denis_berthier wrote:So, let me first make a few remarks and ask a few questions to clarify some of your definitions.

First of all, the expression "all patterns" and its variants ("all patterns with 5 CSP variables", "all patterns restricted to a single plane [or number]") is not well defined. When you write:,logel wrote: P1 is defined by all patterns restricted to a single plane of the sudoku grid.

P1[N1] e.g. is the plane of numbers equal 1 containing all fish patterns with number 1

this is somehow contradictory. We don't know if we should read the first or the second line, i.e. whether P1[N] is the set of standard Fish (2nd line) or the set of g-fish [Franken and Mutant Fish] (first line).

OK, take the first line. The other was only meant as an explanation. I was not aware of the restricted use of the term "fish". Distinguishing "fish" and "g-fish" has not relevance in this context.

denis_berthier wrote:Secondly, the number of "CSP planes" (of types n, r, c, b) cannot be used to measure the size of a pattern (you don't say so, but it may be implicitly understood in some places). You must count the number of CSP variables (of types rc, rn, cn, bn). Thus, a fish on number 1 doesn't have size 1 but 2, 3 or 4.

This will be important when you need to measure the size of the patterns you use in T&E(X).

I never said that I intended to measure the size of patterns. My results just show solvability. Pattern size restrictions are not implemented, but may be an interesting extension. Even if you limit size of patterns in T&E(X) level 1, the size of the resulting level 0 pattern is unlimited.

denis_berthier wrote:logel wrote:Class T&E(S+BI+Pair+T5): solving 17159, remaining 39289

T5 is defined as: all Patterns with exactly 5 CSP variables

Because of the basic construction of sudoku, T5 is always a single number pattern with two rows, two colums and a box link.

Allowing patterns with 1, 2 or 5 but not with 3 or 4 CSP variables seems quite arbitrary.

Moreover, I think "S+BI+Pair+T5" doesn't have the confluence property - and therefore this class is not well defined. When a pattern P built on 5 CSP variables is destroyed by some eliminations before being applied, what remains may have fewer variables; as it is not in your class, it doesn't allow the elimination allowed by P.

However, this doesn't have any impact on the sequel.

Ups, I think I misunderstood the term "CSP variable". BTW it would be nice to have a dictionary of sudoku terms.

Besides the typo "T5" should read "L5", I am using the term line for myself that I thought to be a synonym for CSP variable and is apparently not.

A line is the set of undecided candidates of one of the 324 constraints. So a line is a cell or row/col/box of a fix number.

A plane is the set of undecided candidates that share the same number or row/col/box.

I think the number planes are often called floor in sudoku discussions. They seem to be more important than the others, because of the extra box constraints.

BI pattern consist of two crossing lines and pairs of two by two crossing lines. L5 patterns are rings of five lines. So the above class is actually confluent. You are right that it does not really fit to the others, but I used it for more pragmatic reasons. I wanted to sort out the less hard puzzles quickly.

denis_berthier wrote:logel wrote:Class T&E(S+BI+P1[NBRC]+P2[NBRC]) : remaining 317 unsolved puzzles, mainly residing in the top 3000 of the list

P2[1,2] e.g. contains all patterns restricted to the combined planes P[1] and P[2].

P2[N] denotes all multi-fish patterns in all the 36 pairings of two number planes.

That means patterns of this class all work in pairs of CSP planes, that do not overlap.

This seems to confirm my above reading: this should be written as T&E(P1[N]+P1[ B]+P1[R]+P1[C]+P2[N]+P2[ B]+P2[R]+P2[C]).

Can you confirm?

Yes. But for the P2 its even more clear to write P2[NN] etc. Plane sets P2[NB] or P[RB] are also possible, but are not in the above class.

denis_berthier wrote:This class should solve all the puzzles classified by Champagne as multi fish. Can you confirm?

Can you say more about the (new ???) multi-row, multi-column and multi-block patterns?

Can you say more about the maximum size of the patterns (e.g. multi-fish) needed to reach this result ("only 317 unsolved")?

I am now very careful about answering the "multi fish" question. I did not find a definition on Champagne's web pages. But the answer is probably yes.

Of course I expected the most eliminations in the number plane sets.

From figures of actually applied P2 eliminations in the above class I found:

50% in P2[NN]

25% in P2[BB]

10% in P2[RR]

10% in P2[CC]

The high value for box pairs was a bit of a surprise. Additionally many of P2[RR] and P2[CC] eliminations com from row/col pairs residing in the same box tower. So locality seems to be important.

Eliminations on plane connections are rather rare. These eliminations are the only ones with targets outside the members of the plane set.

All plane sets of used classes work with non-intersecting planes. With P3 there are lots of non-intersecting plane sets, especially P3[BBR] or P3[BBC), I did not yet investigate.

Experiments with two intersecting planes shows very little results.

One can understand this. Parallel planes like P2[NN] or P3[NNN] e.g. cut a 3D block out of the sudoku universe, the others don't.

denis_berthier wrote:It would be interesting to give these 5 puzzles explicitly.

- Code: Select all
`1.......9.5....2....87...4.2...3......48.5....8.6...7...6..4.5.........1....9.3..;11.80;11.80;7.90;elev;11;21;21`

.....6..94...8.2.....7...1.2.9...8....4.3.9...6.....5.3.8.4.......5......7...1...;11.80;1.20;1.20;elev;4;31;21

....56.8...71.....6.....4.......85...3......29...4..6..1.7.......2.....38...9..5.;11.70;11.70;2.60;elev;30;55;21

98.........79..6......5....7..4..9....6.3..2......5..1.7.5..8....4.1..5......2..3;11.60;11.60;2.60;GP;H11;118;22

1......8..567..........3..6..7..5.......2.9...3.6....4....9..1..4.5....78.....2..;11.40;11.40;2.60;elev;246;415;21

denis_berthier wrote:One important point for me is the maximum size of the patterns involved in your results (i.e. the number of CSP variables they rely on)

Some more comments on pattern sizing and more definitions I use.

A general elimination is a set of candidates including the target(s) and a set of connecting lines forming a network where the targets are FALSE in all valid permutations.

That means, no specific structure, just a contradiction of the target to the rest of the set. Because of this there is no useful size property.

In a minimal elimination all of the candidates besides the targets are absolutely necessary. These ones are those we usually talking about and they have properties that allow counting size.

- A minimal elimination has a minimal set of lines where all candidates of those lines are members of the set. These active lines seem to correspond to CSP variables (Confirm ?) and give a basis for counting size.

- All other lines in the elimination network are passive lines.

Taking the number of active lines and ignoring the other properties is not my favorite size definition.

The framework producing the above result uses general eliminations. The basic idea is as simple as stupid. I construct all permutations of the plane sets I intend to use. Subsequent operations are performed on this data.

There is nothing really NEW of this approach, only the focus is on another point. I think Allan Barker used permutation sets of limited size to pragmatically support or speed-up search in his solver.

My intention was not directed to pragmatic solving but to completeness. Getting size information requires more effort.

Of course I can reduce any specific general elimination to a minimal one for each of the targets. These will be of different size and not unique usually.

The implemented process works with greedy progression. Each contradiction on T&E(1) will be immediately applied on level 0. Therefore one can miss easily eliminations of smaller size in terms of CSP variables.

The whole subject is still interesting and I hope to present more details soon.