Are Turing’s “m-configurations” the same as the “means” in his original definition of “computable”?
In the first line of Turing’s paper “On Computable Numbers…”, he defines a “computable” number as follows:
A real number “for which the decimal representation can be calculated by finite means.”
My question is; are “means” to which he refers actually just the number of m-configurations which he proceeds to define shortly after the first line of his paper?
It seems that the “means” he is mentioning are the actual m-configurations, and I’m trying to understand the difference between these, and how the “steps”, “m-configurations”, and “calculations” relate.
As I understanding, it works like this;
A real number R is computable <=> There exists a “machine” with a finite number of “m-configurations” which can be used to print the decimal representation of R, even if the actual quantity of numbers which are printed on the “tape” to represent the decimal value is not finite.
— Therefore, the “means” are the number of m-configurations, and if R is computable, the machine can still perform an infinite number of actions to calculate the decimal representation of R, so long as there are only a finite number of m-configurations which produce the (infinite number) of actions.
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I’m recently trying to study the paper “Graph Partitioning with Natural Cuts”: https://www.microsoft.com/en-us/research/wp-content/uploads/2010/12/punchTR.pdf
There is a term edge-connected components in section 3.1 Detecting Tiny Cuts:
The first pass identifies all edge-connected components of the graph using depth-first search. They form a tree T. We make T rooted by choosing as a root the edge-connected component with maximum size, which on road networks typically corresponds to most of the graph. We then traverse T in a top-down fashion. As soon as we enter a subtree S of total size at most U, we contract it into a single vertex…
First, it seems not to mean “connected component” because in section 2 it says:
To simplify the description of our algorithm, we assume G is connected
And I tried to search this term on Google but only a term k-edge-connected (https://en.wikipedia.org/wiki/K-edge-connected_graph) found and it seems to mean something different.
Could someone tell me what this term means exactly, please? Thank you!
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