[r-t] Question about shortest link methods - does truth make a difference?

[Edmund Mottershead]

I believe you can expand your argument for Dt to Dp.

Suppose Dp(r)=Dpt(r) does not hold for all r.
So there exist some rows r such that Dp(r) < Dpt(r).
Let us take the row from this set such with the shortest link method.
Call this row R and we have that Dp(R) < Dpt(R) and for all r such that
that is true, Dp(R) <= Dp(r)

If we prick out Dp(R) we will find some row k which appears twice.
So in the place notation for Dp(R) we will have a piece of notation X such
that for a row l, X(l) = k
In addition we will have another piece of place notation Y and another row
m such that Y(k) = m

>From the definition of Dp(R), the concatenation of X and Y must create a
long place, else we would just omit the intervening rows.
However I claim that for any X.Y creating long places we can devise a 3
change, true method section accomplishing the same result.

This method section will be shorter than the false section leading from k
to k, since there must be at least two changes between the ks, eg
l,k,*,*,k,m
as opposed to l,*,*m.

This is a contradiction, so Dp(r) = Dpt(r) for all r.

The tricky part is rigorously showing how to do your trick of getting from
34.14 to 12X14 for all concatenations of place notation.
If one of the changes has bells making places in adjacent pairs then we can
replace that change with the 'opposite' change, then the cross change, eg
36.56 becomes 36.1234X.
If both the place are split, then it seems that you can rewrite the changes
to be on a smaller sets of bells, eg 16.36 becomes 14X1256.
I can't get my brain to think about the equivalent scenarios for odd
numbered bell methods at this time of night, but I think that it works.

Best,
Ed


On Fri, Apr 3, 2020 at 10:23 PM Alexander E Holroyd <holroyd at math.ubc.ca>
wrote:

> Let r be a row, and consider possible sequences of non-jump changes that
> get from rounds to r.  Let D(r) be the minimum possible length of such a
> sequence.  So for example,
> D(1234)=0
> D(2314)=2  (the unique minimum length sequence is 34.14)
> D(4321)=4  (one minimum sequence is plain hunt, -14-14).
>
> Now let Dp(r) be the minimum length if no bell is allowed to make 3 or
> more consecutive blows in the same place.  So for example
> Dp(2314)=3 (which is larger than D(2314)=2)
> (34.14 is not allowed because 4 makes 3 blows in 4ths, but we can do
> 12-14 instead).
>
> Also let Dt(r) be the minimum length if the sequence of rows must be
> true, i.e. no row is allowed to be repeated.
>
> Finally let Dpt(r) be the minimum length if it must be true AND have no
> long places.
>
> Obviously for every row r,
> D(r)<=Dp(r)<=Dpt(r) and
> D(r)<=Dt(t)<=Dpt(r)
>
> In fact it is easy to prove that for every row r
> D(r)=Dt(r):
> if some minimum length sequence is not true, we can just omit the rows
> between two identical rows to get a shorter sequence, a contradiction.
>
> We saw above that for some rows r (e.g. 2314)
> D(r)<Dp(r)
> and because both sequences are true, this same example also has
> Dt(r)<Dpt(r).
>
> My question is:
> Does Dp(r)=Dpt(r) hold for every r?
> Or is there a row r for which Dp(r)<Dpt(r)?
> In other words, if forbid long places, is there a row that you can get
> to quicker if you allow falseness than if you don't?
>
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