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Full Answer
What is a busy beaver?
What is a busy beaver function?
How many steps does a busy beaver take?
How many states does a busy beaver function need?
When was the busy beaver game invented?
Can a busy beaver be generalized?
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Does the proof that the busy beaver function is uncomputable ... - Quora
Answer: No, a proof that the Busy Beaver function is uncomputable can be a reduction from the Halting problem. And the Halting problem can be proved uncomputable without diagonalization (or self-reference), as Gregory Chaitin demonstrated. See my answer at What is a proof of the undecidability of...
Is busy beaver the fastest growing function known to man?
The busy beaver function grows faster than any computable function.However, it can be computed by a Turing machine which has been given access to an oracle for solving the halting problem.
Understanding proof for Busy Beaver being uncomputable
Proof: Suppose to the contrary that such a machine existed, and call it T. We will use T to construct a machine which solves the halting problem as follows: On input < T, w >, a description of a Turing machine and its associated input, we can determine in finite time the number of states that T has (it is encoded in the description of T).
Busy Beavers, and the Quest for Big Numbers - Math ∩ Programming
Back in the days before computers (the early 20th century), mathematicians were fiercely interested in the theory of computing things. The common visitor to this blog will recall our primers on Finite Automata, Turing Machines, and such things as the Game of Life.It might not be a surprise to hear that there are many many different models of computation, and before real computers actually came ...
Busy Beavers - Wikipedia
This article may have been created or edited in return for undisclosed payments, a violation of Wikipedia's terms of use.It may require cleanup to comply with ...
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What is a busy beaver?
This infinite sequence Σ is the busy beaver function, and any n -state 2-symbol Turing machine M for which σ ( M) = Σ ( n) (i.e., which attains the maximum score) is called a busy beaver. Note that for each n, there exist at least four n -state busy beavers (because, given any n -state busy beaver, another is obtained by merely changing the shift direction in a halting transition, another by shifting all direction changes to their opposite, and the final by shifting the halt direction of the all-swapped busy beaver).
What is a busy beaver function?
The busy beaver function quantifies the maximum score attainable by a Busy Beaver on a given measure. This is a noncomputable function. Also, a busy beaver function can be shown to grow faster asymptotically than any computable function.
How many steps does a busy beaver take?
The current (as of 2018) 5-state busy beaver champion produces 4098 1s, using 47 176 870 steps (discovered by Heiner Marxen and Jürgen Buntrock in 1989), but there remain 18 or 19 (possibly under 10, see below) machines with non-regular behavior which are believed to never halt, but which have not been proven to run infinitely. Skelet lists 42 or 43 unproven machines, but 24 are already proven. The remaining machines have been simulated to 81.8 billion steps, but none halted. Daniel Briggs also proved some machines. Another source says 98 machines remain unproven. There is an analysis of holdouts. So, it is likely that Σ (5) = 4098 and S (5) = 47176870, but this remains unproven, and it is unknown if there are any holdouts left (as of 2018). At the moment the record 6-state champion produces over 3.515 × 1018 267 1s (exactly (25×4 30341 +23)/9), using over 7.412 × 1036 534 steps (found by Pavel Kropitz in 2010). As noted above, these are 2-symbol Turing machines.
How many states does a busy beaver function need?
It has only been proven for extremely small machines with fewer than five states, while one would presumably need at least 20-50 states to make a useful machine. Furthermore, every known exact value of S ( n) was proven by enumerating every n -state Turing machine and proving whether or not each halts. One would have to calculate S ( n) by some less direct method for it to actually be useful.
When was the busy beaver game invented?
The n-state busy beaver game (or BB-n game ), introduced in Tibor Radó 's 1962 paper, involves a class of Turing machines, each member of which is required to meet the following design specifications:
Can a busy beaver be generalized?
It is possible to further generalize the busy beaver function by extending to more than one dimension.
Overview
In theoretical computer science, the busy beaver game aims at finding a terminating program of a given size that produces the most output possible. Since an endlessly looping program producing infinite output is easily conceived, such programs are excluded from the game.
More precisely, the busy beaver game consists of designing a halting Turing machine with alphabet {0,1} which writes the most 1s on the tape, using only a given set of states. The rules fo…
The game
The n-state busy beaver game (or BB-n game), introduced in Tibor Radó's 1962 paper, involves a class of Turing machines, each member of which is required to meet the following design specifications:
• The machine has n "operational" states plus a Halt state, where n is a positive integer, and one of the n states is distinguished as the starting state. (Typically, the states are labelled by 1, 2, ..., n, …
Related functions
The busy beaver function quantifies the maximum score attainable by a Busy Beaver on a given measure. This is a noncomputable function. Also, a busy beaver function can be shown to grow faster asymptotically than any computable function.
The busy beaver function, , is defined such that Σ(n) is the maximum attainable score (the maximum number of 1s finally on the tape) among all halting 2-symbol n-state Turing machines …
Applications
In addition to posing a rather challenging mathematical game, the busy beaver functions offer an entirely new approach to solving pure mathematics problems. Many open problems in mathematics could in theory, but not in practice, be solved in a systematic way given the value of S(n) for a sufficiently large n.
Consider any conjecture that could be disproven via a counterexample among a countable number o…
See also
• Rayo's number
• Turmite
Notes
1. ^ Radó, Tibor (May 1962). "On non-computable functions" (PDF). Bell System Technical Journal. 41 (3): 877–884. doi:10.1002/j.1538-7305.1962.tb00480.x.
2. ^ Chaitin (1987)
3. ^ Adam Yedidia and Scott Aaronson (May 2016). A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory (Technical Report). MIT. arXiv:1605.04343. Bibcode:2016arXiv160504343Y.
External links
• The page of Heiner Marxen, who, with Jürgen Buntrock, found the above-mentioned records for a 5 and 6-state Turing machine.
• Pascal Michel's Historical survey of busy beaver results which also contains best results and some analysis.
• Definition of the class RTM - Reversal Turing Machines, simple and strong subclass of the TMs.