The New Sudoku Players' Forum

[denis_berthier]

So what ? When i look, if there are givens in 4 cells, that is an extra-logic assumption ?

NO. The extra-logic assumption is saying that the givens are different from the values derived from them, in terms of use in further derivations.
Please read the first pages of any book about logic. Then we can talk. I have no more time to waste with you.

[eleven]

What you are talking about here is not logic, but fundamentalism. You are just blind for anything outside your self-defined sudoku world.

The extra-logic assumption is saying that the givens are different from the values derived from them, in terms of use in further derivations.

??? You really seem to be confused now.

[eleven]

Ah, now i see your problem. Yes, in the final logical consequence givens and derived values will lead to the same result. But you can make shortcuts like UR1.1 (and some unqiueness methods), if you are flexible enough to distinguish between given and derived numbers. Unfortunately you are not.
But that does not allow you to call these methods flawed.
These are no "logical extra-assumptions", but elegant and pure logical methods.

[eleven]

Just saw that again.

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 12 . . | 12  . . | * * 1
 12 . . | 123 . . | * * *
 .  . . | .   . . | 1 * *
 --------------------------
 *  * * | 1   * * | 1 * *

Is possible.

It isn't possible either because of the 12 pair in row 1, but this one (eliminating 1r2c4) would work as well to be able to eliminate 2r2c4 without uniqueness assumption (using the same proof).

Code: Select all

 12 . . | 12  . . | * * *
 12 . . | 123 . . | * * 1
 .  . . | .   . . | 1 * *
 --------------------------
 *  * * | 1   * * | 1 * *

(A more general rule for non uniqueness solvers is: I you can eliminate a candidate a, which is is in UR cells with a candidate b, you can safely remove all b's, which force a pattern a-b-a-b.)

[Serg]

Hi, all!
It looks like Denis asked very special and simple question, but really his question is not simple and is not special, because it concerns all methods of Sudoku solving. I'd like to describe 2 different appoaches to Sudoku solving to clarify my position.

"Rigorous" approach
There are no initial assumption about puzzle to be solved, except for puzzle's solution will obey Sudoku rules (each row/column/box must contain all different 9 digits from the set {1,2,...,9}).
Players use solving methods proved to be true and not requiring any assumptions about puzzle's validity. Uniqness methods are not used because they based on puzzle's validity assumption.
At the end of solving player get exact knowledge - has the puzzle any solution, or not. In case the puzzle has solution(s), the player must determine - has the puzzle only one solution or multiple solutions. If the puzzle has one solution, the player must find this solution. If the puzzle has multiple solutions, the player must find at least 2 different solutions (to prove puzzle's invalidity).
Player don't need any answers to the puzzle to determine - is his solution right or not - he does that check himself. Player is able to check himself the puzzle for validity.

"Not rigorous" approach
There is initial assumption that puzzle to be solved is valid (i.e. it has unique solution).
Players use solving methods proved to be true. Uniqness methods can be used too.
If player encounters contradiction (with Sudoku rules) during solving, the player get exact knowledge that the puzzle is not valid, but it is unknown - has the puzzle any solution or it has multiple solutions. If player finds puzzle's solution during solving, the player get exact knowledge that the puzzle has at least one solution, but it is unknown - has the puzzle one or multiple solutions.
Player needs answer to the puzzle to determine - is the puzzle valid or not. If player's solution coincides with answer, the puzzle is highly likely valid and player's solution is right. If player's solution doesn't coincide with answer, the puzzle is invalid, because it has multiple solutions.

If I am not mistaken, the debates in this thread are about possibility of "UR 1.1" rule's usage in "rigorous" approach. For me, "UR 1.1" rule cannot be used in rigorous approach (similar to other uniqness methods).

Serg

[denis_berthier]

Hi Serg,
Whereas I mostly agree with what you say, I wouldn't use the terms rigorous or not rigorous in the same way as you to qualify the use of uniqueness rules. If one has checked that a puzzle has a unique solution or is absolutely sure the puzzle provider has done it, then there is nothing non rigorous in using them. Sure, not proving uniqueness and relying on external information while solving always leaves a sense of incompleteness. Lack of rigour, on the other hand, can appear whether one uses uniqueness or not.

To re-iterate, the topic of the conversation was: does UR1.1 depend on the assumption of uniqueness or not? It is not: is UR1.1 valid in case of uniqueness.
The answer is now clear, except for eleven, who openly rejects logic and sticks to his old flawed "proof".

