The New Sudoku Players' Forum

[denis_berthier]

Do you consider the following representation of this anti-track to be a sequence?
P'(5r6c38, 8) : (-5r6c38)=> [2r6c8 and (8r6c3->8r3c2->8r1c4)->8r7c5->6r7c9]->2r7c4->5r4c4
Or as a diagram

Code: Select all

                              ->2r6c8-------------------------
                            /                                  \
P'(5r6c38, 8) : (-5r6c38)=> ->8r6c3->8r3c2->8r1c4->8r7c5->6r7c9->2r7c4->5r4c4
                                   \             /      \      /
                                     -----------          -----


Hi Robert,
It belongs to you to define (anti)-tracks as sequences or as sets.
If you define them as sequences, they appear clearly as a disguised version of braids (which Defise's completed notation in his above post makes still clearer). If you define them as sets, it's hard for the reader to find a right order leading to an elimination (and exponentially harder with length).
It seems what you are doing is define them as sets but suppose they are read as sequences.
So please, decide once and for all what you mean and keep consistent.

The diagram is at best a partial order, if left-to-right is interpreted as "before".
The anti-track notation may be a sequence if all the non-candidate signs are discarded and the candidates are read from left to right. But is that what you mean?

[Mauriès Robert]

Hi Denis,

It belongs to you to define (anti)-tracks as sequences or as sets.
If you define them as sequences, they appear clearly as a disguised version of braids (which Defise's completed notation in his above post makes still clearer). If you define them as sets, it's hard for the reader to find a right order leading to an elimination (and exponentially harder with length).
It seems what you are doing is define them as sets but suppose they are read as sequences.
So please, decide once and for all what you mean and keep consistent.

The diagram is at best a partial order, if left-to-right is interpreted as "before".
The anti-track notation may be a sequence if all the non-candidate signs are discarded and the candidates are read from left to right. But is that what you mean?

I don't think you've read (or reread) the papers I've published on TDP, especially this one to the attention of the readers of this forum, where I think I'm very consistent :
Tracks and anti-tracks are sets of candidates answering precise definitions that I recall in a concise form here: set of candidates that one would place with the TB (basic techniques) on the puzzle under generating hypothesis.
These definitions thus explicitly say how these sets are constructed, and this allows anyone who wants to use them to do so without ambiguity on the puzzle by marking.
In doing so, by constructing these sets from the generating hypothesis, it is indeed a sequence of candidates that is executed. So for me track and anti-track are indeed sets of candidates, the associated sequences describe their mode of construction.
Moreover, I make the distinction between the track (or anti-track) and its mode of construction (diagram):
The track is written P={B1, B2, B3, ...} with a "=", its mode of construction P : B1->B2->B3... with ":" and a symbol " ->" whose meaning "is if B1 is placed then B2 is also placed by the TBs"
As for François, he is free to represent the tracks in his own way and frankly it doesn't bother me at all as long as the definition of the track is not changed, which is the case, but from there to say that he makes disguised braids there is a step that I won't take. At most it tries to show you that tracks and braids are the same fight, it's optimized T&E!
Cordialy
Robert

[DEFISE]

deleted post.

[DEFISE]

As you see, I'm not wide awake!
I am going to correct...

[DEFISE]

It’s true that TB produce more eliminations than whips, but they remain more banal and much easier to find, so I don’t count them.

So,here is the solution of a typical puzzle from mith:

Code: Select all

btte

PS: remarks on your K/L/C/B notation: it's a good idea to have added the type of CSP-variable to the tracks, but the notation for it is awful.

It's in a desire for compactness.

At the cost of readability.

What do you call sequence ?

Here, let's say a fully ordered set of candidates: C1, C2, C3...
Not any set without any order, as Robert says.

1) Well I see the problem. In fact I would have to count subsets[>= 3] as a step since they are of a higher level than whips[2] (because I count whips[2] as a step).
This new rule does not change the number of steps (=12) of my “slashes” puzzle resolution.

2) Anyway, with my presentation or yours, I need to visualize the complete grid to follow the sequences.

[DEFISE]

Hi Denis and François,
I read with interest your exchanges about the slashes puzzle. Certainly, François' 12-step resolution counts only the whips[n≥3], but if we count in Denis' resolution only the whips[n>4], we find 21, whereas in François' resolution we count only 10 for the same W. François' resolution is therefore better optimized, it seems to me.
Robert
PS: this discussion would be better placed in the thread dedicated to slashes.

Hi Robert,
It doesn't make sense to compare our “slashes” resolution in terms of the number of steps. Denis's algorithm is “Simple First” regardless of the number of steps. So it's normal that he find more steps than my program which try to minimize the number of steps (Let’s call it "Few Steps").
I have also a “Simplest First” program to calculate the W-rating, that needs obviously more steps than my “Few Steps”.

[denis_berthier]

for me track and anti-track are indeed sets of candidates, the associated sequences describe their mode of construction.
Moreover, I make the distinction between the track (or anti-track) and its mode of construction (diagram):
The track is written P={B1, B2, B3, ...} with a "=", its mode of construction P : B1->B2->B3... with ":" and a symbol whose meaning "is if B1 is placed then B2 is also placed by the TBs"

But the only thing useful for the reader is the sequential version (or the graphical network one).

to say that he makes disguised braids there is a step that I won't take. At most it tries to show you that tracks and braids are the same fight, it's optimized T&E!

