## The concept of a RESOLUTION RULE and its applications

Advanced methods and approaches for solving Sudoku puzzles

THEOREM: NRCZT-CHAINS SUBSUME THE BASIC INTERACTIONS

This is a stronger and much more interesting assertion than the previous one about nrczt-nets subsuming these interactions.
The proof follows the same lines as the previous one, but is simpler.

As this is mainly about nrczt-chains, the details are given in the "fully supersymmetric chains" thread.
denis_berthier
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### A few remarks about TMs and Steve's repeated claims

A few remarks about TMs and Steve's repeated claims

As Steve (Stephen Kurzhals) keeps claiming that TMs (Triangular Matrices) are an "anterior generalisation" of nrczt-chains, let me make things clear.

1) Steve's "definition" of TMs
Steve K wrote:The precise final defintion has been in existence for quite some time. Here is a link: http://www.sudoku.org.uk/SudokuThread.asp?fid=4&sid=9059&p1=3&p2=11
One can also read the forums at sudoku.com.au. Triangular matrices were defined there in late Winter of 2006.
The definition has never been in flux.

The first post of the referenced thread "defines" PMs; TMs are "defined" further down on the page (post dated 19-Jul-2007). The definition given there is as follows:
Steve K wrote:Tri-angular matrix definition:
nxn
Each row contains at least one truth
The top entry of each column is in conflict with each item below it.
For row i, items i+2 and greater are empty. This can be translated, in Booleans, as False.

Notice that "row" and "column" refer to the matrix structure and not to the grid.
Notice also that what the entries of the matrix are is not defined. Very strange for a "precise final definition". Are they candidates? (My question is all but innocent: some posts in the thread suggest that they could be ALSs).

Now, Steve's claim has taken various forms. The last one (as of today) is (23-Jul-2008, http://forum.enjoysudoku.com/viewtopic.php?t=6047)
Steve K wrote:TM's are generalized predecessor to *****tchains

2) Let us first evacuate trivialities:
I'll skip Steve's embarassed comments in a later post in the same thread, as he never defines what a *****t-chain is (unless of course he means an nrczt-chain):
Steve K wrote:In my original post, I was careful to not write nrczt-chains specifically to avoid this type of interchange. Instead I wrote *****t chains. [...] In my next post, I forgot to maintain the distinction between ***** t chains from nrczt-chains.

Steve is forgiven for forgetting to maintain a distinction that never existed.

Steve acknowledges:
Steve K wrote:Never have I suggested, in any post in any forum that you did not arrive at your techniques independently.

Conversely, I've never denied that PMs and TMs existed before (h)xy(z)(t) or nrc(z)(t) chains. And I really don't care for this because the logical relation between the two, if any, still remains undefined.
But all this is only a matter of credit and very far from the points I want to make in this post.
These points are important in the context of this thread, because they are a good illustration of what a resolution rule is based on (a precise factual pattern) and what it can't be reduced to (an almost empty logical skeleton).

3) So, let's take the question seriously.

3a) Firstly, TMs themselves (and PMs for the same reasons), as precise patterns to be found on a real grid, remain undefined (even if we take for granted that elements in the matrix are candidates). Steve's "definition" is not given in factual terms, but in terms of "weak and strong inference sets". The way these inference sets are concretely related to facts on the grid is basically undefined.
The problem is not new and it can be illustrated with AICs: as of today, "strong links" can be either bilocation, bivalue or ALSs - but this has not always been the case.
For AICS, usage has established that "strong link" means either bilocation or bivalue or ALS. I'm not saying that this is no longer a problem: on the contrary, questions from beginners constantly arise from this ambiguity and complexity matters are totally hidden by it. (I won't comment on the reasons why some people need to maintain this ambiguity).
If usage established that "strong links" may also be interpreted contextually and don't have to be symmetrical [two possibilities allowed by the ...(z)(t)-chains], then one could claim that these chains are AICs, that they are a special case of AICs and that they are therefore not new. This amounts to denying any possibility of introducing any new types of chains.
All this is absurd, because a chain rule is mainly defined by its elementary building blocks (bilocations, bivalues, ALSs, conjugacy modulo something, ...) - the "precise factual patterns" one has to find on a real grid - and not by the trivial alternating of "weak" and "strong" inferences - the "almost empty logical skeleton".

The problem is the same with Steve's definition, but it is still worse for matrices, because no usage has established anything about what his "weak and strong inference sets" should be in practice. So that they don't even tell you what you must look for on the grid.

