

Turing Machines and the Busy Beaver Game - mishkinf
http://www.drivenbycode.com/turing-machines-and-the-busy-beaver-game/

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belovedeagle
I believe there is some confusion in the article about whether non-halting
Turing machines are properly called "Turing machines"; that is, the author
seems to use the phrase "Turing machine" to refer only to halting machines.

> Not each of these ways end up being Turing machines since a Turing machine,
> by definition, must have a finite number of operations that end with a
> halting state.

By contrast, a more accepted definition of a Turing machine is given:

> An n-card Turing machine is a machine that has a control unit (as seen
> above) that transitions between n discrete states until it reaches the
> halting-state.

Note that this definition states that the Turing machine no longer transitions
once it has reached the "halting state" [1] but it should not be read to imply
that a machine must actually reach the halting state after finitely many
operations.

The article could be corrected by replacing "Turing machine" in the
problematic statement with "Turing machine which halts"; or by making explicit
the notion of a candidate BBTM as a TM which must halt.

As for the current reading of the statement, it is incorrect given that there
are in fact [4(n+1)]^2n possible binary n-state TMs given the formalization
used in the article, although there are many symmetries which could reduce
that number.

[1] The distinction between "a" halting state and "the" halting state is
immaterial, since you can view states which specify "HALT" either as many
distinct halting states or all leading to a singular state, halting.

