Hardworking beaver
Hardworking beaver (also English busy beaver ) are special Turing machine , the possible write many ones on the tape and after a finite number calculation steps to halt state taking (ie pause ). The Radó function (also hardworking beaver function ) indicates the maximum number of ones that a hardworking beaver can write with a given number of states. Both were first considered in 1962 by the Hungarian mathematician Tibor Radó .
Formal consideration
definition
According to Radó, a hardworking beaver is a Turing machine, which, as usual, can assume states and a halt state. In contrast to the general definition of a Turing machine, however, it is subject to special rules. A hardworking beaver always has to move either to the left or to the right on the belt. So there is no instruction here to remain in a field. A hardworking beaver does not expect empty fields either, only those that already contain a value from the twoelement alphabet he is familiar with . The tape on which the hardworking beaver is placed is completely filled with zeros beforehand. A hardworking beaver has to assume the halt state after a finite number of steps, i.e. it must not get into an endless loop. After stopping , it must have written the maximum number of ones, compared to all other Turing machines that also have states that work according to the same rules. Only Turing machines that do not hold could write more ones, but then would not be hardworking beavers.
Hardworking beaver function
The maximum number of ones which a diligent beaver with write conditions, the value of the Out hard Beaver function (also Radó function) at the location defined .
Unsolvable problem
The determination of the hardworking beavers is a problem that cannot generally (for everyone ) be solved algorithmically. It is not generally possible to decide whether a given Turing machine with states actually writes a maximum number of ones on the tape. However, this is possible for individual Turing machines of low complexity. So the set of values of is neither decidable nor recursively enumerable , although it is welldefined . Since the complement of this set cannot be recursively enumerated either, this set is often chosen as an example for a language that is not in the first level of the arithmetic hierarchy .
Because of these properties of the value set, the function cannot be calculated . It can also be shown that its asymptotic growth is stronger than that of any computable function.
Practical consideration
In practice it has been shown that even realistically speaking, it no longer seems possible to know about the value . To do this, one would have to find out for each individual Turing machine with states after how many steps it holds, or else prove that it does not. For a given number of states (plus a hold state) there are different Turing machines for an alphabet with two characters , because for each of the input states it must be determined for each of the two possible read symbols which of the two symbols is to be written on the tape and in which direction the reading head should be moved and which of the possible states the Turing machine should then assume. Different Turing machines have to be considered even with possible input states.
The Bulgarian Georgi Georgiev published a study in 2003 in which he analyzed diligent beavers to see whether they would stop or not. For only just over 40 hardworking beavers evaded a reliable result, because they could not be finally analyzed with the methods used by Georgiev due to special behavior. Of those determined to be terminating (sustained), none writes more than 4098 ones on the tape.
conditions  Turing machines  source  

1  64  1  (1962; Rado) 
2  20736  4th  (1962; Rado) 
3  16777216  6th  (1965; Lin, Rado) 
4th  2.56 x 10 ^{10}  13  (1972; Weimann, Casper, Fenzel) 
5  ≈ 6.34 × 10 ^{13}  ≥ 240  (1983; Jochen Ludewig) 
≥ 501  (1983; Uwe Schult)  
≥ 1915  (1984; George Uhing)  
≥ 4098  (1989; Jürgen Buntrock and Heiner Marxen)  
6th  ≈ 2.32 × 10 ^{17}  > 3.514 × 10 ^{18267}  (2010; Pavel Kropitz) 
7th  ≈ 1.18 × 10 ^{21}  Estimation unrealistic 
Examples
Hardworking beaver with a condition
The partial transfer function is defined as follows:
current state 
read. symbol 
schr. symbol 
new condition 
Head direction 


0  →  1  R. 
runs through the following states, whereby the current head position is printed in bold:
step  Condition  tape 

1  0 0  
holds  1 0 
Hardworking beaver with 2 states
The transfer function is defined as follows:
current state 
read. symbol 
schr. symbol 
new condition 
Head direction 


0  →  1  R.  
1  →  1  L.  
0  →  1  L.  
1  →  1  R. 
runs through the following states, whereby the current head position is printed in bold:
step  Condition  tape 

