Confluence and rules for uniquenessthat led to it.

denis_berthier wrote:I'm not claiming that some notion of an "isolated sub-puzzle" couldn't be defined. I'm claiming that it has not been.

You are correct, so let me at least define what I mean by decoupled or isolated sub puzzles, ISPs.

Isolated Sub-Puzzles

An isolated sub-puzzle is any group of candidates that clears all other candidates from all sets (row, column, box, and cell) that it occupies. The most common examples are a single and a four-cell UR. After an ISP clears its sets of other candidates, it no longer has any logical connection with the rest of the puzzle. (Here, logic means based on Sudoku's first rule, the original 324 constraints.)

One way to see this is using cover sets. A single is a set with one candidate that can be covered with any of the other 3 sets in which the candidate sits. It therefore clears these other 3 sets. A 4-cell UR forms 4 different continuous nice loops that clear all 16 sets that contain the UR. Note, such patterns are only isolated after they clear their sets.

The Missing Oracle Conjecture Here's what I think everyone is getting at.

Given a puzzle with two solutions and a UR where no candidates have yet been removed, no combination of Sudoku logic can eliminate any candidate in the UR and thus force one of the two solutions.

If the conjecture is true then UR1.1 type logic does not require the oracle of uniqueness rather, the oracle's absence (the puzzle's uniqueness) is discovered by the logical elimination of one or more of the potential UR's candidates. The conjecture also has implications with respect to trial and error, which can eliminate candidates in a UR and produce individual solutions.

I have always assumed the conjecture to be true and have searched extensively for the elusive contrary example. However, an absolute proof also seems elusive. When a UR is isolated from the rest of the puzzle, the proof is trivial. The problem is proving it for all grids. At least the following should be true.

1. A unique puzzle has no logic that can eliminate a candidate belonging to its solution.

2. This must also apply to the individual solutions of a 2-solution puzzle thus, no logic external to the UR can eliminate any of the 8 potential candidates of a UR. This is because any external logic that can remove a UR candidate would destroy the puzzle if the other solution were manually removed first.

3. The only remaining way to eliminate a UR candidate must then be a combination of external logic with some of the UR's own candidates. This is the hard part to disprove. Perhaps someone knows of a simple way.

Proving the missing oracle principle would be very useful (granted, it is already useful). If correct, it could indicate the presence of unknown or unintended trial and error in a set of rules or a solver, when the rules or solver is able to eliminate candidates from a UR or isolate any of the puzzle's solutions. One observation, my set solver can find all the solutions to multiple solution puzzles but it must find them simultaneously, it can't find them sequentially nor find only some of them.

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