3b) Secondly, the way an nrczt-chain should be mapped into a TM is still undefined.
Before claiming that A is more general than B, the least one should do is give the general principles of the mapping from B to A. As of today, I've never seen the slightest indication on how to do this: associating with each nrczt-chain a well defined TM. So this is a challenge to Steve: instead of creating diversions (such as the story about ***t versus nrczt), give these principles (and then we can start a real discussion) or give up your repeated and unsustained claims.
denis_berthier
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denis berthier, after considering the tone of your post, I will be surprised if Steve K responds. Indeed, I would recommend that he not do so.
ronk
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Ronk,
As far as I know, you are not a moderator of this forum and Steve knows what he has to do. As for the tone of my post, could you be more precise about what bothers you? My criticisms are factual. Are you bothered by factual criticisms?

For me, it would be very strange that someone would repeat endlessly the same claims and be unable to answer the precise question about them I'm challenging him with.
denis_berthier
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denis_berthier wrote:Ronk,
As far as I know, you are not a moderator of this forum and Steve knows what he has to do.

What's that got to do with someone expressing an opinion. No one ordered you to cease and desist (which is what moderators generally do). Besides, speaking for myself, I can't understand the arrogant tone of the post in question to someone who hasn't posted in this thread (& afaik elsewhere here) for some time. Why not give it a rest.
DonM
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DonM wrote:I can't understand the arrogant tone of the post in question to someone who hasn't posted in this thread (& afaik elsewhere here) for some time.

...not posted in this thread but modified a previous post in another thread. I just noticed it.
I put my post in this thread because it is about resolution rules, TMs, AICs...

Obviously, we don't have the same notion of arrogance and you missed our previous exchanges on the topic (here and in the unmoderated Eureka).
Steve has been repeating his claims for more than one year, always evading precise questions. I'm very surprised that he can get support in such behaviour.
Anyway, if Steve feels I've really been arrogant, I apologise for it. That wasn't my intention.

Back to the topic of my post and to my final challenge. I've very good reasons to think that Steve will be unable to answer. I'll give them in time. But first, I'll give him a chance to answer.
denis_berthier
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Hi Denis,
I can’t use your nrczt-chains and Steve’s TM for some reasons (maybe I’m not enough smart… ), but it seems that you are trying to beat Steve for answering…
IMO, if I think that I’m right then I would not try to beat someone…
Sorry, if I’m wrong…
ttt
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ttt wrote:Hi Denis,
I can’t use your nrczt-chains and Steve’s TM for some reasons (maybe I’m not enough smart… ), but it seems that you are trying to beat Steve for answering…
IMO, if I think that I’m right then I would not try to beat someone…
Sorry, if I’m wrong…
ttt

Hi ttt
From the solution you provided on the "Almost ER" thread, I'm quite sure you're a very smart player; you're using techniques much more complex than nrczt-chains.
If you want to see nrczt-chains at work everyday, you can have a look at this French forum http://www.sudoku-factory.com/forumsudoku/index.php, section "Tactiques".
(unfortunately, their notation is very local: e.g. b4 instead of r4c2....).
For a progressive approach on how to use these chains, see my answer to andre43 (Fri Apr 18, 2008) in the middle of this page: http://forum.enjoysudoku.com/viewtopic.php?t=5591&start=120. Then, you'll understand that they are a very natural generalisation of xy-chains (and they don't need any reference to abstract matrices).
As for TMs, as they don't define any specific factual patterns, I'm not surprised you can't use them.

I don't know if I'm trying to beat Steve for answering. What I know is that I don't claim something is true if I don't have a proof of it.
denis_berthier
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denis_berthier wrote:
DonM wrote:I can't understand the arrogant tone of the post in question to someone who hasn't posted in this thread (& afaik elsewhere here) for some time.

...not posted in this thread but modified a previous post in another thread. I just noticed it.
I put my post in this thread because it is about resolution rules, TMs, AICs...

Obviously, we don't have the same notion of arrogance and you missed our previous exchanges on the topic (here and in the unmoderated Eureka).

If you're referring to the 'Almost ER' thread, I was aware of it, but Steve's last response there was almost exactly 3 weeks ago. Usually in these situations that is long enough for adults to cool off and moderate their responses.
DonM
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DonM

Your insistence on blaming me is puzzling.

Why aren't you blaming Steve for not providing any evidence for his repeated claims????
denis_berthier
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Denis, I have no interest in an extended argument. If, in your opinion, there is no relationship between TM's and any type of t chain, that is fine with me.

IMO, where as you would like to constrict the definition of TM's into a finely detailed mapping of sudoku facts into deductions one can make, I have no interest whatsoever in such a finely detailed mapping.