There's no fight: I've proven long ago the clear relationship between T&E and braids and I never tried to hide it. On this forum, there must still be somewhere an "Abominable T&E and lovely braids" thread. As disguised braids missing crucial information (the CSP-Variables re-added by Defise), your tracks straightforwardly inherit this relationship.

But my T&E vs braids theorem doesn't mean that T&E and braids are the same thing, let alone that "braids are optimised T&E". They are much more than this.
From the very start, like all the chains I introduced, braids have an inherent notion of length (which your tracks didn't have at the beginning). A braid can be defined by a precise first order formula in the elementary language of Sudoku (or, more generally, in the elementary language of CSPs). As a result, a braid rating can be defined. None of this is meaningful for T&E.
.

[denis_berthier]

1) Well I see the problem. In fact I would have to count subsets[>= 3] as a step since they are of a higher level than whips[2] (because I count whips[2] as a step).
This new rule does not change the number of steps (=12) of my “slashes” puzzle resolution.

I'd say you would have to count subsets[>= 2], but anyway the comparison of steps is like comparing apples and potatoes.

2) Anyway, with my presentation or yours, I need to visualize the complete grid to follow the sequences.

Yes, of course. I think this is true of any patterns.

[denis_berthier]

Hi Robert,
It doesn't make sense to compare our “slashes” resolution in terms of the number of steps. Denis's algorithm is “Simple First” regardless of the number of steps. So it's normal that he find more steps than my program which try to minimize the number of steps (Let’s call it "Few Steps").
I have also a “Simplest First” program to calculate the W-rating, that needs obviously more steps than my “Few Steps”.

We've cross posted. 100% on the same line here.

One thing original in your few steps approach is, you first compute the W-rating and only then you try to find a smaller number of steps, with no claim of optimality.
What I had seen before is (IMO hopeless) attempts to combine all the individual lengths of all the individual steps into a single rating.

[DEFISE]

Denis and Robert,
About the "track is a set or a sequence" debate, I choose “sequence”.
But it is not my responsibility to define a track in all its generality.
On the other hand, I would be able to define precisely the tracks that I use to simulate the whips and the braids. But these are very specific tracks.

[DEFISE]

One thing original in your few steps approach is, you first compute the W-rating and only then you try to find a smaller number of steps, with no claim of optimality.
...

Right, "Simplest First" then "Few Steps".

[Ajò Dimonios]

Hi Denis, Robert, François.
Regarding the debate on whether a track is a sequence or a set, I prefer a set, because in the construction of a track the order of application of TB can be different but you always get, if the track is invalid, a contradiction, vice versa if it is not possible to obtain a contradiction with the application of TB, the set obtained is always the same and does not depend on the order of application of TB. I also remember that the tracks can be used even when there is no contradiction, see example diagonals-centres-t38366.html, see the tracks T (1r4c8) and T (1r7c8).

Paolo

[denis_berthier]

I would be able to define precisely the tracks that I use to simulate the whips and the braids. But these are very specific tracks.

Do they have anything specific other than not including g-candidates or Subsets?

[Mauriès Robert]

Hi Denis,

From the very start, like all the chains I introduced, braids have an inherent notion of length (which your tracks didn't have at the beginning).

Indeed the notion of length is not introduced in the definition of a track or anti-track, but nothing obliges, during a resolution to develop the tracks or anti-tracks in their totality to obtain results, everything depends on the results sought. It is to clarify this point that I added the notion of length by writing P(A,n) or P'(A,n) to signify that the construction sequence was stopped at n elements, which then represents only a part of the track or anti-track.
In other words, P(A)={B1, B2, ...,Bn, ...} contains all the candidates of P(A), whereas P(A,n)={B1, B2, ...,Bn} contains only n. P(A,n) is a restriction of P(A).
In a more ordinary language, in TDP, I suggest resolutions by successive steps, and this from the beginning, so to limit the "length" of the tracks to the simplest, this is not new. But it is sometimes interesting to do otherwise by pushing the construction of tracks as far as possible in order to increase eliminations and validations (see Paolo's intervention on this thread).
Contrary to you, whose motto is "the simplest first" and no matter how many steps, which is easy when a software does it, my motto would rather be "the most efficient way possible" with as few steps as possible, by hand it's more affordable.
Finally, if on this forum, I only present resolutions with tracks or anti-tracks with limited restrictions (thus in short successive steps), it is to conform as much as possible to the habits of its participants who use patterns, and in doing so it was interesting (for me, and for François I think) to compare sometimes this approach with yours (the simplest first).
Cordialy
Robert

[DEFISE]

I would be able to define precisely the tracks that I use to simulate the whips and the braids. But these are very specific tracks.

Do they have anything specific other than not including g-candidates or Subsets?

No they have not anything specific.
These tracks T(C) can be defined as a set of candidates that could be validated if C was validated, using only uniqueness rule in an entity (=CSP variable).
This set of candidates can be considered as a sequence since each candidate must respect the constraints imposed by all the preceding ones.
Add the fact that I stop the process if a contradiction appears (if I want to associate a braid or a whip to the track).

N.B: The process of creating these tracks is an algorithm for finding patterns (whips and braids).