The New Sudoku Players' Forum

[denis_berthier]

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Previously, I have spoken of the ultra-persitency of ORk-relations (see the Augmented User Manual own GitHub or on ResearchGate https://www.researchgate.net/publication/365186265_Augmented_User_Manual_for_CSP-Rules-V21).

This post is about splitting ORk-relations. The idea of doing such things was more or less inspired by solutions proposed by other people in the Puzzles section of this forum. What I've done is find a generic way to express some of the ad hoc arguments made there.

First define a c-chain[n] as a sequence of n+1 different candidates C1... Cn+1 such that for each 1 ≤ i ≤ n, Ci and Ci+1 are related by a (3D) bivalue relation.
It is obvious that, if n is even then C1 and Cn+1 have the same truth value.
This is a generic definition and a generic property.

Theorem (generic): given an ORk relation between candidates Z1, ... Zk, if there is any even c-chain[n] between two of the Zi's, say Za and Zb, then the original ORk-relation can be split into two ORk-1 relations, between k-1 candidates, respectively {Z1, ... Zk}-Za and {Z1, ... Zk}-Zb.
[Edit, added]:Corollary: if k=2, both Z1 and Z2 can be asserted.

Proof: obvious

Note: "between two of the Zi's, say Za and Zb" means that C1 = Za and Cn+1 = Zb

For an example of using the theorem, see second post here: http://forum.enjoysudoku.com/51007-in-63137-t-e-3-min-expands-t40721-3.html and next post.
For an example of using the corollary, see here: http://forum.enjoysudoku.com/other-hard-one-from-mith-s-trivalue-oddagon-list-t40727-4.html.
For an example of using the theorem repeatedly in conjunction with ORk-reduction (ultra-persistency, see here: http://forum.enjoysudoku.com/1182-in-63137-list-of-t-e-3-min-expands-t40739-3.html

Note: the way such splitting is coded in CSP-Rules is still experimental.

[denis_berthier]

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How do you know that c-chains of even lengths are the only way ORk-relations can be reduced?

1) ORk relations are reduced when a guardian is eliminated (this is part of their ultra-persistency in CSP-Rules ).
But your question (about my previous post) is about a totally different topic.

2) Indeed, I don't know what you claim. On the contrary, there are probably many kinds of chain patterns that could be used as a basis for reducing ORk-relations.

What I've been looking for is rules that:
- allowed to reduce the number of guardians; it appeared that trying to split ORk-relations (instead of trying to reduce them, i.e. to eliminate guardians from them) was the most natural thing to do;
- involved only generic conditions (with no ad-hoc reasoning);
- were simple enough to be integrated in the hierarchy of resolution rules (though they are not resolution rules).
The last two conditions are trivially satisfied by c-chains (a c-chain[n] being posited at level Ln, just before the ORk-chains[n] that might rely on their results).

Now, can these splitting rules take into account the different ad hoc reductions that have been proposed for anti-tridagon puzzles in the Puzzles section? I haven't checked and it's not that important here for the following reason:
my approach has always been to find universal rules/techniques that can be applied in a systematic way. They may not be the smartest ones for a particular puzzle, but they aim at being the smartest ones in the mean.
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