Many examples exist, both in my blog and elsewhere which show examples of TM's that are also t chains. If the examples do not make it obvious, I will repeat what I have said long ago: If the arguments in a TM are restricted to native sis and native weak inferences, then all t chains are easily written as a TM. I have defined precisely what I mean by native long ago, and have little interest in doing so again, as I believe that such definitions will be ignored by you, as they were before. Of course, I have also pointed out long ago, and more than once, that native sis are equivalent to your definition of cells. If one only considers native weak inferences between such "cells", then this suffices to create the mapping you insist does not exist.

Many of your questions confuse me, such as "are ALS allowed as arguments in a TM?" It seems to be rhetorical, and I assume it must be. Clearly, one can use any sis, and any weak inference, within a TM.

The road map, therefor, to create a set of specific resolution rules, if one wishes, from any matrix form is quite clear. If it is not clear to you, then I can only assume that you have not been paying attention - as it is clear that you are intellectually superior to me.

Of course I agree that the set of resoltution rules that one may then write directly from any matrix form is large. If the set is too large for your taste, that is fine with me. I repeat: I have no actual interest in writing resolution rules. I have merely pointed out that the path to do so from matrices is both easy and obvious.

I am not clear as to precisely what claim, nor which claims, I have made which you find to be false. If you were to specifically list claims that I have made which you find fallacious, that would be helpful. I have noted often, however, that you have misinterpreted what I have written and then ascribed to me a claim that I have not made. Some of these incorrently ascribed claims are false.

Finally, as I have noted many times - TM's were introduced to me by Andrei Z. For whatever reason, it seems that noting the similarities between Andrei's TM's and t chains offends you. If it makes you happy, I shall desist in noting that in threads in which you are already participating.
Last edited by Steve K on Sat Aug 16, 2008 1:00 am, edited 1 time in total.
Steve K

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Steve K wrote:Denis, I have no interest in an extended argument. If, in your opinion, there is no relationship between TM's and any type of t chain, that is fine with me.

I've never said there's no relationship.
I've said that the relationship you're claiming, i.e. that TMs are a generalisation of nrczt-chains, is not proven(and is probably false).

Steve K wrote:IMO, where as you would like to constrict the definition of TM's into a finely detailed mapping of sudoku facts into deductions one can make, I have no interest whatsoever in such a finely detailed mapping.

Understood: you may claim whatever you want and you have no interest in proving it.

Steve K wrote:Many examples exist, both in my blog and elsewhere which show examples of TM's that are also t chains.

Again, this is besides the point. "Some cases of Bs are As" doesn't prove that "all Bs are As".

Steve K wrote:If the examples do not make it obvious, I will repeat what I have said long ago: If the arguments in a TM are restricted to native sis and native weak inferences, then all t chains are easily written as a TM. I have defined precisely what I mean by native long ago, and have little interest in doing so again, as I believe that such definitions will be ignored by you, as they were before. Of course, I have also pointed out long ago, and more than once, that native sis are equivalent to your definition of cells. If one only considers native weak inferences between such "cells", then this suffices to create the mapping you insist does not exist.

All this is empty. If "native sis" are cells why introduce this notion here if not for diversion?

I've no time right now but I'll soon give an example of an nrczt-chain that can't be written as a TM in this way.
denis_berthier
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Steve K wrote:
Many examples exist, both in my blog and elsewhere which show examples of TM's that are also t chains.
If the examples do not make it obvious...

Denis B
Again, this is besides the point. "Some cases of Bs are As" doesn't prove that "all Bs are As".

Thanks, Denis - I must have never known that. Clearly, you have decided to, once again, take statements out of context and respond to them as if they had no context. Perhaps if, a long long time ago, you had actually read my posts carefully, our conversation would be less tiresome and more meaningful.

For example, I did not say that native sis are cells. You said that. I said, paraphrasing, that native sis are equivalent to your "cells".

The proof of a mapping of t chains with all their extensions into a TM is so trivial that although I have written before, it hardly seems necessary: Here it is again:

t chains, with all their extensions are chains of what Denis defines as "cells". Each of these "cells" is a sis. Denis' "links" between "cells" are native weak inferences. This maps directly into a TM. QED.

If one needs the actual mapping, that is quite trivial and obvious:
A chain of n "cells" maps into an nxn TM. "cell" m maps into TM row m. Right-most "cell" content for all "cells" except "cell" n maps into TM row m, column m+1. "Cell" content that "sees" the target are placed in TM column 1. "Cell" content for row j (j>m)that is "linked" to right-most "cell" content for "cell" m is placed in TM column m+1. TM entries which have not been filled are left empty. In this fashion, all "cell" entries are placed into the TM, and all weak/strong inference relationships that a TM required have been met. Thus, for every one of Denis' "t" chains of "cells" there exists a TM that can be written directly from the ordered content of the chain of "cells".

