The New Sudoku Players' Forum

[denis_berthier]

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)

A given (partial-)braid has a fixed length but a partial-braid[n] can be extended to a (generally large) number of partial-braids[n+1]. More importantly, a (partial-)braid[n] is not defined as the restriction of a longer one or of some hypothetical, totally virtual (partial-)braid of unlimited length that has not yet been effectively obtained ("in its totality", as you say).
Saying that P(A, n) is a restriction of P(A) is nonsense: you will generally use some P(A, n) before you can find any effective P(A) "in its totality".

After amending your definitions for their inconsistencies and vagueness, the real differences of your approach appear to be:
- in the way you use braids;
- in the way you define forcing-braids (your track/anti-track): my forcing-braids follow only two streams of reasoning, yours can follow more (not really a plus for manual solvers, IMO);
- in your acceptation of OR-branching (using T&E(2) even when it's not necessary) when a (partial-)braid can't be extended;
- in your use of Subsets as rlcs (i.e. your use of S-braids) when it's not necessary.
The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

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.

I think this is now clear for everybody.
Note that I always said that simplest-first is what SudoRules does but that it lets manual solvers free to use my chains in different, less systematic, ways. It's clear that most manual solvers will take whatever they find.

[Mauriès Robert]

Hi Denis,
Sorry to contradict you.

Saying that P(A, n) is a restriction of P(A) is nonsense. P(A, n) is the restriction of an undefined (generally very large) number of potential P(A)'s. And you will generally use some P(A, n) before you can find any effective P(A) "in its totality".

There is only one track P(A) associated with generator A, but many restrictions P(A,n). So it is not nonsense to talk about restrictions.
- If P(A) is valid because A is the solution of its cell (which we do not know a priori), then P(A)≡ensemble of the solutions candidates of all the cells.
- If P(A) is invalid because A is not the solution of its cell (which we do not know a priori), then P(A)≡ is the set of all the candidates of the puzzle, the solutions candidates and the non-solutions candidates of all the cells.
The definition I give of a track does not say that we are always able, with the TB alone, to find all the candidates that make it up, so it implies that one or more restrictions must be built to achieve this.

After amending your definitions for their inconsistencies and vagueness, the real differences of your approach appear to be:
- in the way you use braids;
- in the way you define forcing-braids (your track/anti-track): my forcing-braids follow only two streams of reasoning, yours can follow more (not really a plus for manual solvers, IMO);
- in your acceptation of OR-branching (using T&E(2) even when it's not necessary) when a (partial-)braid can't be extended;
- in your use of Subsets as rlcs (i.e. your use of S-braids) when it's not necessary.
The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

I have never changed my definitions of tracks and anti-pistes, they are perfectly precise and consistent, the properties that come from them and the way I use them too. You can't say the contrary because the logic behind my concepts is not yours, as it obviously isn't on this forum, which has led me to present them in a pedagogical way and not as a questioning of the TDP.
Tracks and anti-tracks are based on the use of TB at each stage of their construction, so yes it differs from your logic, it is no less logical for all that. You want to dissect into different types of patterns, free to you, not me. Why, not me, simply because only one way to do it is easier to practice than lots of patterns to memorize and identify. That's my point of view and what comes out of it.
Cordialy
Robert

[denis_berthier]

Saying that P(A, n) is a restriction of P(A) is nonsense: you will generally use some P(A, n) before you can find any effective P(A) "in its totality".

There is only one track P(A) associated with generator A, but many restrictions P(A,n). So it is not nonsense to talk about restrictions.
[...]
The definition I give of a track does not say that we are always able, with the TB alone, to find all the candidates that make it up, so it implies that one or more restrictions must be built to achieve this.

Exactly what I said: only the P(A, n) are useful in practice. P(A) is a useless idea.

Tracks and anti-tracks are based on the use of TB at each stage of their construction

Yes, this is not a revelation: each stage is a step of T&E(Subsets, A); this has always been clear.

- If P(A) is valid because A is the solution of its cell (which we do not know a priori), then P(A)≡ensemble of the solutions candidates of all the cells.
- If P(A) is invalid because A is not the solution of its cell (which we do not know a priori), then P(A)≡ is the set of all the candidates of the puzzle, the solutions candidates and the non-solutions candidates of all the cells.

The clear confirmation that P(A) is useless nonsense: either the full solution or everything; but which of the two is never known before the solution is obtained.

After amending your definitions for their inconsistencies and vagueness, the real differences of your approach appear to be:
- in the way you use braids;
- in the way you define forcing-braids (your track/anti-track): my forcing-braids follow only two streams of reasoning, yours can follow more (not really a plus for manual solvers, IMO);
- in your acceptation of OR-branching (using T&E(2) even when it's not necessary) when a (partial-)braid can't be extended;
- in your use of Subsets as rlcs (i.e. your use of S-braids) when it's not necessary.
The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

You want to dissect into different types of patterns, free to you, not me. Why, not me, simply because only one way to do it is easier to practice than lots of patterns to memorize and identify.

