How Hendrik,
hendrik_monard wrote:denis_berthier wrote:.Has anyone tried using the complete grid?.
This may not be a direct reply to your question, but I would like to submit the idea anyway.
I have been working for a while with grids. In my two previous contributions (of new B7B+ puzzles) of May 7th and June 13th, the puzzles (in fact min-expands) were already grouped per grid. But since then, I have focused on grids almost exclusively. This has the advantage that when you produce a B7B puzzle and you want to know if it is new, you check its solution (grid) with the list of already known B7B+ grids.
That's exactly the idea. Two puzzles for two different complete grids (or even with two different BRT-expansions) can't be identical or isomorphic, so they don't have to be checked against each other. As non-iso checks are quadratic in the number of puzzles to be checked, the advantage is gigantic. I think mith is doing something like that in his non-iso checks for his TE3 database; otherwise, they'd be impossible.
That's why I suggested splitting the search for "hard" puzzles into two parts (which the currently available softwares can't do):
- explore the puzzles for each complete grid, grid by grid;
- mutate minimals by -p+q search in a restricted way that only produces puzzles with new complete grids.
This is also justified by the unique topology that can be defined on the set of consistent puzzles so as to make the T&E-depth, B, BxB and BxBB classifications continuous: the BRT topology. In it, different complete grids are isolated from each other.
Note that the separation idea can be extended to BRT-expansions (puzzles with different BRT-expansions are isolated from each other) and to other kinds of expansions.
hendrik_monard wrote:Once you have a new minimal B7B+ puzzle, it is possible to 'produce' all other related B7B+ puzzles from that grid with repeated expanding, minimizing, expanding the new minimals, minimize these expands and so on, until exhaustion.
I'll take it as the definition of "all the other related puzzles".
I also take "expansion" to mean "expansion by Singles" (i.e. in my more precise terminology BRT-expansion - because not only Singles but the full Basic Resolution Theory BRT is applied in order to get the expansion).
hendrik_monard wrote:Therefore, my recently corrected, completed and restructured local database contains only one line per grid. Each line contains the grid, x (in BxB), the first published puzzle from this grid (the original isomorph, minimal or not), its publishing date, and finally a reference to the publisher. This list contains now 527 grids and is based on a collection of all B7B+ puzzles (minimals or expanded) published in this thread. It is sorted by the date of the first published puzzle.
As I understand it, all the known minimal puzzles in B7B+ can be rebuild from this list by "repeated expanding, minimizing....". Or do you mean only by minimising?
Is this list minimal in any sense?
hendrik_monard wrote:I even risk to propose the following draft hypothesis:
"When a new grid (solution of a minimal B7B+ puzzle) is identified, it is in many cases possible to derive new minimal puzzles with B7B+ qualification through a straightforward process of consecutive expansions (keeping B7B+ qualification) of the initial minimal(s), followed by identifying new minimals from the expanded puzzles and repeating this process until exhaustion.
As you write "in many cases", I think this part of the conjecture has been largely verified by the recent results reported in this thread.
hendrik_monard wrote:There cannot be other B7B+ puzzles within the same grid than those obtained through this process."
Of course, an hypothesis only stands until falsified. This could f.i. happen if a B7B+ puzzle is found within the same grid but totally unrelated to the already identified B7B+ puzzles within that grid.
I would be cautious about this part of the conjecture.
In addition to BRT-expansion, I've studied 1-expansion (add 1 clue from the solution grid) and the BRT+1-combination. This allows to find more minimals than only BRT-expansion. But I can't say if BRT+1 expansion can be reduced (at least "in many cases") to "repeated expanding, minimizing....". No doubt this is worth exploring.
Two points worth considering here are:
- all the known minimal B7B+ puzzles (except 3 old ones) have a (non-degenerate) tridagon;
- the (non-degenerate) tridagon is very stable under changes in the clues.
This may explain the success of "repeated expanding, minimizing" in "many cases". However, there are puzzles with more than one tridagon. In all the cases I've checked, the tridagons are somehow related (same digits, same blocks, many cells in common). At this point, I can't tell if such relations can be explained by "repeated expanding, minimizing". But this is also worth exploring. I'm sure mith would have lots of things to say on this topic, but we've had no news of him for a long time.
hendrik_monard wrote:PS. It is my intention to relaunch, after a long break, my revisited generating scripts focused on the identification of new B7B+ grids.
That's good news.
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