All of this is of course painfully obvious.

Denis B. wrote:
I've no time right now but I'll soon give an example of an nrczt-chain that can't be written as a TM in this way

It is assumed that Denis did not read my post carefully, or he would not have actually made this statement. I do look forward, however, to Denis' example of a chain that cannot be written as TM following such an obvious and trivial mapping.
Steve K

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Steve
I just wanted to make sure you didn't have any mapping other than the "obvious" one.
denis_berthier
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### Trying to map an nrczt-chain into a TM

Trying to map an nrczt-chain into a TM

As Steve (Stephen Kurzhals) keeps claiming that TMs (Triangular Matrices) are a generalisation of nrczt-chains, but never gave any proof of this (and this post will show that his previous answer is not a proff), let me try to see if one can map any nrczt-chain into a well defined TM.

For completeness:
The definition of an nrczt-chain is given here: http://forum.enjoysudoku.com/viewtopic.php?t=5591
The definition of a TM is given here: http://www.sudoku.org.uk/SudokuThread.asp?fid=4&sid=9059&p1=3&p2=11
I repeat it here:
Steve K wrote:Tri-angular matrix definition:
nxn
Each row contains at least one truth
The top entry of each column is in conflict with each item below it.
For row i, items i+2 and greater are empty. This can be translated, in Booleans, as False.

Notice that "row" and "column" refer to the matrix structure and not to the grid.
Notice also that what an entry of the matrix should be is not defined.

So let x1, y1, x2, y2, x3, y3 ... be the successive left- and right- linking candidates of a given nrczt-chain based on target z

First principle of the mapping:
each entry of a TM is a candidate
- Notice that this is not explicitly stated in Steve's definition but I can't see what else they could be.

Second principle of the mapping:
the successive diagonal elements of the TM are z, y1, y2, y3 ...
the successive sub-diagonal elements of the TM are x1, x2, x3 ....

Third principle of the mapping:
columns of the matrix can only contain candidates that are nrc-linked to the (unique) candidate at the meet of the column and the diagonal
- Notice that this is not specified in Steve's definition - which a priori allows indirect (and therefore undefined) contradictions between the first and any other element of the column

Fourth principle of the mapping:
all the entries of the matrix are on or below the diagonal

Fifth principle of the mapping:
apart from row 1 which contains only z, rows of the matrix contain all the candidates (left- and right- linking, additional z- and t-) present in the nrc-link between xk and yk (remember that an nrczt-chain is fully specified only when such links have been chosen)
Thus row k+1 of the matrix should fully express the conjugacy of xk and yk modulo z and the previous yi
Row k+1 should therefore contain:
in column 1: z-candidates relevant for this conjugacy
in column i+1 (i < k): t-candidates relevant for this conjugacy, which are justified by right-linking candidate yi
- Notice that each row now expresses in a well defined factual way, Steve's unspecified condition "Each row contains at least one truth".

Sixth principle of the mapping:
the matrix contains no other entry than those defined above

It seems that this is more or less equivalent to Steve's definition above.

Notice that the principles of this mapping entail lots of specific properties of the TM, not implied by their general definition, e.g., in addition to those already mentioned: there's no empty cell in the diagonal and sub-diagonal.

At first sight, it seems that this mapping works and this is what may have misled Steve. BUT it doesn't.

The problem is that for a given xk yk conjugacy modulo something to hold, we may need several additional candidates to be justified by the same z- or previous t-candidate.
Which entails that there should be several candidates in the same cell of the matrix. (and the same problem arises with Steve's definition of the mapping).

Here is an example, in the full nrczt notation (all candidates displayed):
n7r7{c6 c1} – n7{r2c1 r1c3} – n7r9{c3 c7 c1#n7r7c1} – n7r8{c7 c6 c1#n7r7c1 c2#n7r7c1} – n8{r8c6 r9c4} – {n8 n3}r9c3 – n3{r9c5 r7c6} ==> r7c6 <> 5

In the 4th cell (of the chain), two candidates (n7r8c1 and n7r8c2) are justified by the same first right-linking candidate n7r7c1 and should occupy the same cell in the matrix.

Once you've understood this example, it is very easy to devise lots of others.

Conclusion: the natural tentative mapping form nrczt-chains to TMs that could justify Steve's claims does not work.

Remarks:
- This is not a full proof that TMs are not a generalisation of nrczt-chains, but it makes Steve's claims very unlikely and it shows that, if he was to maintain them, a detailed proof would be required of him.
- You can always try to extend the definition of TMs to include subsets in the cells of the matrix, but that makes the general TMs still many more light years away from nrczt-chains.
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