Sure, you don't dissect anything. As a result, you don't prove anything new.
I don't have lots of patterns, only a handful of types of chains.
You are not answering what I said about the differences in our approaches. Is there anything in my 4 points you don't agree upon?

[Mauriès Robert]

There is only one track P(A) associated with generator A, but many restrictions P(A,n). So it is not nonsense to talk about restrictions.

Exactly what I said: only the P(A, n) are useful in practice. P(A) is a useless idea.

It is not on the fact that in practice one develops a P(A) track only partially (thus that one builds P(A,n) ) that I took you back with explanations, it is on your affirmation: "Saying that P(A,n) is a restriction of P(A) is nonsense". So no bad faith please!

Tracks and anti-tracks are based on the use of TB at each stage of their construction

Yes, this is not a revelation: each stage is a step of T&E(Subsets, A); this has always been clear.

If you take my sentence out of context, you'll answer anything. You can't reduce track technique to T&E that involves arbitrary essays.

- If P(A) is valid because A is the solution of its cell (which we do not know a priori), then P(A)≡ensemble of the solutions candidates of all the cells.
- If P(A) is invalid because A is not the solution of its cell (which we do not know a priori), then P(A)≡ is the set of all the candidates of the puzzle, the solutions candidates and the non-solutions candidates of all the cells.

The clear confirmation that P(A) is useless nonsense: either the full solution or everything; but which of the two is never known before the solution is obtained.

What is absurd is to take a sentence out of its explanatory context to suggest that what I write is absurd. There is nothing absurd about my definitions and their consequences. As I do not doubt your great intelligence in understanding what exactly TDP is, I see this again as bad faith.

After amending your definitions for their inconsistencies and vagueness, the real differences of your approach appear to be:
- in the way you use braids;
- in the way you define forcing-braids (your track/anti-track): my forcing-braids follow only two streams of reasoning, yours can follow more (not really a plus for manual solvers, IMO);
- in your acceptation of OR-branching (using T&E(2) even when it's not necessary) when a (partial-)braid can't be extended;
- in your use of Subsets as rlcs (i.e. your use of S-braids) when it's not necessary.
The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

These are the only questions I haven't answered yet, and I must confess that I was hesitant to do so in the face of the bad faith you seem to show. But in the end, as I have never dodged, here are my answers.

After amending your definitions for their inconsistencies and vagueness, ...

On this statement I have already answered you. For the rest ...

...the real differences of your approach appear to be:
- in the way you use braids;
- in the way you define forcing-braids (your track/anti-track): my forcing-braids follow only two streams of reasoning, yours can follow more (not really a plus for manual solvers, IMO);
- in your acceptation of OR-branching (using T&E(2) even when it's not necessary) when a (partial-)braid can't be extended;
- in your use of Subsets as rlcs (i.e. your use of S-braids) when it's not necessary.
The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

- For the first two points which are linked, it seems to me. Of course the difference is real, since we use different definitions to build candidate sequences. I use tracks, you use braids, whips, S-braids etc.... I won't say that tracks are braids since there is no question of pattern with tracks, we can just compare the effects of our processes in given circumstances. Concerning tracks and anti-tracks, only one reasoning is followed to solve : the interactions of two conjugated tracks. This is precisely very simple for the manual solver. On the other hand, one can express this reasoning differently depending on whether one develops a single track or both.
- For the third point, there is no logical contraindication to use OR, so this is expected with TDP. From there to say that I am misusing it ... is to disregard all the resolutions I have given on this forum without OR. You not use OR, it is a choice without logical obligation. It definitely differentiates us, but not to the advantage of one more than the other.
- For the last point, the definition of the tracks being made from the basic techniques (TB), indeed the subsets of the TB are used naturally and it is not more complicated than trying to avoid them since everyone knows how to use the TB. So the problem is not in terms of necessity, since everything you solve with your whips, braids, etc... I can do it with my tracks without subsets, but in terms of choice.

Finally on your conclusion :

The latter two reflect your T&E-ish approach, while mine is based only on logical definitions (I mean concretely that SudoRules has no code other than the logical definitions given in PBCS).

I don't see how using OR or subsets has anything to do with T&E. If the track technique is T&E, your whips, braids, etc... are too. There is no less logic in my approach than in yours, and like you it is possible to develop a solver program based only on the logical definitions of TDP.

