SpAce wrote:[about the confluence property]denis_berthier wrote:In short, it says that the final result (all the candidates eliminated or asserted as values) is independent of the order in which you apply the rules of T. As a counter-example, uniqueness techniques generally break the confluence property: if you don't apply them when you can, they may not be applicable later.

That's intriguing.

Yes, I know. And I can tell you it has been a shock to me when I first discovered a case of non-confluence, years ago.

SpAce wrote:Do you have a concrete example of your counter-example in mind?

Probably the most famous example on this forum is UR1:

UR1: consider the following template

ab ab

ab abc

where the 4 cells span 2 rows, 2 column and 2 blocks

Then c had to be the value of the 4th cell.

The proof is quite easy and doesn't even involve uniqueness. If c is not the value, we are left with the following template

ab ab

ab ab

But it is then easy to see that a and be can be permuted in any solution.

(Needless to say, there are many actual examples of this pattern in real puzzles. See the thread about "deadly patterns".)

As an aside, I never use uniqueness (though some U-rules are programmed in CSP-Rules), because I consider it better to prove it along with finding the solution.

Now, there are two very interesting things about this pattern:

- the UR1 rule is not provable constructively from the 4 rules of Sudoku

- if you haven't applied it when the above pattern was available and, say, the first a gets eliminated by some other rule, then you have no means of concluding that c has to be in the solution. On the contrary, you get the following remaining pattern, which can in no way forclude b in the 4th cell:

b a

a bc

SpAce wrote:I only know that extra clues can truly break uniqueness patterns and thus make a puzzle harder (for those who accept uniqueness methods).

No, extra clues can't make a puzzle harder. That can only happen in some inconsistent views of solving.

SpAce wrote:I can't imagine a case where normal solving steps applied in any order would result in the same.

And this is now the reason why I was so shocked when I discovered non-confluence. For several months after introducing whips, I believed that whip resolution theories had the confluence property. But I found a counter-example and that led me to find a point I had overlooked in my proof. That also led me to define braids.

Braid resolution theories do have the confluence property; indeed, they have been designed from whips in order to have it; (and braids are related in a very precise way to T&E); braids are the good concept for theoretical analyses. But whips are much simpler than braids. This is where the miracle happens and where I had my second shock: whips are statistically an excellent approximation of braids (i.e. they almost always give the same rating to a puzzle).

Now, if you want a really detailed example of non-confluence for whips, you'll have to open PBCS1 at section 5.10.3

Non-confluence is rarely observed, but it does happen.

The good news is, most of the usual rules do not disrupt the confluence property. This includes pointing/claiming (i.e. whips[1]) and Subsets (naked, hidden and super-hidden) and probably most of the aquarium (but I never checked). This also includes properly defined reversible chains, S-braids and B-braids. I have proven all this in detail in PBCS.

SpAce wrote:Sure, uniqueness patterns might get more difficult to spot if they're partly solved or missing candidates, but their solving powers are still available and applicable. In fact, some of such degenerate patterns can even be used without assuming uniqueness. What am I missing? (I'm assuming we're talking about single-solution puzzles.)

You got your counter-example with UR1.