denis_berthier wrote:logel wrote:An Universal Elimination Pattern is a set of candidates along with a set of links connecting the candidates according to basic sudoku constraints, where

(1) exactly one designated candidate is the elimination target ( single target )

(2) the target value is false for all valid permutations of candidate values ( elimination )

(3) all candidates and all links in the set are absolutely necessary for (2) ( minimal set )

(4) there exists no subset that is a separate universal elimination pattern ( no cascading )

(4): subset of what? Of candidates, of links or (more properly) of CSP variables?

(3) and (4) are somehow redundant

(3): "all links" is impossible to guarantee (see the NP example below)

Its certainly not redundant, but I had some problem to find the right wording. Meanwhile I edited a correction (see above).

Without (4) you face the following problem: Assume any two eliminations where the second depends on the result of the first one. Without (4) nothing prevents to unify both and declare the union as elimination of the second target. The condition (3) holds because if you remove any of the candidate, the pattern does not work. I want to exclude this kind of cascaded eliminations and I am fine with any other wording you might propose.

denis_berthier wrote:logel wrote:In many cases where lines in the set do overlap some of the lines (not their candidates) are dispensable.

This contradicts your point (3) above.

See the example in my reply to champagne. It make more clear what is meant with dispensable lines.

denis_berthier wrote:logel wrote:This implies: Any (universal) elimination pattern is fully determined by enumerating its base set.

This implies also that the link structure (net or queue) and other properties depend on the base set.

In other words: The base set is defining the elimination, but stop … wait …

This requires the base set to be unique for a particular elimination pattern. That would be really great. But a proof seems not so easy. Ideas welcome.

So I put the uniqueness of the base set of an elimination pattern as a conjecture here.

The conjecture is true for all practical cases, but I have some doubts.

The conjecture is trivially false. Take a Naked Pairs based on two cells in a mini-row. You can consider it both as a Pairs in a row or as Pairs in a block. In the 3rd cell in the mini-row, any NP elimination can be done by any of the two different patterns.

Your example is confusing to me. First I am not familiar with the term mini-row, but this may not be important.

Naked Pairs have always two cells as base lines each containing the two candidates of the associated cell. The universal elimination pattern contains the four candidates plus one dedicated target. This remains true even in the case that the core candidates happen to share the same row and box(block). The base set is unique.

If you meant a hidden pair instead the situation is slightly different. You can name the two lines that make up the base set either row or box line. But a line is defined as set, and the identity of sets is absolutely defined by their members, not by their label. So the base set is also unique. Lines are not bound to CSP variables or two a super-set they are made from. I always suspected that there is a subtle difference between my line and your CSP variable.

If there is any counter example for base set uniqueness is must be very large.

Note that I defined the target to be part of the elimination set. One reason is connected to the base set uniqueness. Any pattern that fits in a Dirichlet matrix (rank-0) has of course two line sets that cover the core after all eliminations are completed. The core network still exists but without target its not a universal elimination any more.

denis_berthier wrote:logel wrote:I will use the number of lines in the base set as pattern size. Its also equal to the maximal number of true candidates in the pattern core. This is not my favourite size function because it has some disadvantages. But it is easy computable and easy comparable with other results. Discussion of alternatives would overload this post and is saved for later.

See my suggestion for this definition of size in the "Pattern-based classification of (hard) puzzles" thread. This is the definition I use for all "my" patterns: whips, braids, extended whips or braids, Subsets, g-Subsets. So, at least, we have compatibility on this important point.

Yes, bus there is a decisive difference. Lets name the assumed candidate of a level-one sequence start candidate. As my eliminations are candidate sets, I can build the union of all eliminations that appear in that level-one sequence. This large set certainly contains all objects necessary for the elimination of the start candidate, but usually many more. Now its possible to reduce this set to a minimal configuration in a similar way as I reduce planes. The result is a universal elimination pattern with the start candidate as target. I can enumerate the base set and know the size. So the size propagates from eliminations to aggregated eliminations, and the nice part of this is that it works for any size function defined on properties of a universal elimination. The aggregated elimination is of course far from "optimal", but at least I have one.

denis_berthier wrote:My overall conclusion: you should re-formulate your definition with the base set as the primary data.

I will think about this, but its not instantly obvious that it makes anything easier. The base line set is a good descriptor of an elimination, please note that I do not use any link sets like Allan Barker concept. Links are there in the form of common knowledge and used when needed. The view on the pattern as candidate set is also required, because permutations operate on candidate values, not base lines. I am open to any formulation as long as the intention is preserved.