In the end, there is indeed a big difference between your approach and mine, that's your opinion and mine.
So I think it would be reasonable to stop this discussion there, since obviously everyone remains on these positions!
Cordialy
Robert

[denis_berthier]

Robert,
If you can't see the huge problem in "defining" P(A) as something that can only be the full solution or every candidate, there's indeed no point of talking more about your approach.

[Mauriès Robert]

Hi Denis,

Robert,
If you can't see the huge problem in "defining" P(A) as something that can only be the full solution or every candidate, there's indeed no point of talking more about your approach.

I only suggested stopping this discussion because, as it stands, it leads to nothing but a systematic denigration on your part and the cutting of what I write.
If you wish to continue it, be objective: what others do that is different from what you do is not necessarily an absurdity, a nonsense, a problem,... (I use your words!).
For example, reread precisely my definition of an anti-track and say "objectively, intrinsically" how it poses a "huge problem", not how it poses a huge problem for "you".
To avoid language confusions due to my poor English, I would prefer that exchanges be in French, so by private messages.
Robert

[denis_berthier]

Robert,
If you can't see the huge problem in "defining" P(A) as something that can only be the full solution or every candidate, there's indeed no point of talking more about your approach.

I only suggested stopping this discussion because, as it stands, it leads to nothing but a systematic denigration on your part and the cutting of what I write.
[...]
For example, reread precisely my definition of an anti-track

It seems you can't accept any criticism and your only response is to call it denigration. Let's see if you have any real answer to the following:

Definition :
- E = {Ai, i=1,2,...,n} being a set of n candidates Ai of the puzzle G, an anti-track P'(E) = {Bj, j=1,2,...,m} is the set of m candidates Bj that would be placed with the basic techniques (TB) in the cells of G as (true) solution IF the candidates of E were eliminated (false) from G. It is said that E generates P'(E)

In this definition, P'(E) is exactly the set of placements done within T&E(E, TB, 1). The condition "placed with the basic techniques" doesn't allow using DFS or further levels of T&E.

There is only one track P(A) associated with generator A, but many restrictions P(A,n). So it is not nonsense to talk about restrictions.
- If P(A) is valid because A is the solution of its cell (which we do not know a priori), then P(A)≡ensemble of the solutions candidates of all the cells.
- If P(A) is invalid because A is not the solution of its cell (which we do not know a priori), then P(A)≡ is the set of all the candidates of the puzzle, the solutions candidates and the non-solutions candidates of all the cells.

In this view, P(A) includes all the logical consequences of A, whatever techniques one has to use to find them. In particular, it can include DFS at any depth.

The two are contradictory. Fullstop. There's no problem of English or French language here. Do I really need to explain why this is a huge problem?

The representation of an anti-piste is usually done by colour marking [...], but it is also useful to explain the construction process by making a diagram connecting the candidates to each other as follows:

Code: Select all

P'(E) : (-E)->B1->B2->[(B3->B4)->B5]->B6 ...
                    \
                      ->B7->B8 ...


which here means: the removal of candidates from E imposes the placement of B1, which imposes the placement of B2, which imposes the placement of B3, B4 and B5, which imposes the placement of B6, but also the placement of B2 imposes the placement of B7 and then B8, etc.

So, a track is a set but whenever you use it, be it for producing it or understanding it, it has to be a sequence (or, at least, a partially ordered set). Certainly a wonderful example of logical consistency.

There's no bad faith on my part and no will of denigration. It's just that it's difficult to guess the intended meaning of all this.

[Mauriès Robert]

Hi Denis,
Since my English is not a problem, I answer here as clearly as possible, point by point.
I asked you for your objective opinion like this:

For example, reread precisely my definition of an anti-track and say "objectively, intrinsically" how it poses a "huge problem", not how it poses a huge problem for "you".

To this you answer in the following order:

In this definition, P'(E) is exactly the set of placements done within T&E(E, TB, 1). The condition "placed with the basic techniques" doesn't allow using DFS or further levels of T&E.

So you're not saying that my definition is nonsense or problematic because you identify it with another of your definitions. That's already better!
As far as DFS or other levels of T&E are concerned, it is by concatenating tracks or anti-tracks that one achieves this, which I have defined elsewhere and noted P(A).P(B) for two tracks P(A) and P(B), or P'(A).P(B) for an anti-track and a track. So this does not call into question the definition of tracks and anti-tracks.

Next ...

There is only one track P(A) associated with generator A, but many restrictions P(A,n). So it is not nonsense to talk about restrictions.
- If P(A) is valid because A is the solution of its cell (which we do not know a priori), then P(A)≡ensemble of the solutions candidates of all the cells.
- If P(A) is invalid because A is not the solution of its cell (which we do not know a priori), then P(A)≡ is the set of all the candidates of the puzzle, the solutions candidates and the non-solutions candidates of all the cells.

