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Is there a difference between the end state of a Turing machine and the halt state? Especially, for example the Busy Beaver 3. It is said that it is with 3 states but there is also a halt. Is the end state $q_2$ or the $halt$?

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    $\begingroup$ Busy Beaver is a famous Turing machine problem, which aims of determining the maximum number of steps, done by a TM, when computing the number of 1s starting from an empty tape. For one 1 (one state) - one step, for four 1s (2 states) - six steps, for six 1s (3 states) - fourteen steps. So it grows exponentially. Therefore, it is proved that it`s not computable. The problem however, dates back from 1960s. $\endgroup$ – user8 Jan 28 '15 at 20:13
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    $\begingroup$ The growth is not exponential, it is super-exponential. Also, exponential growth (and even super-exponential growth) doesn't imply non-computable. However it happens to be the case that the busy beaver function growths faster than every computable function. $\endgroup$ – kasperd Feb 22 '15 at 22:04
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"End(ing) state" and "halt(ing) state" are just two (four) different ways of saying the same thing.

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In the specific case of the Busy Beaver , the end state is the halt state, al other states (q0,q1,q2) must have a transition defined for every symbol in the alphabet of the TM so the machine doesn't halt on them.

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  • $\begingroup$ Also, considering the definition of the busy beaver problem, the halting state is not considered a "regular" state, probably because it is not optional. I believe you are making the common mistake of thinking that the state with the higher index ($q_2$, in your example) has some special attribute, but it doesn't. $\endgroup$ – André Souza Lemos Apr 24 '15 at 4:05

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