1  … 000 0 000…  
2  … 0001 0 00…  
3  ... 000 1 100 ...  
4th  … 00 0 1100…  
5  … 0 0 11 100…  
6th  … 01 1 1100…  
holds  … 011 1 100… 
Hardworking 3 condition beaver
The transfer function is defined as follows:
current state 
read. symbol 
schr. symbol 
new condition 
Head direction 


0  →  1  R.  
1  →  1  R.  
0  →  0  R.  
1  →  1  R.  
0  →  1  L.  
1  →  1  L. 
runs through the following states, whereby the current head position is printed in bold:
step  Condition  tape 

1  0 0 0000  
2  01 0 000  
3  010 0 00  
4th  01 0 100  
5  0 1 1100  
6th  0 11100  
7th  1 1 1100  
8th  11 1 100 
step  Condition  tape 

9  111 1 00  
10  1111 0 0  
11  11110 0  
12  1111 0 1  
13  111 1 11  
14th  11 1 111  
holds  111 1 11 
The function S
Radó also defined a function whose value is the maximum number of steps of a holding Turing machine with a twoelement alphabet and states. This function cannot be calculated either; if it were, the stopping problem with an empty input tape could be resolved , because a Turing machine with states that does more than steps never stops.
Since a maximum of one can be written in each step, the following applies
 .
The following relationship still exists between the functions and .
 .
Also not calculable function
Another noncalculable function results if the additional restriction is introduced that all ones must form a coherent chain.
The name for it has become common.
In 1965, C. Dunham specified another variant of the function of the hardworking beaver. is defined as the maximum number of ones that a Turing machine with a twoelement alphabet and states can write if it is put on a tape with a block of ones and holds it. If this function were calculable, there would also be a Turing machine M with a twoelement alphabet that calculates. This Turing machine has states. Then , where the equal sign is precisely the definition of M, and the sign stems from the fact that M is a machine with states and is attached to (i.e. a block of ones) and therefore, according to the definition of D , must satisfy the inequality . This contradiction shows the unpredictability of D.
literature
 AK Dewdney : The (new) Turing Omnibus. 66 Excursions in Computer Science. Computer Science Press, New York NY 1993, revised 1996, ISBN 0716782715 .
 Jochen Ludewig, Uwe Schult, Frank Wankmüller: Chasing the Busy Beaver. Notes and Observations on a competition to find the 5state Busy Beaver. University of Dortmund  Department of Computer Science, Dortmund 1983 ( Department of Computer Science, University of Dortmund. Report 159).
 Heiner Marxen, Jürgen Buntrock: Attacking the Busy Beaver 5 . In: Bulletin of the EATCS. 40, February 1990, ISSN 02529742 , pp. 247251.
Web links
 W. rooms: Busy Beaver (PDF, 97 KiB ), exact introduction to literature examples, at 10 pages for basic course Theoretical computer science at the CusanusGymnasium Wittlich shown
 Reinhard Völler: Turing computability from theoretical computer science , brief description of Rado's busy beaver problem with some examples from literature, Hamburg University of Applied Sciences
 Heiner Marxen: Busy Beaver (English)  known results, many links (to research) up to 2010, including:
 In search of hardworking beavers  Turing machine simulators, University of Stuttgart 1996
Individual evidence
 ↑ ^{a } ^{b} T. Radó: On noncomputable functions ( Memento of March 27, 2014 in the Internet Archive ) (PDF; 3.6 MB; WebArchive of March 27, 2014), In The Bell System Technical Journal , Volume 41, No. 3, pp. 877884, May 1962
 ↑ Busy Beaver nonregular machines for class TM (5)
 ↑ AM BenAmram, BA Julstrom, U. Zwick: A Note on Busy Beavers and Other Creatures , In Mathematical Systems Theory , 29 (4), pp. 375386, July / August 1996, doi: 10.1007 / BF01192693
 ^ C. Dunham: A Candidate for the simplest uncomputable function In: Communications of the Association for Computing Machinery (Letter to the Editor) 8, 4, 1965, ISSN 00010782 , p. 201