In this view, P(A) includes all the logical consequences of A, whatever techniques one has to use to find them. In particular, it can include DFS at any depth.

I understand what you're asking. It is not the definition of a track that is the problem, but the two assertions that you recall on P(A). I wrote this in an attempt to show you, certainly too quickly (Errare humanum est,) that P(A,n) is indeed a restriction of P(A). What I have written is correct only if P(A) leads to the solution or leads to a contradiction. It is therefore necessary to forget it.
The correct justification that only one track P(A) can be associated with a candidate A, thus justifying the name of restriction for P(A,n), is the following:
A track P(A) can only have two statuses, valid or invalid. The purpose of solving a puzzle with tracks is to establish this status, which is not known a priori.
- For a track P(A) whose status would be "valid", we would necessarily have P(A) ⊆ S where S is the set of candidate solutions of the puzzle. There could thus be only one valid track associated with this candidate A.
- For a track P(A) whose status is "invalid", i.e. for which a contradiction appears in its construction, it is possible to construct (with the TBs) several sequences of different candidates all leading to contradiction. Each of these sequences corresponds to a subset of P(A) which is well formed from the union of all these subsets since each of them is obtained by applying the definition. This means that there can be only one invalid track associated with candidate A.

Next...

The representation of an anti-piste is usually done by colour marking [...], but it is also useful to explain the construction process by making a diagram connecting the candidates to each other as follows:

Code: Select all

P'(E) : (-E)->B1->B2->[(B3->B4)->B5]->B6 ...
                    \
                      ->B7->B8 ...


which here means: the removal of candidates from E imposes the placement of B1, which imposes the placement of B2, which imposes the placement of B3, B4 and B5, which imposes the placement of B6, but also the placement of B2 imposes the placement of B7 and then B8, etc.

So, a track is a set but whenever you use it, be it for producing it or understanding it, it has to be a sequence (or, at least, a partially ordered set). Certainly a wonderful example of logical consistency.

The most natural representation of the tracks is done by marking their candidates on the puzzle. This is enough to solve. On this forum, the habit is to describe the sequences (chains, AIC, etc...) both textually (diagram) and on the puzzle (marking). So I followed this habit by adding a text description myself. I don't see any inconsistency in this.
Cordialy
Robert

[denis_berthier]

To this you answer in the following order:

In this definition, P'(E) is exactly the set of placements done within T&E(E, TB, 1). The condition "placed with the basic techniques" doesn't allow using DFS or further levels of T&E.

So you're not saying that my definition is nonsense or problematic because you identify it with another of your definitions. That's already better!

Not as much better as you may think. By providing my re-interpretation of P(A) in terms of my definition of T&E, I was making you a favour. There remains a large difference: I have some theory of T&E behind the scenes. In spite of the title "Theory of tracks", I can't see any theory in your paper. Oh yes, there are a few "definitions" and trivial theorems, but where is there anything really new or not trivial?

And if P(A) is a set - and not a sequence - what is P(A, n), in terms of restriction of P(A)?

So, a track is a set but whenever you use it, be it for producing it or understanding it, it has to be a sequence (or, at least, a partially ordered set). Certainly a wonderful example of logical consistency.

The most natural representation of the tracks is done by marking their candidates on the puzzle. This is enough to solve. On this forum, the habit is to describe the sequences (chains, AIC, etc...) both textually (diagram) and on the puzzle (marking). So I followed this habit by adding a text description myself. I don't see any inconsistency in this.

Which doesn't answer my point: if the sequential/net representation is useful, it's not because of I don't know what fake reason related to this forum. It's because it's the only useful one.
The set "definition" is never useful, neither during the real construction of a track nor for the reader to understand it.

[Mauriès Robert]

Hi Denis,

Not as much better as you may think. By providing my re-interpretation of P(A) in terms of my definition of T&E, I was making you a favour. There remains a large difference: I have some theory of T&E behind the scenes. In spite of the title "Theory of tracks", I can't see any theory in your paper. Oh yes, there are a few "definitions" and trivial theorems, but where is there anything really new or not trivial?

Since your opinion on the TDP is this one, I see no reason to continue this discussion. So end.
Robert

[denis_berthier]

Hi Denis,

Not as much better as you may think. By providing my re-interpretation of P(A) in terms of my definition of T&E, I was making you a favour. There remains a large difference: I have some theory of T&E behind the scenes. In spite of the title "Theory of tracks", I can't see any theory in your paper. Oh yes, there are a few "definitions" and trivial theorems, but where is there anything really new or not trivial?

Since your opinion on the TDP is this one, I see no reason to continue this discussion. So end.

It's wonderful how you always avoid answering the hard questions!