | 1LA | 1LD |\n| E | --- | 0LA |\n\n
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\n\nEach entry of the table is called a **transition** and instructs us what to do when reading a 0 or a 1 at the current head's position on the memory tape in a given state. For instance, in the above machine,
reading a 0 in state A will:\n\n1. write a 1 at the current head position (i.e replacing the read 0 with a 1)\n2. move the head to the right\n3. jump to state B for the next instruction\n\nThe machine will **halt** (i.e. cease functioning) if it ever tries to execute an **undefined transition** denoted by **---**. Here, it would happen only if ever
reading a 0 in state E .\n\nIn the context of the Busy Beaver Challenge, machines are always executed starting in state A and with a memory tape that is initially all 0 (i.e. all memory cells are 0).\n\n
\n\n### Interactive simulator\n\nAs with probably any programming language, the best way to understand Turing machines is to play with them:\n\n
\n\nA more detailed simulator is available at
https://turingmachine.io. Here is the code of the above machine in their format:\n\n
{ theCode }\n\n
\n\n### Space-time diagrams\n\nSpace-time diagrams provide a condensed way to visualise the behavior of [Turing machines](#turing-machines). The space-time diagram of a machine is a 2D image where the i
th row represents the memory tape of the machine at the i
th iteration. Black pixels are used for memory cells containing 0 and white for 1.\n\nHere is the space-time diagram of the first 10,000 iterations of the
5-state busy beaver champion:\n\n
\n\n\n\n
\n\nAdditional green and red colors are used to track the head position and its movement: green when the head has moved to the left and red when it has moved to the right.\n\nBy default, these space-time diagrams are re-scaled to fit a 400x500 canvas, hence they can be inexact due to the scaling algorithm, especially at small scales (i.e. few simulation steps).\n\nIf you tick **Explore mode** you will enter a more precise visualisation of space-time diagrams that are accurate at small scales and that will give you additional coloured information about the state of the machine at each step.\n\n
\n\n#### Machine ID\n\nThe Busy Beaver Challenge is based on a
seed database of 88,664,064 undecided 5-state machines, see
Method. We can consequently refer to undecided machines with their ID in this database.\n\nFor instance:
https://bbchallenge.org/55897188\n\n\n\n
\n\n### Will it halt or not?\n\nTuring machines have an important property: starting from a given memory tape (all-0 in our case), they either **halt** or don't. By halting we mean that the machine tries to execute an undefined transition and, since it is undefined, stops functioning. Here is a machine that halts after 4 steps:\n\n
\n\nHere is another machine that halts after 105 steps:\n\n\n\nIf a machine has no undefined transition it is sure that it will never halt as it cannot ever encounter an undefined transition.\n\nHowever, it is not because a machine has an undefined transition that it will halt one day. The most simple example to support this statement is the following machine that will never halt starting from all-0 tape although it has plenty undefined transitions:\n\n\n\n\n\nIn this case it is easy to convince ourselves that the machine will never halt starting from all-0 tape. However, if we take our earlier example:\n\n\n\nWill this one halt or not starting from all-0 tape?\n\nAnswering this question does not look simple. Here, patience can answer it for us because the machine **does halt**, after 47,176,870 steps. However, this fact would have been quite difficult to predict just from looking at the code of the machine.\n\nTo this day, no 5-state Turing machine is known to halt after more than 47,176,870 steps.\n\nWith the Busy Beaver Challenge, we hope to discover if that 47,176,870 record can be beaten or not among 5-state machines. This implies to decide, for all machines with 5 states, whether they halt or not.\n\nBut why focusing on 5 states? Let's first reformulate the problem in terms of busy beavers.\n\n\n\n## The busy beaver function BB\n\n\n\n### Definition of BB\n\nWe can now properly define the busy beaver function BB (called S originally) as introduced in [[Rado, 1962]](https://cs.famaf.unc.edu.ar/~hoffmann/cc18/Rado-On-non-computable.pdf):\n\n\n
\nBB(n) = \n
Maximum number of steps done by a halting Turing machine with n states starting from all-0 memory tape
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