The proof that UR1.1 depends on some extra-logical assumption is clear. It only requires to adhere to the standard practices of logic:
1) there can't be any difference between a given and a value deduced from the givens, in terms of their consequences - this is universal logic and was explained in my first post
2) the two versions of the UR1.1 rule (with and without the restriction of having givens in the 4 UR cells) are therefore equivalent in terms of which assumptions they require and in terms of the conclusions they allow
3) I have found a multi-solution puzzle(*) with the UR1.1 pattern (where there are givens in three of the four UR cells and the value of the 4th UR cell in a solution (in fact, in all of them) is not that predicted by UR1.1
4) this strongly suggests that the extra-logical assumption necessary for UR1.1 is uniqueness

The example is::

Code: Select all

12345678.
4......1.
5.9......
2.76.4..5
3...1....
69.5231..
7.1..28..
8......2.
9.......3


The content of cells r1r4 x c1c2 is:
1 2
2 18
UR1.1 would assign the value 8 to r4c2. But all the solutions found by Sudoku Explainer have value 1 for that cell.

[champagne]

For me the proof given is not a problem, but...

What says the UR1.1 proof as far as I understand it:

A_ in a solution grid, assume that we find a pattern

Code: Select all

ab
ba       of 2 digits a,b

then exchanging ‘a’ and ‘b’, we get another valid solution grid.
Is it a fact or a corollary of the basic rules, no matter, we have here the smallest unavoidable set.
In a solution grid, any unavoidable set (we have billions of them) must hit a clue to have a sudoku (not a rubbish sudoku).

B_ take now a puzzle supposed to be a sudoku or a rubbish sudoku.

You get during the solving process the pattern

Code: Select all

1 2
2 13

none of the four cells is a given

Theorem, with such a pattern the solution is ‘3’

note: if the puzzle has no valid solution, whatever you take will not change the final conclusion.

I accept as hidden condition that the set of rules used to reach this point has no uniqueness assumption.

Code: Select all

proof

If the cell having the pm13 is filled with ‘1’
Applying the corollary A_, we have a second solution
2 1
1 2
but this in contradiction  with the solved cells,
so ‘1’ is not valid

No problem in the logic IMO, but I have doubts that this situation will show up, so it could be a theorem of no interest.
And the proof is valid for a rubbish puzzle as well, again, the challenge is to reach the pattern during the solving process.

[denis_berthier]

Hi champagne,
Repeating RedEd and eleven's flawed proof with no new argument exposes your beliefs but doesn't make it better.
The flaw is that the condition "none of the 4 cells is a given" cannot be expressed in logic. But you've probably not read my first post.

[champagne]

The example is::

Code: Select all

12345678.
4......1.
5.9......
2.76.4..5
3...1....
69.5231..
7.1..28..
8......2.
9.......3


The content of cells r1r4 x c1c2 is:
1 2
2 18
UR1.1 would assign the value 8 to r4c2. But all the solutions found by Sudoku Explainer have value 1 for that cell.

This is not a UR1.1, 3 of the four cells are given.
in a UR1.1 none of the four cells can be a given

[denis_berthier]

champagne,
Read my whole argument
And there's no need to shout in red. That doesn't make your point better.

[denis_berthier]

I have doubts that this situation will show up,[...] the challenge is to reach the pattern during the solving process.

I agree on this part.
And I think the urgent task for believers in the UR1.1 rule with no givens is to provide an example.

[champagne]

Hi champagne,
Repeating RedEd and eleven's flawed proof with no new argument exposes your beliefs but doesn't make it better.
The flaw is that the condition "none of the 4 cells is a given" cannot be expressed in logic. But you've probably not read my first post.

Hi Denis,

The flaw is that the condition "none of the 4 cells is a given" cannot be expressed in logic.

This is not a logic sentence. The proof that no 16 clues exists done by Gary Mc Guire is based on this logic, same for the scan to have a full list of the 17 clues puzzles.

[champagne]

I have doubts that this situation will show up,[...] the challenge is to reach the pattern during the solving process.

I agree on this part.
And I think the urgent task for believers in the UR1.1 rule with no givens is to provide an example.

Even if no example is produced, this does not make the logic false. But I stop here, this topic is anyway of small interest

[denis_berthier]

this topic is anyway of small interest

And your contributions empty of any new content don't make it more interesting.

[denis_berthier]

The flaw is that the condition "none of the 4 cells is a given" cannot be expressed in logic.

This is not a logic sentence. The proof that no 16 clues exists done by Gary Mc Guire is based on this logic, same for the scan to have a full list of the 17 clues puzzles.

As usual, you're mixing everything. McGuire's proof is based on unavoidable sets, not on expressing non-givens in logic.

Now, if you are able to express "rc=n is not a given" in logic, we're all ears.